Algorithm and Insurgency in Khyber Pakhtunkhwa: Difference between pages

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[[Image:LampFlowchart.svg|thumb|right|[[Flowchart]]s are often used to graphically represent algorithms.]]
{{Infobox War
|image=[[Image:NWFP FATA.svg|300px]]
|caption=District map of [[NWFP]] and [[Federally Administered Tribal Areas|FATA]].
|conflict=War in North-West Pakistan
|partof= the [[War on Terrorism]]
|date=March 2004–ongoing
|place=[[Federally Administered Tribal Areas]] and [[North-West Frontier Province]] in [[Pakistan]]
|result=ongoing
|combatant1={{flag|Pakistan}},<br>{{flag|United States}}
|combatant2={{flagicon|Afghanistan|Taliban}} [[Tehrik-i-Taliban Pakistan]],<br>[[Tehreek-e-Nafaz-e-Shariat-e-Mohammadi]],<br>[[Lashkar-e-Islam]],<br>[[Image:Flag of Jihad.svg|22px]] [[al-Qaeda]], <br> [[Islamic Movement of Uzbekistan]] (until 2007)
|commander1={{flagicon|Pakistan}} [[Asif Ali Zardari]],<br>{{flagicon|Pakistan}} [[General]] [[Ashfaq Parvez Kayani|Ashfaq Kayani]],<br>{{flagicon|Pakistan}} [[Lt Gen]] [[Masood Aslam]],<br>{{flagicon|Pakistan}} Maj Gen Javed Iqbal,<br>{{flagicon|Pakistan}} Maj Gen Tariq Khan,<br>{{flagicon|Pakistan}} Maj Gen Nasser Janjua,<br>{{flagicon|Pakistan}} Maj Gen Khalid Rabbani<br>''Former commanders''<br>{{flagicon|Pakistan}} [[General|Gen]] [[Pervez Musharraf]],<br>{{flagicon|Pakistan}} Lt Gen Safdar Hussain,<br>{{flagicon|Pakistan}} [[Maj Gen]] [[Alam Khattak]],<br>{{flagicon|Pakistan}} [[Maj Gen]] [[Ahmad Shuja Pasha|Ahmad S Pasha]]
|commander2={{flagicon|Afghanistan|Taliban}} [[Baitullah Mehsud]],<br>{{flagicon|Afghanistan|Taliban}} [[Hafiz Gul Bahadur]],<br>{{flagicon|Afghanistan|Taliban}} [[Faqir Mohammed]],<br>[[Maulana Fazlullah]],<br>[[Mangal Bagh]]<br>''Fugitives''<br>[[Image:Flag of Jihad.svg|22px]] [[Osama bin Laden]],<br>[[Image:Flag of Jihad.svg|22px]] [[Ayman al Zawahiri]],<br>{{flagicon|Afghanistan|Taliban}} Mullah [[Mohammed Omar]],<br>{{flagicon|Afghanistan|Taliban}} [[Jalaluddin Haqqani]],<br>[[Tohir Yo‘ldosh]]
|strength1=80,000 Pakistani troops<ref>[http://www.bloomberg.com/apps/news?pid=20601087&sid=aCkMAj8lVUB8&refer=home Bloomberg.com: Worldwide<!-- Bot generated title -->]</ref>,<br>~80,000 [[Frontier Corps]]
|strength2=Unknown
|casualties1=1,370 soldiers and policemen killed (official)<ref>http://ap.google.com/article/ALeqM5hkiMxbHNH0BqgpWA2ZG6VD6wVTmAD93GH45O1</ref><br><br>Other estimates:<br>1,815 soldiers and policemen killed<br>112 tribesmen killed<small><ref name=autogenerated4>http://in.news.yahoo.com/071114/139/6n8m0.html</ref></small>,<br>2,259 soldiers wounded<small><ref name=autogenerated4 /></small>,<br>810 soldiers and policemen missing or captured<ref>[http://www.thenews.com.pk/top_story_detail.asp?Id=9870 Over 200 soldiers in captivity of militants By Mushtaq Yusufzai & Sailab Mahsud] [[September 3]], [[2007]] [[The News International, Pakistan]]</ref>
|casualties2=4,594+ [[militant]]s killed<small><ref>http://www.thenews.com.pk/top_story_detail.asp?Id=15771</ref><ref>http://news.xinhuanet.com/english/2008-09/26/content_10118035.htm</ref></small><br>(488 foreign fighters <ref name=autogenerated2 />)
|casualties3=5,320 civilians killed,<br>25 foreigners killed<br>(11 Frenchmen, 9 Americans, 4 Chinese, 1 Czech, 1 Dane, 1 Turk and 1 Vietnamese)<ref>[http://www.defenselink.mil/news/newsarticle.aspx?id=15292</ref><ref>http://ap.google.com/article/ALeqM5hkiMxbHNH0BqgpWA2ZG6VD6wVTmAD93GH45O1</ref><ref>http://www.gmanews.tv/story/123833/Suicide-attacks-soar-in-Pakistan-since-2007</ref>
}}
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{{Campaignbox War on Terrorism}}
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The '''War in North-West Pakistan''' is an armed conflict between the [[Pakistani Army]] and Islamist militants made up by local tribesmen, the [[Taliban]] and foreign extremists.<ref>[http://www.washingtonpost.com/wp-dyn/content/article/2006/01/24/AR2006012401528.html The War in Pakistan - washingtonpost.com<!-- Bot generated title -->]</ref><ref>[http://wiredispatch.com/news/?id=176983 FEATURE-Pakistan lifts veil on not-so-secret Waziristan war<!-- Bot generated title -->]</ref><ref>[http://news.bbc.co.uk/2/hi/south_asia/3645114.stm BBC NEWS | South Asia | Pakistan's undeclared war<!-- Bot generated title -->]</ref> It began in 2004 when tensions rooted in the Pakistani Army's search for [[al-Qaeda]] members in [[Pakistan]]'s mountainous [[Waziristan]] area (in the [[Federally Administered Tribal Areas, Pakistan|Federally Administered Tribal Areas]]) escalated into armed resistance by local tribesmen.
In [[mathematics]], [[computing]], [[linguistics]] and related subjects, an '''algorithm''' is a sequence of finite instructions, often used for [[calculation]] and [[data processing]]. It is formally a type of [[effective method]] in which a list of well-defined instructions for completing a task will, when given an initial state, proceed through a well-defined series of successive states, eventually terminating in an end-state. The transition from one state to the next is not necessarily [[deterministic]]; some algorithms, known as [[probabilistic algorithms]], incorporate randomness.


Clashes erupted between the Pakistani troops and al-Qaeda's and other militants joined by local rebels and pro-Taliban forces. The Pakistani actions were presented as a part of the [[War on Terrorism]], and had connections to the [[War in Afghanistan (2001–present)|war]] and [[Taliban insurgency]] in [[Afghanistan]].<ref>{{cite web|title=Killing scares media away from Waziristan|url= http://www.csmonitor.com/2006/0622/p07s02-wosc.html|publisher=[[Christian Science Monitor]]|date=2006-06-22|author=David Montero|accessdate=2008-08-25}}</ref><ref>{{cite web|title=Pakistan attacks Waziristan compound|url= http://english.aljazeera.net/archive/2006/06/2008410114854246429.html |publisher=[[Al Jazeera]]|date=2006-03-16|accessdate=2008-08-25}}</ref>
A partial formalization of the concept began with attempts to solve the [[Entscheidungsproblem]] (the "decision problem") posed by [[David Hilbert]] in 1928. Subsequent formalizations were framed as attempts to define "[[effective calculability]]" (Kleene 1943:274) or "effective method" (Rosser 1939:225); those formalizations included the Gödel-Herbrand-Kleene [[Recursion (computer science)|recursive function]]s of 1930, 1934 and 1935, [[Alonzo Church]]'s [[lambda calculus]] of 1936, [[Emil Post]]'s "Formulation I" of 1936, and [[Alan Turing]]'s [[Turing machines]] of 1936–7 and 1939.


After a [[Waziristan Accord|truce]] in September 2006, clashes escalated again in 2007, with local tribesmen ousting militant Uzbeks and attacking the Pakistani security forces again in July 2007.
==Etymology==
[[Muhammad ibn Mūsā al-Khwārizmī|Al-Khwārizmī]], [[Persian people|Persian]] [[astronomer]] and [[mathematician]], wrote a [[treatise]] in [[Arabic]] in 825 AD, ''On Calculation with Hindu Numerals''. (See [[algorism]]). It was translated into [[Latin]] in the 12th century as ''Algoritmi de numero Indorum'' (al-Daffa 1977), which title was likely intended to mean "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's rendition of the author's name; but people misunderstanding the title treated ''Algoritmi'' as a Latin plural and this led to the word "algorithm" (Latin ''algorismus'') coming to mean "calculation method". The intrusive "th" is most likely due to a [[false cognate]] with the [[Greek language|Greek]] {{lang|grc|ἀριθμός}} (''arithmos'') meaning "number".


Since the conflict began, Pakistan has lost more than twice the number of its soldiers compared to the number of US troops killed in Afghanistan. However, as of September 2008, the number of militants killed by the Pakistan Army reached the 4,500 mark.<ref>http://www.timesonline.co.uk/tol/news/world/asia/article4111277.ece</ref>
== Why algorithms are necessary: an informal definition ==


==Background==
No generally accepted ''formal'' definition of "algorithm" exists yet.
===July 2002: Initial troop movements===
In July 2002, Pakistani troops entered the [[Tirah|Tirah Valley]] in the [[Khyber Agency]] for the first time since Pakistan independence in 1947. They proceeded to move into the [[Shawal Valley]] of [[North Waziristan]], and later [[South Waziristan]]. This was made possible after long negotiations with various tribes, who reluctantly agreed to allow the military's presence on the assurance that it would bring in funds and development work.


However, once the military action started in South Waziristan a number of Waziri sub-tribes viewed the action as an attempt to subjugate them. As attempts to persuade them to hand over the foreign militants failed, and missteps by the authorities increased feelings of ill-will, the security campaign against suspected al-Qaeda militants turned into an undeclared war in 2004 between the Pakistani military and the rebel tribesmen.
An informal definition could be "an algorithm is a computer program that calculates something." For some people, a program is only an algorithm if it stops eventually. For others, a program is only an algorithm if it stops before a given number of calculation steps.


===December 2003: Waziri attempts on President Musharraf's life===
A prototypical example of an "algorithm" is Euclid's algorithm to determine the maximum common divisor of two integers greater than one: "subtract the smaller number from the larger one; repeat until you get a zero or a one." This procedure is known to stop always and the number of subtractions needed is always smaller than the larger of the two numbers.
In December 2003, two assassination attempts against President [[Pervez Musharraf]] were traced to Waziristan. The government responded by intensifying military pressure on the area, however the fighting was costly and government forces would sustain heavy casualties throughout 2004 and into early 2005 when the government switched to a tactic of negotiation instead of direct conflict.<ref name=rohde>{{cite news | title = Al Qaeda Finds Its Center of Gravity | url = http://www.nytimes.com/2006/09/10/weekinreview/10rohde.html | first = David | last = Rohde | publisher = New York Times | date = [[2006-09-10]] | accessdate = 2006-09-12 }}</ref>


==Timeline==
We can derive clues to the issues involved and an informal meaning of the word from the following quotation from {{Harvtxt|Boolos|Jeffrey|1974, 1999}} (boldface added):
===March 2004–September 2006 phase===
====2004: Fighting breaks out====
{{main|Battle of Wana}}
In March 2004, heavy fighting broke out at Azam Warsak, near the South Waziristan town of [[Wana (Pakistan)|Wana]]. Pakistani troops faced an estimated 400 militants holed up in several fortified settlements. It was speculated at the time that [[Osama bin Laden]]'s deputy [[Ayman al-Zawahiri]] was among those trapped by the Pakistani Army, but he either escaped or was never among these fighters.


====Peace deals with Taliban====
<blockquote>No human being can write fast enough or long enough or small enough to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give '''explicit instructions for determining the nth member of the set''', for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which '''they could be followed by a computing machine''', or by a '''human who is capable of carrying out only very elementary operations on symbols''' {{Harv|Boolos|Jeffrey|1974, 1999|p=19}}</blockquote>
In April 2004 the [[Government of Pakistan]] signed the first of three peace agreements with militants in [[South Waziristan]]. It was signed with [[Taliban]] commander [[Nek Muhammad Wazir]], but was immediately abrogated once Nek Muhammad was killed by American [[Hellfire missile]] in June 2004. The second was signed in February 2005 with Nek's successor [[Baitullah Mehsud]], which brought relative calm in the [[South Waziristan]] region. This deal will be later mimicked in the neighboring [[North Waziristan]] territory in September 2006 as the third and final truce between the government and the militants. However, all of these truces would not have a substantial effect in reducing bloodshed.<ref name=rohde /> The later two deals were officially broken in August 2007 after the [[Lal Masjid siege|Lal Masjid episode]] raising the suicide attacks on forces and civilians [[Terrorism in Pakistan#July - September|ten-fold throughout the country]].


====2005====
The words "enumerably infinite" mean "countable using integers perhaps extending to infinity." Thus Boolos and Jeffrey are saying that an algorithm ''implies'' instructions for a process that "creates" output integers from an ''arbitrary'' "input" integer or integers that, in theory, can be chosen from 0 to infinity. Thus we might expect an algorithm to be an algebraic equation such as '''y = m + n''' — two arbitrary "input variables" '''m''' and '''n''' that produce an output '''y'''. As we see in [[Algorithm characterizations]] — the word algorithm implies much more than this, something on the order of (for our addition example):
On [[May 4]], [[2005]], Pakistani commandos captured [[Abu Faraj al-Libbi]] after a raid outside the town of [[Mardan]], 30 miles north of Peshwar. Abu Farraj al-Libbi was a high ranking al-Qaeda official, rumored to be third after Osama bin Laden and Ayman al-Zawahiri. Al-Libbi replaced [[Khalid Shaikh Mohammed]] after his arrest in March 2003 in connection with the [[September 11, 2001 attacks|September 11th attacks]]. The Pakistani government arrested al-Libbi and held him on charges in relation to being a chief planner in two assassination attempts on the life of President Pervez Musharraf in December 2003.<ref>{{cite news|url=http://www.usatoday.com/news/world/2005-05-04-pakistan-arrest_x.htm?POE=NEWISVA |title=Pakistan reports arrest of Osama bin Laden's operations chief | author=John Diamond | publisher=USA Today |date=2005-05-04}}</ref>
:Precise instructions (in language understood by "the computer") for a "fast, efficient, good" ''process'' that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally-contained information and capabilities) to find, decode, and then munch arbitrary input integers/symbols '''m''' and '''n''', symbols '''+''' and '''=''' ... and (reliably, correctly, "effectively") produce, in a "reasonable" [[time]], output-integer '''y''' at a specified place and in a specified format.


====2006====
The concept of ''algorithm'' is also used to define the notion of [[decidability (logic)|decidability]]. That notion is central for explaining how [[formal system]]s come into being starting from a small set of [[axiom]]s and rules. In [[logic]], the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of ''algorithm'' that suits both concrete (in some sense) and abstract usage of the term.
[[Image:Pakistan-Waziristan-Map.png|200px|thumb|left]]
On [[January 13]], [[2006]], the U.S. launched an [[Damadola airstrike|airstrike on the village of Damadola]]. The attack occurred in the [[Bajaur]] tribal area, about 7 km (4.5 miles) from the Afghan border, and killed at least 18 people. The attack again targeted Ayman al-Zawahiri, but later evidence suggests he was not there. [[DNA analysis]] showed that at least five high-ranking al-Qaeda foreigners were killed, including [[Midhat Mursi]], a bomb building expert, for whom a $5 million award was offered.{{Fact|date=February 2007}}


On [[March 4]], [[2006]], Pakistani forces started a massive assault on pro-Taliban elements in the region. Pakistani officials said 46 militants and five soldiers died after fighting erupted, although some reports put the death toll at over 70.{{Fact|date=March 2007}}
:''For a detailed presentation of the various points of view around the definition of "algorithm" see [[Algorithm characterizations]]. For examples of simple addition algorithms specified in the detailed manner described in [[Algorithm characterizations]], see [[Algorithm examples]].''


On [[June 21]], [[2006]], pro-Taliban militants in the [[Bannu]] region of North Waziristan stated they shot down a military [[helicopter]] that was reported to have crashed. The government denied missile fire as the cause, stating it was due to technical faults. On the same day militants killed an inspector and two constables on a road connecting Bannu and the main town of [[Miranshah]]; the police officers were reportedly ambushed and killed while praying alongside their vehicle.<ref>{{cite news | title = 'Policemen killed' in Waziristan | url = http://news.bbc.co.uk/2/hi/south_asia/5104784.stm | publisher = BBC News | date = [[2006-06-22]] | accessdate = 2006-08-07 }}</ref>
== Formalization of algorithms ==
<!-- If you change this heading's title, [[Computer program]] links here. -->
Algorithms are essential to the way [[computer]]s process information. Many [[computer program]]s contain algorithms that specify the specific instructions a computer should perform (in a specific order) to carry out a specified task, such as calculating employees’ paychecks or printing students’ report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a [[Turing completeness|Turing-complete]] system. Authors who assert this thesis include Savage (1987) and Gurevich (2000):


====June 2006: Ceasefire====
<blockquote>...Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine (Gurevich 2000:1)...according to Savage [1987], an algorithm is a computational process defined by a Turing machine. (Gurevich 2000:3)</blockquote>
Also on [[June 21]], [[2006]] the military head of the Taliban in Waziristan, [[Sirajuddin Haqqani]], issued a decree that it was no longer Taliban policy to fight the Pakistan Army. This marked the end of significant fighting in South Waziristan, however the Taliban intentionally did not circulate the decree in North Waziristan thereby keeping pressure on the Government as the terms for a comprehensive accord were worked out.<ref>{{cite news|url=http://www.dawn.com/2006/06/23/top2.htm|title=Forces, militants heading for truce|publisher=[[Dawn (newspaper)|Dawn]]|date=2006-06-22}}</ref>


On [[June 26]], [[2006]], a [[suicide bomber|suicide car bomb]]er killed nine Pakistani soldiers. Officials say that the explosives-laden vehicle detonated about six kilometres (four miles) east of Miranshah, the main town in the North Waziristan region.<ref>{{cite news | title = 'Suicide attack' on Pakistan army | url = http://news.bbc.co.uk/2/hi/south_asia/5117472.stm | publisher = BBC News | date = [[2006-06-26]] | accessdate = 2006-08-07 }}</ref>
Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and/or stored for further processing. Stored data is regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more [[data structure]]s.


====September 2006: Waziristan peace accord signed====
For any such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).
On [[September 5]] [[2006]], the [[Waziristan Accord]], an agreement between tribal leaders and the Pakistani government was signed in Miranshah, North Waziristan.<ref>[http://www.jpost.com/servlet/Satellite?cid=1154525988190&pagename=JPost%2FJPArticle%2FShowFull Pakistan, Taliban militants sign peace agreement]</ref> to end all fighting. The agreement includes the following provisions:<ref>[http://www.dawn.com/2006/09/06/top2.htm Waziristan accord signed]</ref>


* The Pakistani Army will help reconstruct infrastructure in tribal areas of North and South Waziristan.
Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom", an idea that is described more formally by ''[[control flow|flow of control]]''.
* The Pakistani Military will not tolerate any assistance to intruders in North Waziristan, and will monitor actions in the region.
* The Pakistan government is to compensate tribal leaders for the loss of life and property of innocent tribesmen.
* “Foreigners” (informally understood to be foreign [[jihad]]ists) are not allowed to use Pakistani territory for any terrorist activity anywhere in the world.
* 2,500 foreigners who were originally held on suspicion of having links to al-Qaeda or the Taliban<ref>[http://www.telegraph.co.uk/news/main.jhtml?xml=/news/2006/09/15/wpak15.xml US outraged as Pakistan frees Taliban fighters] - [[Daily Telegraph]]</ref> were to be detained for necessary action against them.
<!---* The Taliban is required to refrain from violence anywhere using Pakistan's territory; the agreement does not stipulate refraining from violence specifically in [[Afghanistan]].<ref name=thefourthrail>{{cite news | title = Talibanistan: The Establishment of the Islamic Emirate of Waziristan| url = http://billroggio.com/archives/2006/09/talibanistan_the_est.php | publisher = The Fourth Rail| date = [[2006-09-05]] | accessdate = 2006-09-05 }}</ref> --->


The agreement, dubbed the Waziristan accord, has been viewed by some political commentators as a success for Pakistan.<ref>[http://www.foxnews.com/wires/2006Sep13/0,4670,PakistanTruceorSurrenderLH1,00.html Some See Pakistan's Truce As a Defeat] - [[Fox news]]</ref> Further details of the agreement, as well as comments on the agreement made by US, Pakistani, and Taliban spokesmen is available in the Waziristan accord article.
So far, this discussion of the formalization of an algorithm has assumed the premises of [[imperative programming]]. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the [[assignment operation]], setting the value of a variable. It derives from the intuition of "[[memory]]" as a scratchpad. There is an example below of such an assignment.


====October 2006: The madrassa air strike====
For some alternate conceptions of what constitutes an algorithm see [[functional programming]] and [[logic programming]] .
On [[October 30]], [[2006]], the Pakistani army [[Chenagai airstrike|conducted an air strike]], with possible U.S. involvement, against a [[madrassa]] in the [[Bajaur]] region bordering [[Afghanistan]]. The madrassa was destroyed killing 70 to 80 people.


In retaliation for the attack the militants conducted a suicide bombing on an army camp on [[November 8]], [[2006]], killing 42 Pakistani soldiers and wounding 20.
=== Termination ===
Some writers restrict the definition of ''algorithm'' to procedures that eventually finish. In such a category Kleene places the "''decision procedure'' or ''decision method'' or ''algorithm'' for the question" (Kleene 1952:136). Others, including Kleene, include procedures that could run forever without stopping; such a procedure has been called a "computational method" (Knuth 1997:5) or "''calculation procedure'' or ''algorithm''" (Kleene 1952:137); however, Kleene notes that such a method must eventually exhibit "some object" (Kleene 1952:137).


===March–April 2007 phase===
Minsky makes the pertinent observation, in regards to determining whether an algorithm will eventually terminate (from a particular starting state):
In March, Pakistan signed a peace treaty with [[Faqir Mohammed]], the Taliban leader in Bajaur.<ref>{{cite news | title =Pakistan signs the Bajaur Accord| author =Roggio B| publisher =The Long War Journal| date=17 March 2007| url=http://www.longwarjournal.org/archives/2007/03/pakistan_signs_the_b.php}} Retrieved on 14 January 2008.</ref><ref>{{cite news | title =Bajaur: When 'Peace' Yields War| author =Stakelbeck E| publisher =CBN News| date=3 April 2007| url=http://www.cbn.com/CBNnews/130966.aspx}} Retrieved on 14 January 2008.</ref> The Taliban now held three districts in the [[Federally Administered Tribal Areas]]: [[South Waziristan]], [[North Waziristan]] and [[Bajaur Agency]].<ref>{{cite news | author =Roggio B| title =The fall of northwestern Pakistan: An online history| publisher =The Long War Journal| url=http://www.longwarjournal.org/archives/2006/09/the_fall_of_wazirist.php}} Retrieved on 13 January 2008.</ref>
<blockquote>But if the length of the process is not known in advance, then "trying" it may not be decisive, because if the process does go on forever — then at no time will we ever be sure of the answer (Minsky 1967:105).</blockquote>


====Waziri-Uzbek tensions====
As it happens, no other method can do any better, as was shown by [[Alan Turing]] with his celebrated result on the undecidability of the so-called [[halting problem]]. There is no algorithmic procedure for determining of arbitrary algorithms whether or not they terminate from given starting states. The analysis of algorithms for their likelihood of termination is called [[termination analysis]].
Reportedly, the fighting sparked by the killing of [[Saiful Adil]], an [[al-Qaeda]]-linked Arab, blamed on the [[Uzbeks]] by [[Maulavi Nazir]], described as a top pro-Taliban militant commander in the region. According to the other version, fighting started after Mullah Nazir, whom the government says has come over to its side, ordered the Uzbek followers of [[Tohir Yo‘ldosh]], formerly a close confidant of [[Osama bin Laden]], to disarm. It was also preceded by the clashes between the [[Islamic Movement of Uzbekistan|IMU]] and a pro-government tribal leader in [[Azam Warsak]], in which 17 to 19 people died before a [[ceasefire]] was announced.<ref>{{cite news | title =Rival militants clash in Pakistan | publisher =BBC| date=20 March 2007| url=http://news.bbc.co.uk/2/hi/south_asia/6470299.stm}}{{Failed verification|date=January 2008}}</ref>


====Defeat of the Islamic Movement of Uzbekistan====
See the examples of (im-)"proper" subtraction at [[partial function]] for more about what can happen when an algorithm fails for certain of its input numbers — e.g., (i) non-termination, (ii) production of "junk" (output in the wrong format to be considered a number) or no number(s) at all (halt ends the computation with no output), (iii) wrong number(s), or (iv) a combination of these. Kleene proposed that the production of "junk" or failure to produce a number is solved by having the algorithm detect these instances and produce e.g., an error message (he suggested "0"), or preferably, force the algorithm into an endless loop (Kleene 1952:322). Davis does this to his subtraction algorithm — he fixes his algorithm in a second example so that it is proper subtraction (Davis 1958:12-15). Along with the logical outcomes "true" and "false" Kleene also proposes the use of a third logical symbol "u" — undecided (Kleene 1952:326) — thus an algorithm will always produce ''something'' when confronted with a "proposition". The problem of wrong answers must be solved with an independent "proof" of the algorithm e.g., using induction:
[[Sirajuddin Haqqani]], the son of Taliban leader [[Jalaluddin Haqqani]], tried to stop the fighting but failed. Local Taliban militants allied to the tribesmen were reported attacking and seizing the IMU's private jail in Azam Warsak. The Pakistan Army said did not intend to step in, but witnesses say government artillery fired on the Uzbek bunkers they set up to fight the tribesmen.
<blockquote>We normally require auxiliary evidence for this (that the algorithm correctly defines a [[mu recursive function]]), e.g., in the form of an inductive proof that, for each argument value, the computation terminates with a unique value (Minsky 1967:186).</blockquote>


Heavy fighting resumed on March 29, ending a week-long ceasefire between tribal fighters and foreign militants. According to initial reports, tribesmen attacked a checkpoint manned by Uzbek militants and captured two of them. The clashes also left one tribal fighter dead and three wounded. The following day, a senior Pakistani official announced that 52 people were killed during the past two days, 45 of them Uzbeks and the rest tribesmen. One of Maulvi Nazir's aides put the death toll at 35 Uzbeks and 10 tribal fighters. However, residents in the area said that the death toll on both sides was inflated.
=== Expressing algorithms ===
Algorithms can be expressed in many kinds of notation, including [[natural language]]s, [[pseudocode]], [[flowchart]]s, and [[programming language]]s. Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode and flowcharts are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements, while remaining independent of a particular implementation language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a [[computer]], but are often used as a way to define or document algorithms.


The conflict further escalated on April 2 when a council of elders declared jihad against foreign militants and started to raise an army of tribesmen. According to Pakistani security officials, heavy fighting concentrated in the village of Doza Ghundai left more than 60 people dead, including 50 foreigners, 10 tribal fighters and one Pakistani soldier. He also said that "dozens" of Uzbeks had surrendered to tribal forces and that many bunkers used by militants were seized or destroyed.
There is a wide variety of representations possible and one can express a given [[Turing machine]] program as a sequence of machine tables (see more at [[finite state machine]] and [[state transition table]]), as flowcharts (see more at [[state diagram]]), or as a form of rudimentary [[machine code]] or [[assembly code]] called "sets of quadruples" (see more at [[Turing machine]]).


On [[April 12]] [[2007]] the army general in charge of South Waziristan said that tribal fighters had cleared the Uzbeks out of the valleys surrounding [[Wana (Pakistan)|Wana]] and the foreign militants had been pushed back into the mountains on the Afghan border.<ref>{{cite news | title =Tribesmen 'oust' foreign fighters | author =Plett B| publisher =BBC| date=12 April 2007| url =http://news.bbc.co.uk/2/hi/south_asia/6547783.stm}} Retrieved on 16 January 2008.</ref> Four days later, the local tribesmen has urged [[Islamabad]] to resume control of law and order in the area.<ref>{{cite news | title =Tribe in Pakistan security plea | publisher =BBC| date=16 April 2007| url =http://news.bbc.co.uk/2/hi/south_asia/6559365.stm}} Retrieved on 16 January 2008.</ref>
Sometimes it is helpful in the description of an algorithm to supplement small "flow charts" (state diagrams) with natural-language and/or arithmetic expressions written inside "[[block diagram]]s" to summarize what the "flow charts" are accomplishing.


===July 2007–February 2008 phase===
Representations of algorithms are generally classed into three accepted levels of Turing machine description (Sipser 2006:157):
{{main|2007 timeline of the War in Waziristan}}
*'''1 High-level description''':
:: "...prose to describe an algorithm, ignoring the implementation details. At this level we do not need to mention how the machine manages its tape or head"
*'''2 Implementation description''':
:: "...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level we do not give details of states or transition function"
*'''3 Formal description''':
:: Most detailed, "lowest level", gives the Turing machine's "state table".


====Lal Masjid siege====
:''For an example of the simple algorithm "Add m+n" described in all three levels see [[Algorithm examples]].''
On [[July 3]], [[2007]], the militant supporters of Lal Masjid and Pakistani security forces clashed in Islamabad after the students from the mosque attacked a nearby government ministry building. The Pakistani security forces immediately put up a [[Lal Masjid siege|siege around the mosque complex]] which lasted until July 11 and resulted in 108 deaths. This represented the main catalyst for the conflict and eventual breakdown of the truce that existed between Pakistan and the Taliban in the northwest. Already during the siege there were several attacks in Waziristan in retaliation for the siege.


====Truce in Waziristan broken====
=== Implementation ===
As the siege in Islamabad ensued, several attacks on Pakistani troops in Waziristan were reported. On [[July 14]], [[2007]], a suicide bomber attacked a Pakistani Army convoy killing 26 soldiers and wounding 54. On [[July 15]], [[2007]], two suicide bombers attacked another Pakistani Army convoy killing 16 soldiers and 5 civilians and wounding another 47 people. And in a separate incident a fourth suicide bomber attacked a police headquarters killing 28 police officers and recruits and wounding 35 people.<ref>[http://www.foxnews.com/story/0,2933,289340,00.html FOXNews.com - Two Days of Homicide Attacks Kill 70 in Pakistan - International News | News of the World | Middle East News | Europe News<!-- Bot generated title -->]</ref><ref>[http://fairuse.100webcustomers.com/fairenough/nytA31.html Suicide Bombers Kill 49 in Pakistan<!-- Bot generated title -->]</ref>
Most algorithms are intended to be implemented as [[computer programs]]. However, algorithms are also implemented by other means, such as in a biological [[neural network]] (for example, the [[human brain]] implementing [[arithmetic]] or an insect looking for food), in an [[electrical circuit]], or in a mechanical device.


The assault on the Red Mosque prompted pro-Taliban rebels along the border with Afghanistan to scrap the controversial [[Waziristan Accord]] with the government. <ref>[http://news.bbc.co.uk/2/hi/south_asia/6905808.stm BBC NEWS | South Asia | Scores killed in Pakistan attacks<!-- Bot generated title -->]</ref>
== Example ==
One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm, which can be stated in a high-level description [[English language|English]] prose, as:


====The new war in Waziristan====
'''High-level description:'''
The Army moved large concentration of troops into Waziristan and engaged in fierce clashes with militants in which at least 100 militants were killed including wanted terrorist and former [[Guantanamo Bay detention camp|Guantanamo Bay]] detainee, [[Abdullah Mehsud]].<ref>http://news.yahoo.com/s/ap/20070724/ap_on_re_as/pakistan_militant_leader</ref> The militants also struck back by attacking Army convoys, security check points and sending suicide bombers killing dozens of soldiers and police and over 100 civilians. In one month of fighting during the period from [[July 24]] to [[August 24]], [[2007]], 250 militants and 60 soldiers were killed.
# Assume the first item is largest.
# Look at each of the remaining items in the list and if it is larger than the largest item so far, make a note of it.
# The last noted item is the largest in the list when the process is complete.


On [[September 2]], [[2007]], just a few dozen militants led by [[Baitullah Mehsud]] managed to ambush a 17-vehicle army convoy and captured an estimated 247 soldiers in it, without a shot being fired; an event that shocked the nation.<ref>[http://news.bbc.co.uk/2/hi/south_asia/6978240.stm Pakistan crisis 'hits army morale' By Ahmed Rashid] [[6 September]] [[2007]] - [[BBC]]</ref> Several officers were among the captured.
'''(Quasi-)formal description:'''
Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in [[pseudocode]] or [[pidgin code]]:


After the army returned to Waziristan, they garrisoned the areas and set up check-points, but the militants hit hard. In mid-September Taliban forces attacked a number of Pakistani army outposts all across North and South Waziristan. This resulted in some of the heaviest fighting of the war. On [[September 12]], [[2007]], the first outpost was attacked and overrun by the Taliban resulting in the capture of 12 Pakistani soldiers. The next day on [[September 13]], [[2007]], a suicide bomber in [[Ghazi Tarbela]] attacked a Pakistani army base, destroying the main mess hall and killing 20 members of the [[Karar]] commando group; Pakistan's most elite army unit. Another 29 soldiers were wounded. A series of attacks ensued and by [[September 20]], [[2007]] a total of five Pakistani Army military outposts had been overrun and more than 25 soldiers captured. More than 65 soldiers were either killed or captured and almost 100 wounded.
{{algorithm-begin|name=LargestNumber}}
Input: A non-empty list of numbers ''L''.
Output: The ''largest'' number in the list ''L''.
''largest'' ← ''L''<sub>0</sub>
'''for each''' ''item'' '''in''' the list ''L<sub>≥1</sub>'', '''do'''
'''if''' the ''item'' > ''largest'', '''then'''
''largest'' ← the ''item''
'''return''' ''largest''
{{algorithm-end}}


A little over two weeks later, the Army responded with helicopter gunships, jet fighters and ground troops. They hit militant positions near the town of [[Mir Ali]]. In [[Battle of Mir Ali|heavy fighting]] over four days between [[October 7]] and [[October 10]] [[2007]], 257 people were killed, including 175 militants, 47 soldiers and 35 civilians.
For a more complex example of an algorithm, see [[Euclid's algorithm]] for the [[greatest common divisor]], one of the earliest algorithms known.
[[Image:FATA (4).PNG|300px|thumb|left|With the fall of the [[Swat Valley]], the fighting spread to the whole FATA.]]


=== Algorithm analysis ===
====Battle of Swat Valley====
By the end of October fighting erupted in the Swat district of the North-West Frontier Province, with a large Taliban force, under the command of [[Maulana Fazlullah]], trying to impose Sharia law. Around 3,000 paramilitary soldiers were sent to confront them. After almost a week of [[Battle of Swat|heavy fighting]] the battle came to a standstill with both sides suffering heavy casualties. Then on [[November 1]] and [[November 3]], 220 paramilitary soldiers and policemen surrendered or deserted after a military position on a hill-top and two police stations were overrun. This left the Taliban in control of most of the Swat district.
As it happens, it is important to know how much of a particular resource (such as time or storage) is required for a given algorithm. Methods have been developed for the [[analysis of algorithms]] to obtain such quantitative answers; for example, the algorithm above has a time requirement of O(''n''), using the [[big O notation]] with ''n'' as the length of the list. At all times the algorithm only needs to remember two values: the largest number found so far, and its current position in the input list. Therefore it is said to have a space requirement of ''O(1)'', if the space required to store the input numbers is not counted, or O (log ''n'') if it is counted.


The fighting in Swat is the first serious insurgent threat from pro-Taliban forces in what is known as a settled area of Pakistan. Forces loyal to Maulana Fazlullah, including some foreign fighters, after taking control of a series of small towns and villages, tried to implement strict Islamic law in November 2007. In mid-November the regular army was deployed with the help of helicopter gunships to crush the uprising. The Pakistan Army deployed over 2,500 men. By the beginning of December the fighting had ended and the Army recaptured Swat. Almost 400 pro-Taliban fighters were dead along with 15 Pakistani soldiers and 20 civilians in the military offensive.<ref>[http://www.dawn.com/2007/11/17/top7.htm 12-hour curfew clamped on Swat -DAWN - Top Stories; November 17, 2007<!-- Bot generated title -->]</ref>
Different algorithms may complete the same task with a different set of instructions in less or more time, space, or effort than others. For example, given two different recipes for making potato salad, one may have ''peel the potato'' before ''boil the potato'' while the other presents the steps in the reverse order, yet they both call for these steps to be repeated for all potatoes and end when the potato salad is ready to be eaten. <!-- poor example ... who would boil each potato separately? and making a salad in general requires no cooking ... and the algorithms aren't even truly equivalent, as the flavor and texture changes if the potatoes are peeled before boiling! -->


====The Rawalpindi attacks====
The [[analysis of algorithms|analysis and study of algorithms]] is a discipline of [[computer science]], and is often practiced abstractly without the use of a specific [[programming language]] or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually [[pseudocode]] is used for analysis as it is the simplest and most general representation.
The city of [[Rawalpindi]], which is the military headquarters of the Pakistan Armed Forces, was an attractive target for the militants and they were planning to hit the city.{{Fact|date=January 2008}} On [[September 3]], two suicide bombers targeted a military intelligence (ISI) bus and a line of cars carrying ISI officers. The [[September 2007 bombings in Rawalpindi|bus attack]] killed a large number of Defence Ministry workers and the other attack killed an Army colonel. In all 31 people, 19 soldiers and 12 civilians, were killed.


Over two months later on [[November 24]], in a similar fashion a pair of suicide bombers struck. Again, one of the targets was a military intelligence bus. Almost everyone on the bus was killed. The other bomber blew up at a military checkpoint. 35 people were killed, almost all military.{{Fact|date=November 2007}}
== Classes ==
There are various ways to classify algorithms, each with its own merits.


=== Classification by implementation ===
==== State of emergency ====
The [[2007 Pakistani state of emergency]] was declared by [[Pervez Musharraf]] on [[2007-11-03]] and lasted until [[2007-12-15]]. During this time the constitution of the country was suspended.<ref name="Dawn">{{cite web | title= Musharraf imposes emergency rule | publisher=Dawn | url=http://www.dawn.com/2007/11/04/top1.htm|date=[[2007-11-03]]| accessdate=2007-11-03 }}</ref><ref name="CNN">{{cite web | title= Martial law declared in Pakistan|publisher=CNN | url=http://www.cnn.com/2007/WORLD/asiapcf/11/03/pakistan.emergency/index.html |date=[[2007-11-03]]| accessdate=2007-11-03}}</ref> This action and its responses are generally related to the controversies surrounding the re-election of Musharraf during the [[Pakistani presidential election, 2007|presidential election]] that had occurred on [[2007-10-06]], and also was claimed by the government to be the reaction to the actions by Islamic militants in Waziristan.<ref>{{cite news| url = http://www.timesonline.co.uk/tol/news/world/asia/article2800993.ece| title =Pervez Musharraf spoils for a fight as he declares emergency rule| date=[[2007-11-04]] |accessdate= 2007-11-04 |publisher= The Times}}</ref>
One way to classify algorithms is by implementation means.


====Benazir Bhutto's assassination====
* '''Recursion''' or '''iteration''': A [[recursive algorithm]] is one that invokes (makes reference to) itself repeatedly until a certain condition matches, which is a method common to [[functional programming]]. [[Iteration|Iterative]] algorithms use repetitive constructs like [[Control flow#Loops|loops]] and sometimes additional data structures like [[Stack (data structure)|stacks]] to solve the given problems. Some problems are naturally suited for one implementation or the other. For example, [[towers of hanoi]] is well understood in recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.
{{main|Benazir Bhutto assassination}}
On [[27 December]] [[2007]], Pakistani opposition leader Benazir Bhutto was killed upon leaving a political rally for the [[Pakistan Peoples Party]] (PPP) in [[Rawalpindi]], Pakistan.<ref>{{cite web |url=http://english.aljazeera.net/NR/exeres/737B73AE-EE5D-4C41-8CA2-E04B356FBCBC.htm |title=Bhutto killed in suicide attack |accessdate=2007-12-27 |format=HTML |work=Al Jazeera English }}</ref><ref>[http://www.reuters.com/news/video?videoId=73377&videoChannel=1 Benazir Bhutto is dead] Reuter's video</ref> A suicidal assassin reportedly fired shots in Bhutto's direction just prior to detonating an explosive pellet-laden vest, killing approximately 24 people and wounding many more.<ref>{{cite news |title=Benazir Bhutto Assination NBC News Coverage |url=http://test.redlasso.com/service/svc/clip/playClip?fid=7ed304c1-5472-469b-a209-a9a696368d89 |publisher=NBC |date=2007-12-27 |accessdate=2007-12-27}}</ref><ref>{{cite news |title=Benazir Bhutto Assination CBS News Coverage |url=http://test.redlasso.com/service/svc/clip/playClip?fid=cc91d5dd-83e9-444d-9624-ae1816a31612 |publisher=CBS |date=2007-12-27 |accessdate=2007-12-27}}</ref><ref>{{cite news |title=Benazir Bhutto Assination ABC News Coverage |url=http://test.redlasso.com/service/svc/clip/playClip?fid=67553fa0-c7ba-48cb-8a3a-2152ec570595 |publisher=ABC |date=2007-12-27 |accessdate=2007-12-27}}</ref>
Musharraf and the army blamed the attack on Al-Qaida, but the following day a statement by Commander Baitullah Mehsud was sent to the media saying that he and Al-Qaida had no involvement in the murder of the former Prime Minister, he briefed that these were the crimes of Musharraf and the army. The killing was followed by a wave of violence across the country that left 58 people dead, including four police officers. Most of the violence was directed at Musharraf and the pro-Musharraf political party, [[Pakistan Muslim League (Q)]].
The public chanted slogans against the army and Musharraf: "Musharraf Dog", "General is a murderer", "uniform (army) wearing murderers", etc.
Bhutto had previously survived [[2007 Karachi bombing|an assassination attempt]] made on her life during her homecoming which left 139 people dead and hundreds wounded.<ref name=nytimes_karachi>{{cite news|first=Carlotta|last=Gall|url=http://www.nytimes.com/2007/10/19/world/asia/19pakistan.html|title=Bomb Attack Kills Scores in Pakistan as Bhutto Returns|coauthors=Masood, Salman|publisher=The New York Times|date=2007-10-19|accessdate=2007-12-27}}</ref>


====More fighting in South Waziristan====
* '''Logical''': An algorithm may be viewed as controlled [[Deductive reasoning|logical deduction]]. This notion may be expressed as: '''Algorithm = logic + control''' (Kowalski 1979). The logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms. This is the basis for the [[logic programming]] paradigm. In pure logic programming languages the control component is fixed and algorithms are specified by supplying only the logic component. The appeal of this approach is the elegant [[Formal semantics of programming languages|semantics]]: a change in the axioms has a well defined change in the algorithm.
In January 2008, pro-Taliban militants [[Sararogha Fort raid|overran Sararogha Fort]], and may have overrun a fort in Ladah as well. Both forts are in South Waziristan, and were held by the Pakistani army.<ref>{{cite news | title =Pakistani troops 'flee border post'|publisher =Al Jazeera | date=17 January 2008| url =http://english.aljazeera.net/NR/exeres/EA53A69D-1973-4C86-8064-D4FCB4C40FB1.htm}} Retrieved on 18 January 2008.</ref> Gunmen fired on a children's aid group in in the [[North West Frontier Province]] [[February 25]], [[2008]], leaving four staffers dead. On the same day as the children's aid group attack, another suicide bomber struck in the garrison-town of Rawalpindi killing Pakistani Lt. Gen. Mushtaq Baig along with two more soldiers and five civilians. Baig was the army's surgeon general and the highest-level military official to be assassinated since Pakistan joined the U.S.-led war on terror.<ref>[http://edition.cnn.com/2008/WORLD/asiapcf/02/25/pakistan.blast/index.html Pakistan attacks hit aid group, military HQ - CNN.com<!-- Bot generated title -->]</ref>


At least 45 people died and 82 were wounded in a suicide attack on the funeral [[February 29]] [[2008]] of a district superintendent of police – killed earlier in the day in a separate attack – in northwest Pakistan's [[Swat (Pakistan)|Swat]] province.<ref>[http://edition.cnn.com/2008/WORLD/asiapcf/03/01/pakistan.explosion/index.html 45 killed in Pakistan funeral blast - CNN.com<!-- Bot generated title -->]</ref>
* '''Serial''' or '''parallel''' or '''distributed''': Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. An algorithm designed for such an environment is called a serial algorithm, as opposed to [[parallel algorithm]]s or [[distributed algorithms]]. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a [[Computer Network|network]]. Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable. Some problems have no parallel algorithms, and are called inherently serial problems.


====Peace agreement====
* '''Deterministic''' or '''non-deterministic''': [[Deterministic algorithm]]s solve the problem with exact decision at every step of the algorithm whereas [[non-deterministic algorithm]] solve problems via guessing although typical guesses are made more accurate through the use of [[heuristics]].
On [[7 February]], [[2008]], a leading militant group led by [[Baitullah Mehsud]] offered a truce and peace negotiations resulting in a suspension of violence.<ref>[http://news.bbc.co.uk/2/hi/south_asia/7232203.stm BBC NEWS | South Asia | Pakistani militants 'call truce'<!-- Bot generated title -->]</ref>


On [[May 21]], [[2008]] Pakistan signed a peace agreement with Taliban fighters. Pakistan's government promised to "gradually" pull out troops from the northwestern valley of Swat. In return the Taliban were due to shut down training camps, hand over foreign fighters and halt suicide attacks on government installations and security forces under the 15-point pact.<ref name=autogenerated3>[http://english.aljazeera.net/NR/exeres/D1AC8679-0D78-4829-952A-004CCA5DA283.htm Al Jazeera English - News - Pakistan Troops To Vacate Swat<!-- Bot generated title -->]</ref>
* '''Exact''' or '''approximate''': While many algorithms reach an exact solution, [[approximation algorithm]]s seek an approximation that is close to the true solution. Approximation may use either a deterministic or a random strategy. Such algorithms have practical value for many hard problems.


===June 2008 - Present===
=== Classification by design paradigm ===
{{main|2007 timeline of the War in Waziristan}}
Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories will include many different types of algorithms. Some commonly found paradigms include:


Despite the agreement sporadic fighting continued until late June and escalated with the takeover of the town of Jandola on [[June 24]], by the militants. 22 pro-government tribal fighters were captured and executed by the Taliban at that time.<ref>[http://www.dawn.com/2008/06/24/top8.htm Baitullah men storm Jandola -DAWN - Top Stories; June 24, 2008<!-- Bot generated title -->]</ref>
* '''Divide and conquer'''. A [[divide and conquer algorithm]] repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually [[recursion|recursively]]), until the instances are small enough to solve easily. One such example of divide and conquer is [[mergesort|merge sorting]]. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in conquer phase by merging them. A simpler variant of divide and conquer is called '''decrease and conquer algorithm''', that solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so conquer stage will be more complex than decrease and conquer algorithms. An example of decrease and conquer algorithm is [[binary search algorithm]].
* '''[[Dynamic programming]]'''. When a problem shows [[optimal substructure]], meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems, and [[overlapping subproblems]], meaning the same subproblems are used to solve many different problem instances, a quicker approach called ''dynamic programming'' avoids recomputing solutions that have already been computed. For example, the shortest path to a goal from a vertex in a weighted [[graph (mathematics)|graph]] can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and [[memoization]] go together. The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems. By using memoization or maintaining a [[Mathematical table|table]] of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.
* '''The greedy method'''. A [[greedy algorithm]] is similar to a [[dynamic programming|dynamic programming algorithm]], but the difference is that solutions to the subproblems do not have to be known at each stage; instead a "greedy" choice can be made of what looks best for the moment. The greedy method extends the solution with the best possible decision (not all feasible decisions) at an algorithmic stage based on the current local optimum and the best decision (not all possible decisions) made in previous stage. It is not exhaustive, and does not give accurate answer to many problems. But when it works, it will be the fastest method. The most popular greedy algorithm is finding the minimal spanning tree as given by [[kruskal's algorithm|Kruskal]].
* '''Linear programming'''. When solving a problem using [[linear programming]], specific [[inequality|inequalities]] involving the inputs are found and then an attempt is made to maximize (or minimize) some linear function of the inputs. Many problems (such as the [[Maximum flow problem|maximum flow]] for directed [[graph (mathematics)|graphs]]) can be stated in a linear programming way, and then be solved by a 'generic' algorithm such as the [[simplex algorithm]]. A more complex variant of linear programming is called integer programming, where the solution space is restricted to the [[integers]].
* '''[[Reduction (complexity)|Reduction]]'''. This technique involves solving a difficult problem by transforming it into a better known problem for which we have (hopefully) [[asymptotically optimal]] algorithms. The goal is to find a reducing algorithm whose [[Computational complexity theory|complexity]] is not dominated by the resulting reduced algorithm's. For example, one [[selection algorithm]] for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as ''transform and conquer''.
* '''Search and enumeration'''. Many problems (such as playing [[chess]]) can be modeled as problems on [[graph theory|graphs]]. A [[graph exploration algorithm]] specifies rules for moving around a graph and is useful for such problems. This category also includes [[search algorithm]]s, [[branch and bound]] enumeration and [[backtracking]].
* '''The probabilistic and heuristic paradigm'''. Algorithms belonging to this class fit the definition of an algorithm more loosely.
# [[Probabilistic algorithm]]s are those that make some choices randomly (or pseudo-randomly); for some problems, it can in fact be proven that the fastest solutions must involve some [[randomness]].
# [[Genetic algorithm]]s attempt to find solutions to problems by mimicking biological [[evolution]]ary processes, with a cycle of random mutations yielding successive generations of "solutions". Thus, they emulate reproduction and "survival of the fittest". In [[genetic programming]], this approach is extended to algorithms, by regarding the algorithm itself as a "solution" to a problem.
# [[Heuristic]] algorithms, whose general purpose is not to find an optimal solution, but an approximate solution where the time or resources are limited. They are not practical to find perfect solutions. An example of this would be [[local search (optimization)|local search]], [[tabu search]], or [[simulated annealing]] algorithms, a class of heuristic probabilistic algorithms that vary the solution of a problem by a random amount. The name "[[simulated annealing]]" alludes to the metallurgic term meaning the heating and cooling of metal to achieve freedom from defects. The purpose of the random variance is to find close to globally optimal solutions rather than simply locally optimal ones, the idea being that the random element will be decreased as the algorithm settles down to a solution.


There had been growing concern about threats to [[Peshawar]] from Taliban fighters. In early June, a Taliban force from [[Khyber]] entered the city and seized 16 Christians, before later releasing them. Mahmood Shah, a former security chief in the tribal regions, said: "The situation is such that [the Taliban] are all around Peshawar. They are on our doorstep."<ref>[http://english.aljazeera.net/news/asia/2008/06/2008628841648797.html Al Jazeera English - CENTRAL/S. ASIA - Pakistan forces in Khyber offensive<!-- Bot generated title -->]</ref>
=== Classification by field of study ===
{{Seealso|List of algorithms}}
Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are [[search algorithm]]s, [[sorting algorithm]]s, [[merge algorithm]]s, [[numerical analysis|numerical algorithms]], [[graph theory|graph algorithms]], [[string algorithms]], [[computational geometry|computational geometric algorithms]], [[combinatorial|combinatorial algorithms]], [[machine learning]], [[cryptography]], [[data compression]] algorithms and [[parsing|parsing techniques]].


On [[June 28]], [[2008]], Pakistan's Army started an offensive against Taliban fighters in [[Khyber]]. The military took control of a key town and demolished an insurgent group's building. 1 militant was reportedly killed while 2 soldiers died in Swat valley.<ref>[http://www.france24.com/en/20080629-pakistan-broadens-anti-militant-offensive France 24 | Pakistan broadens anti-militant offensive | France 24<!-- Bot generated title -->]</ref> The operation was halted in early [[July]].
Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was originally invented for optimization of resource consumption in industry, but is now used in solving a broad range of problems in many fields.


On [[July 6]], [[2008]], a suicide bomber attacked a police station in Islamabad killing 12 policemen and seven civilians. Another 53 people were wounded. The police were part of a protection cordon set up by the government for a rally which was attended by about 12,000 people who were marking the first anniversary of government forces storming Islamabad's Red Mosque, or Lal Masjid.<ref>[http://www.cnn.com/2008/WORLD/asiapcf/07/06/pakistan.blast/index.html Suicide blast targeting police kills 16 at Pakistan rally - CNN.com<!-- Bot generated title -->]</ref>
=== Classification by complexity ===
{{See also|Complexity class| Parameterized Complexity}}
Algorithms can be classified by the amount of time they need to complete compared to their input size. There is a wide variety: some algorithms complete in linear time relative to input size, some do so in an exponential amount of time or even worse, and some never halt. Additionally, some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them.


On [[July 12]], [[2008]], militants ambushed a military convoy killing 17 soldiers.{{Fact|date=July 2008}}
=== Classification by computing power ===


On [[July 19]], [[2008]], clashes erupted between the Taliban and a rival faction of alleged pro-government Taliban militants. 10-15 of the pro-government fighters were killed and another 120 were captured. Among the captured were two commanders who were tried under "Islamic" law by the Taliban and then executed.
Another way to classify algorithms is by computing power. This is typically done by considering some collection (class) of algorithms. A recursive class of algorithms is one that includes algorithms for all Turing computable functions. Looking at classes of algorithms allows for the possibility of restricting the available computational resources (time and memory) used in a computation. A subrecursive class of algorithms is one in which not all Turing computable functions can be obtained. For example, the algorithms that run in [[P (complexity)|polynomial time]] suffice for many important types of computation but do not exhaust all Turing computable functions. The class of algorithms implemented by [[primitive recursive function]]s is another subrecursive class.


On [[July 21]], [[2008]], heavy fighting in Baluchistan killed 32 militants, 9 soldiers and 2 civilians. More than two dozen militants were captured and a large weapons cache was found.<ref>[http://news.yahoo.com/s/nm/20080721/wl_nm/pakistan_baluchistan_dc_1 Fighting flares in Pakistan's Baluchistan; 43 killed - Yahoo! News<!-- Bot generated title -->]</ref>
Burgin (2005, p. 24) uses a generalized definition of algorithms that relaxes the common requirement that the output of the algorithm that computes a function must be determined after a finite number of steps. He defines a super-recursive class of algorithms as "a class of algorithms in which it is possible to compute functions not computable by any Turing machine" (Burgin 2005, p. 107). This is closely related to the study of methods of [[hypercomputation]].


Between [[July 28]] and [[August 4]] [[2008]], heavy fighting flared up in the northwestern Swat valley leaving 94 militants and 22 soldiers and policemen dead. Another 28 civilians were also killed.<ref>[http://news.yahoo.com/s/nm/20080804/wl_nm/pakistan_violence_dc_2 Ninety-four militants, 14 troops killed in NW Pakistan - Yahoo! News<!-- Bot generated title -->]</ref>
== Legal issues ==
:''See also: [[Software patents]] for a general overview of the patentability of software, including computer-implemented algorithms.''


Heavy fighting erupted on [[August 6]] [[2008]], in the Loisam area of Bajaur district. Loisam lies on the strategically important road leading towards the main northwestern city of Peshawar. The fighting started when hundreds of militants poured into the area attacking government forces. After four days of fighting on [[August 10]] the military withdrew from the area. 100 militants and 9 soldiers were confirmed killed and another 55 soldiers were missing, at least three dozen of them captured by the militants.<ref>[http://news.yahoo.com/s/afp/20080810/wl_afp/pakistanafghanistanunrest_080810132035 100 militants, nine soldiers killed in Pakistan clashes - Yahoo! News<!-- Bot generated title -->]</ref><ref>[http://www.latimes.com/news/nationworld/world/la-fg-pakistan10-2008aug10,0,6869936.story Pakistan fighting ends as troops withdraw - Los Angeles Times<!-- Bot generated title -->]</ref> While the fighting was going on in Bajaur, in the Buner area of North West Frontier Province militants killed at least nine policemen in an attack on a check post.<ref>[http://in.news.yahoo.com/43/20080809/876/twl-taliban-militants-kill-nine-policeme_1.html Taliban militants kill nine policemen in Pakistan - Yahoo! India News<!-- Bot generated title -->]</ref> The checkpoint was then abandoned, and the local Pakistani forces withdrew to Khar, the main town of Bajaur Agency. There were reports that the town of Khar was then besieged by tribal militants. <ref>[http://news.bbc.co.uk/2/hi/south_asia/7552425.stm BBC NEWS | South Asia | 'Dozens die' in Pakistan clashes<!-- Bot generated title -->]</ref>
Algorithms, by themselves, are not usually patentable. In the [[United States]], a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals do not constitute "processes" (USPTO 2006) and hence algorithms are not patentable (as in [[Gottschalk v. Benson]]). However, practical applications of algorithms are sometimes patentable. For example, in [[Diamond v. Diehr]], the application of a simple [[feedback]] algorithm to aid in the curing of [[synthetic rubber]] was deemed patentable. The [[Software patent debate|patenting of software]] is highly controversial, and there are highly criticized patents involving algorithms, especially [[data compression]] algorithms, such as [[Unisys]]' [[Graphics Interchange Format#Unisys and LZW patent enforcement|LZW patent]].


On 23 August 2008, at least 15 people were killed in a suicide attack at a police check post in Char Bagh area of Swat. Also, three persons including two kids were killed in a bomb blast in Abuha.<ref>[http://thenews.jang.com.pk/updates.asp?id=52931 15 feared dead in Swat suicide bombings]</ref>
Additionally, some cryptographic algorithms have export restrictions (see [[export of cryptography]]).


====Tribesmen declare war against the Taliban====
== History: Development of the notion of "algorithm" ==
By the beginning of September 2008, Pakistani tribal elders began organising a private army of approximately 30,000 tribesmen to fight the taliban. A lashkar, or private army, comprised of Pakistani tribesmen, began torching the houses of Taliban commanders in Bajaur, near the Afghan border, vowing to fight them until they are expelled.
=== Origin of the word ===


A local jirga decided to form the lashkar in the wake of the increasing presence of the local Taliban in the area. The lashkar began torching houses, including the house of a local Taliban commander named Naimatullah, who had occupied several government schools and converted them into seminaries.
{{Seealso|Timeline of algorithms}}
The word ''algorithm'' comes from the name of the 9th century [[Persian people|Persian]] mathematician [[al-Khwarizmi|Abu Abdullah Muhammad ibn Musa al-Khwarizmi]] whose works introduced Indian numerals and algebraic concepts. He worked in [[Baghdad]] at the time when it was the centre of scientific studies and trade. The word ''[[algorism]]'' originally referred only to the rules of performing [[arithmetic]] using [[Hindu-Arabic numeral system|Arabic numerals]] but evolved via European Latin translation of al-Khwarizmi's name into ''algorithm'' by the 18th century. The word evolved to include all definite procedures for solving problems or performing tasks.


A tribal elder named Malik Munsib Khan, who heads the lashkar, said that tribesmen would continue their struggle until the Taliban were expelled from the area, adding that anyone found sheltering Taliban militants would be fined one million rupees and their houses will be torched. The tribesmen also torched two important centres of the Taliban in the area and gained control of most of the tehsil.
=== Discrete and distinguishable symbols ===


The main reasons for this was that the operations that were taking place in the [[Federally Administered Tribal Areas]] had displaced some 300,000 people while dozens of citizens have been killed in clashes between the militants and military. This showcases why the Taliban have become highly unpopular with the tribal elders and their tribesmen.<ref>http://www.csmonitor.com/2008/0901/p99s01-duts.html</ref><ref>http://www.mcclatchydc.com/226/story/53151.html</ref><ref>http://www.mcclatchydc.com/226/story/53151.html</ref>
'''Tally-marks''': To keep track of their flocks, their sacks of grain and their money the ancients used tallying: accumulating stones or marks scratched on sticks, or making discrete symbols in clay. Through the Babylonian and Egyptian use of marks and symbols, eventually [[Roman numerals]] and the [[abacus]] evolved (Dilson, p.16–41). Tally marks appear prominently in [[unary numeral system]] arithmetic used in [[Turing machine]] and [[Post-Turing machine]] computations.


====US support and aid for Pakistani tribesmen====
=== Manipulation of symbols as "place holders" for numbers: algebra ===
Recent American military proposals outlines an intensified effort to enlist tribal leaders in the frontier areas of Pakistan in the fight against Al Qaeda and the Taliban, as part of a broader effort to bolster Pakistani forces against militancy in the region.
The work of the ancient [[Greek mathematics|Greek geometers]], [[Islamic mathematics|Persian mathematician]] [[Muhammad ibn Mūsā al-Khwārizmī|Al-Khwarizmi]] (often considered the "father of [[algebra]]" and from whose name the terms "[[algorism]]" and "algorithm" are derived), and Western European mathematicians culminated in [[Leibniz]]'s notion of the [[calculus ratiocinator]] (ca 1680):
:"A good century and a half ahead of his time, Leibniz proposed an algebra of logic, an algebra that would specify the rules for manipulating logical concepts in the manner that ordinary algebra specifies the rules for manipulating numbers" (Davis 2000:1)


The proposal is modeled in part on a similar effort by American forces in Iraq that has been hailed as a great success in fighting foreign insurgents there. But it raises the question of whether such partnerships can be forged without a significant American military presence in Pakistan. And it is unclear whether enough support can be found among the tribes. Small numbers of United States military personnel have served as advisers to the Pakistani Army in the tribal areas, giving planning advice and helping to integrate American intelligence. Under this new approach, the number of advisers would increase.
=== Mechanical contrivances with discrete states ===


American officials said these security improvements complemented a package of assistance from the Agency for International Development and the State Department for the seven districts of the tribal areas that amounted to $750 million over five years, and would involve work in education, health and other sectors. The State Department's Bureau of International Narcotics and Law Enforcement Affairs is also assisting the Frontier Corps with financing for counternarcotics work.<ref>http://www.iht.com/articles/2007/11/19/america/19policy.php</ref><ref>http://www.nytimes.com/2007/11/19/washington/19policy.html?_r=1&adxnnl=1&ref=world&adxnnlx=1222717017-fvB/geZ6uJD4OkQF6LCPSg&oref=slogin</ref><ref>http://216.239.59.104/search?q=cache:Fda3_xPQYTkJ:www.state.gov/p/sca/rls/2008/106495.htm+US+support+pakistani+tribes&hl=en&ct=clnk&cd=19&gl=uk</ref>
'''The clock''': Bolter credits the invention of the weight-driven [[clock]] as “The key invention [of Europe in the Middle Ages]", in particular the [[verge escapement]] (Bolter 1984:24) that provides us with the tick and tock of a mechanical clock. “The accurate automatic machine” (Bolter 1984:26) led immediately to "mechanical [[automata]]" beginning in the thirteenth century and finally to “computational machines" – the [[difference engine]] and [[analytical engine]]s of [[Charles Babbage]] and Countess [[Ada Lovelace]] (Bolter p.33–34, p.204–206).


====Islamabad Marriott Hotel Bombing====
'''Jacquard loom, Hollerith punch cards, telegraphy and telephony — the electromechanical relay''': Bell and Newell (1971) indicate that the [[Jacquard loom]] (1801), precursor to [[Hollerith cards]] (punch cards, 1887), and “telephone switching technologies” were the roots of a tree leading to the development of the first computers (Bell and Newell diagram p. 39, cf Davis 2000). By the mid-1800s the [[telegraph]], the precursor of the telephone, was in use throughout the world, its discrete and distinguishable encoding of letters as “dots and dashes” a common sound. By the late 1800s the [[ticker tape]] (ca 1870s) was in use, as was the use of [[Hollerith cards]] in the 1890 U.S. census. Then came the [[Teletype]] (ca 1910) with its punched-paper use of [[Baudot code]] on tape.
On 23 September 2008, the Pakistani Army, backed by helicopter gunships and artillery killed more than 60 insurgents in northwest Pakistan in offensives as the response to the [[Islamabad Marriott Hotel bombing]] over the weekend at the Marriott hotel in the capital Islamabad that killed 53 people. In the nearby Bajur tribal region, the Army killed at least 10 militants during an ongoing offensive there, according to government officials.<ref>[http://www.foxnews.com/story/0,2933,426207,00.html]</ref>


The Bajur operations, which the army says has left more than 700 suspected militants dead, has won praise from U.S. officials who are worried about rising violence in Pakistan and Afghanistan.<ref>[http://ap.google.com/article/ALeqM5hkiMxbHNH0BqgpWA2ZG6VD6wVTmAD93CC4R00]</ref>
Telephone-switching networks of electromechanical [[relay]]s (invented 1835) was behind the work of [[George Stibitz]] (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the “burdensome’ use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea.... When the tinkering was over, Stibitz had constructed a binary adding device". (Valley News, p. 13).


====Renewed Bajaur offensive====
Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" ''open'' and ''closed''):
Pakistani President [[Asif Ali Zardari]] publicly vowed revenge in response to the [[Islamabad Marriott Hotel bombing]]. By 26 September 2008, Pakistani troops had successfully conducted and completed a major offensive in the Bajaur and the Tang Khata regions of the [[Federally Administered Tribal Areas]]. Pakistani troops had killed over 1,000 militants in a huge offensive, a day after President Asif Ali Zardari lashed out at US forces over a clash on the Afghan border.
: It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned." (Davis, p. 14).


Tariq Khan, Inspector General of the Paramilitary Frontier Corps, mentioned to journalists that since the beginning of the Bajaur operations, there were up to 2,000 militant fighters including hundreds of foreigners who were fighting with the soldiers and the security forces. The overall death toll was over 1,000 militants and also adding that 27 Pakistani soldiers had also been killed with 111 soldiers seriously wounded.<ref>http://www.thestandard.com.hk/breaking_news_detail.asp?id=6788&icid=4&d_str=20080926</ref><ref>http://timesofindia.indiatimes.com/World/Pakistan/Pakistan_says_five_top_militants_among_1000_dead_in_offensive/articleshow/3530363.cms</ref>
=== Mathematics during the 1800s up to the mid-1900s ===
'''Symbols and rules''': In rapid succession the mathematics of [[George Boole]] (1847, 1854), [[Gottlob Frege]] (1879), and [[Giuseppe Peano]] (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's ''The principles of arithmetic, presented by a new method'' (1888) was "the first attempt at an axiomatization of mathematics in a symbolic language" (van Heijenoort:81ff).


Five top Al-Qaeda and Taliban commanders were among those killed in a month-long operation in Bajaur. Of the five militant commanders killed, four appeared to be foreigners: Egyptian Abu Saeed Al-Masri; Abu Suleiman, an Arab; an Uzbek commander named Mullah Mansoor; and an Afghan commander called Manaras. The fifth was a Pakistani commander named only Abdullah, a son of ageing hardline leader Maulvi Faqir Mohammad who is based in Bajaur and has close ties to Al-Qaeda second-in-command Ayman al-Zawahiri.<ref>http://afp.google.com/article/ALeqM5iGTgxqoTldD4lBtlbyfU5MSNowjg</ref><ref>http://in.reuters.com/article/southAsiaNews/idINIndia-35671020080926</ref>
But Heijenoort gives Frege (1879) this kudos: Frege’s is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a ''lingua characterica'', a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed from specific symbols that are manipulated according to definite rules" (van Heijenoort:1). The work of Frege was further simplified and amplified by [[Alfred North Whitehead]] and [[Bertrand Russell]] in their [[Principia Mathematica]] (1910–1913).


On 27 September 2008, Pakistani troops killed at least 16 militants after coming under attack in a tribal region near the Afghan border. Militants attacked three military posts near Khar, the main town in the restive Bajaur tribal district but soldiers repulsed them with artillery and mortar fire. There were no reports of millitary casualties.<ref>http://afp.google.com/article/ALeqM5gqbV97erFMfXlFjngT7SYUGUxHdQ</ref><ref>http://www.dawn.com/2008/09/28/top4.htm</ref>
'''The paradoxes''': At the same time a number of disturbing paradoxes appeared in the literature, in particular the [[Burali-Forti paradox]] (1897), the [[Russell paradox]] (1902–03), and the [[Richard Paradox]] (Dixon 1906, cf Kleene 1952:36–40). The resultant considerations led to [[Kurt Gödel]]’s paper (1931) — he specifically cites the paradox of the liar — that completely reduces rules of [[recursion]] to numbers.


On 3 October 2008, Pakistani troops, backed by artillery, killed 25 Taliban militants after militants attacked security checkposts in four villages in the restive region. Separately, the local government ordered Afghan refugees in Bajaur to leave the area within three days. In other violence, a suicide bomber blew himself up as he tried to storm the house of [[Asfandyar Wali Khan]], head of the [[Awami National Party]] and who is also a member of Pakistan's ruling coalition in the western town of Charsadda, killing four people but missing the politician.<ref>http://www.gulf-daily-news.com/Story.asp?Article=230579&Sn=WORL&IssueID=31197</ref><ref>http://afp.google.com/article/ALeqM5iQP3xOQVBIz1uiIMsTcCOVVAMNiA</ref>
'''Effective calculability''': In an effort to solve the [[Entscheidungsproblem]] defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" (i.e., a calculation that would succeed). In rapid succession the following appeared: [[Alonzo Church]], [[Stephen Kleene]] and [[J.B. Rosser]]'s [[λ-calculus]], (cf footnote in [[Alonzo Church]] 1936a:90, 1936b:110) a finely-honed definition of "general recursion" from the work of Gödel acting on suggestions of [[Jacques Herbrand]] (cf Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene (1935-6:237ff, 1943:255ff). Church's proof (1936:88ff) that the [[Entscheidungsproblem]] was unsolvable, [[Emil Post]]'s definition of effective calculability as a worker mindlessly following a list of instructions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yes-no decision about the next instruction (cf "Formulation I", Post 1936:289-290). [[Alan Turing]]'s proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"(Turing 1936-7:116ff) -- in effect almost identical to Post's "formulation", [[J. Barkley Rosser]]'s definition of "effective method" in terms of "a machine" (Rosser 1939:226). [[S. C. Kleene]]'s proposal of a precursor to "[[Church thesis]]" that he called "Thesis I" (Kleene 1943:273–274), and a few years later Kleene's renaming his Thesis "Church's Thesis" (Kleene 1952:300, 317) and proposing "Turing's Thesis" (Kleene 1952:376).


====Intensified US. strikes====
=== Emil Post (1936) and Alan Turing (1936-7, 1939)===
Since the end of August, the United States has stepped up its attacks in the Federally Administered Tribal Areas, following the change of the country's President.<ref>[http://www.voanews.com/english/2008-09-08-voa15.cfm Suspected US Missile Strike Hits Taliban Commander's House] - Voice of America, September 8, 2008 </ref> On September 3 a [[Angoor Ada raid|commando attack]] took place in a village near the Afghan border in [[South Waziristan]], and there have been 3 strikes from unmanned drones in [[North Waziristan]], culminating on the morning of 8 September 2008, when a [[United States Air Force]] [[Unmanned aerial vehicle|drone aircraft]] fired a number of [[missiles]] at a [[madrassa]] (religious school). The [[Daande Darpkhel airstrike|airstrike]], which unsuccessfully targeted [[Jalaluddin Haqqani]], killed 23 people.<ref name=IHT20080909>Perlez, J. & Shah, P.Z. 2008, [http://www.iht.com/articles/2008/09/09/asia/09pstan.php 'US attack on Taliban kills 23 in Pakistan'], ''International Herald Tribune'', 9 September. Retrieved on 10 September 2008.</ref>


On 25 September 2008, following exchanges of gunfire between US and Pakistani forces on the frontier on Thursday, President Zardari told the United Nations that Pakistan would not tolerate violations of its sovereignty, even by its allies. The incident happened after two US military helicopters came under fire from the Pakistani side, a US military spokesman said, insisting that they had been about a mile and a half inside Afghanistan.
Here is a remarkable coincidence of two men not knowing each other but describing a process of men-as-computers working on computations — and they yield virtually identical definitions.


President Zardari told the United Nations, "Just as we will not let Pakistani's territory to be used by terrorists for attacks against our people and our neighbours, we cannot allow our territory and our sovereignty to be violated by our friends," he said, without citing the United States or the border flareup.<ref>http://afp.google.com/article/ALeqM5iGTgxqoTldD4lBtlbyfU5MSNowjg</ref>
[[Emil Post]] (1936) described the actions of a "computer" (human being) as follows:
:"...two concepts are involved: that of a ''symbol space'' in which the work leading from problem to answer is to be carried out, and a fixed unalterable ''set of directions''.


==Casualties==
His symbol space would be
{|class="wikitable" align="{{{align|right}}}" style="margin:{{#ifeq:{{{align}}}|right|0 0 0.5em 1em|0 1em 0.5em 0}};"
:"a two way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.
|+'''Casualties of the War in Pakistan''' <small>(According to news reports)</small>
|-
! Period
! Pakistani forces
! Militants
! Civilians
|-
! December 2001–March 2004<br>(prelude)
| 29 killed
| Unknown
| 120+
|-
! March 2004–September 2006
| 700 killed,<br>11 captured (all released)
| 1,000
| 1,000
|-
! October 2006–April 2007
| 78 killed
| 100-300
| N/A
|-
! July 2007–June 2008
| 882 killed,<br>40 missing,<br>559 captured (439 released)
| 1,928
| 1,827
|-
! June 2008–present
| 296 killed,<br>200 captured (116 released)
| 1,327
| 462
|-
! colspan="4" | <small>Sources:{{Fact|date=July 2008}} </small>
|}


There has been no conclusive reports on the casualties of the war, though some authors, especially Pakistani writers, have estimated that the total casualties on both sides to be more than 1,000 by 2006.<ref>[http://pakobserver.net/200606/26/Articles02.asp?txt=Pacifying%20Waziristan Pacifying Waziristan by Muhammad Munir]</ref>
:"One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e., INPUT] being marked with a stroke. Likewise the answer [i.e., OUTPUT] is to be given in symbolic form by such a configuration of marked boxes....


On [[September 13]] [[2007]], then-U.S. Deputy Secretary of State [[John Negroponte]] noted during a visit to Islamabad that Pakistan lost 1,000 soldiers fighting the militants since joining the war on terrorism.<ref name=autogenerated5>http://uasos.com/world/fierce-fighting-in-waziristan-as-negroponte-visits-pakistan-reuters.html</ref>
:"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process will terminate only when it comes to the direction of type (C ) [i.e., STOP]." (U p. 289–290) See more at [[Post-Turing machine]]


On [[November 14]], [[2007]], senior Pakistan Army officials told at a news conference that a total of 28 suicide attacks killed some 600 Pakistani security men, in addition to 1,300 civilians in the period after the Lal Masjid siege. It also said that from 2001 till November 14, 2007, at least 966 military men were killed and 2,259 others were injured; 488 foreign extremists were killed, 24 others were arrested and 324 foreign extremists were injured<ref name=autogenerated2>[http://www.topnews.in/600-pakistan-security-men-killed-28-suicide-attacks-after-lal-masjid-operation-26116 600 Pakistan security men killed in 28 suicide attacks after Lal Masjid operation | Top News<!-- Bot generated title -->]</ref>
[[Alan Turing]]’s work (1936, 1939:160) preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. Turing’s biographer believed that Turing’s use of a typewriter-like model derived from a youthful interest: “Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter; and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'" (Hodges, p. 96). Given the prevalence of Morse code and telegraphy, ticker tape machines, and Teletypes we might conjecture that all were influences.


Some have speculated that the unofficial number of Pakistani soldiers [[killed in action]] to be somewhere around 3,000 by the late 2006.<ref>[http://billroggio.com/archives/2006/09/the_fall_of_wazirist.php The Fall of Waziristan: An Online History]</ref><ref>[http://www.cbn.com/CBNnews/56510.aspx Waziristan: Terror Haven for Jihadists? by Erick Stakelbeck] November 15, 2006 [[Christian Broadcasting Network|CBN news]]</ref> A Pakistan writer, [[Ayaz Amir]] states that the army's "Casualties were high, perhaps unsustainable, although we’ll never know the exact figures, the Pakistan army not given to embarrassing disclosures."<ref>[http://www.dawn.com/weekly/ayaz/20060915.htm War and peace, army style] - September 15, 2006 [[Dawn (newspaper)]]</ref>. It should be noted however that Pakistani military provides substantial benefits to families of fallen soldiers<ref>[http://www.strategypage.com/htmw/htlead/articles/20080422.aspx Year of the Soldier; Strategy Page]</ref>, and thus it is unlikely that higher end numbers are correct since it would be difficult to hide fatalities from families. Also Ayaz Amirs and Bill Roggio's claims are spurious to say the least; their assertion that Pakistan Army hides casualties is belied by the fact that the PR arm of the Pakistani military gives a daily update on the operations, complete with casualty figures; broken down by units.<ref>[http://www.ispr.gov.pk/Archive&Press/LatestUpdate/latestUpdate.htm Latest Update from Pakistan Military Webpage, contradicting the claims made by Ayaz Amir and Bill Roggio]</ref>
Turing — his model of computation is now called a [[Turing machine]] — begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers (Turing 1936-7:116).


==United States role==
:"Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book....I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite....
Pakistan received about $4 billion from the [[United States]] for the [[logistic]]al support it provided for the [[counter-terrorism]] operations from 2002 to 2006, and for its own military operation mainly in Waziristan and other tribal areas along the [[Durand line]], according to a report of the [[Asian Development Bank]]. The [[George W. Bush administration|Bush administration]] also offered a $3 billion five-year aid package to Pakistan for becoming a frontline ally in its 'war on terror'. Annual instalments of $600 million each split evenly between military and economic aid, began in 2005.<ref name=1billion>{{Cite web| url=http://www.adnki.com/index_2Level_English.php?cat=Security&loid=8.0.359974956&par=0 | title=Pakistan: $1 billion from U.S. to fight terror | publisher=Aki/[[Dawn (newspaper)|Dawn]] |date=2006-11-14 | accessdate=2006-11-24}}</ref>


In his autobiography, President Musharraf wrote that the United States had paid millions of dollars to the Pakistan government as [[bounty]] money for capturing al-Qaeda operators from tribal areas bordering Afghanistan. About 359 of them were handed over to the US for prosecution. <ref name=1billion />
:"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite...

:"Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided" (Turing 1936-7:136).

Turing's reduction yields the following:

:"The simple operations must therefore include:
::"(a) Changes of the symbol on one of the observed squares
::"(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares.
"It may be that some of these change necessarily invoke a change of state of mind. The most general single operation must therefore be taken to be one of the following:
::"(A) A possible change (a) of symbol together with a possible change of state of mind.
::"(B) A possible change (b) of observed squares, together with a possible change of state of mind"

:"We may now construct a machine to do the work of this computer." (Turing 1936-7:136)

A few years later, Turing expanded his analysis (thesis, definition) with this forceful expression of it:
:"A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Although it is fairly easy to get an intuitive grasp of this idea, it is neverthessless desirable to have some more definite, mathematical expressible definition . . . [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing and Post] . . . We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability † with effective calculability . . . .
::"† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculabile" refer to the intuitive idea without particular identification with any one of these definitions."(Turing 1939:160)

=== J. B. Rosser (1939) and S. C. Kleene (1943) ===

'''[[J. Barkley Rosser]]''' boldly defined an ‘effective [mathematical] method’ in the following manner (boldface added):
:"'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps. With this special meaning, three different precise definitions have been given to date. [his footnote #5; see discussion immediately below]. The simplest of these to state (due to Post and Turing) says essentially that '''an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer'''. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–6)

Rosser's footnote #5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular Church's use of it in his ''An Unsolvable Problem of Elementary Number Theory'' (1936); (2) Herbrand and Gödel and their use of recursion in particular Gödel's use in his famous paper ''On Formally Undecidable Propositions of Principia Mathematica and Related Systems I'' (1931); and (3) Post (1936) and Turing (1936-7) in their mechanism-models of computation.

'''[[Stephen C. Kleene]]''' defined as his now-famous "Thesis I" known as the [[Church-Turing thesis]]. But he did this in the following context (boldface in original):
:"12. '''Algorithmic theories'''... In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independent variables, which procedure necessarily terminates and in such manner that from the outcome we can read a definite answer, "yes" or "no," to the question, "is the predicate value true?”" (Kleene 1943:273)

=== History after 1950 ===
A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, [[foundations of mathematics]] (especially the [[Church-Turing Thesis]]) and [[philosophy of mind]] (especially arguments around [[artificial intelligence]]). For more, see [[Algorithm characterizations]].


==See also==
==See also==
*[[Sectarian violence in Pakistan]]
{{wiktionarypar|algorithm}}
*[[Islamic Emirate of Waziristan]]
{{wikibooks|Algorithms}}
*[[Taliban insurgency]]
{{WVD}}
*[[War on Terrorism]]
<div style="-moz-column-count:2; column-count:2;">
*[[List of wars 2003–current]]
* [[Abstract machine]]
* [[Algorithm characterizations]]
* [[Algorithm design]]
* [[Algorithmic efficiency]] (describes ways of estimating, measuring and improving an algorithms speed)
* [[Algorithm engineering]]
* [[Algorithm examples]]
* [[Algorithmic music]]
* [[Algorithmic trading]]
* [[Computability theory (computer science)]]
* [[Computational complexity theory]]
* [[Data structure]]
* [[Heuristics]]
* ''[[Introduction to Algorithms]]''
* [[List of important publications in computer science#Algorithms|Important algorithm-related publications]]
* [[List of algorithms]]
* [[List of algorithm general topics]]
* [[List of terms relating to algorithms and data structures]]
* [[Partial function]]
* [[Parameterized Complexity]]
* [[Performance analysis]] measuring the actual performance of an algorithm
* [[Run-time analysis]] (non-intuitive) estimation of run times, not analysis at run-time! (see [[Performance analysis]] above
* [[Theory of computation]]
</div>


==References==
==References==
<!--See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how to generate footnotes using the <ref(erences/)> tags-->
{{reflist|2}}

==External links==
*[http://news.bbc.co.uk/2/hi/south_asia/3645114.stm Pakistan's undeclared war] by Zaffar Abbas, [[BBC News]], Friday, 10 September, 2004
*[http://news.bbc.co.uk/2/hi/south_asia/4774762.stm Dozens killed in Pakistan clashes], [[BBC News]], Friday, 5 March, 2006
*[http://www.historyguy.com/waziristan_war.html A brief description of the Waziristan War which began in 2004 in Northwest Pakistan.] From the History Guy Website
*[http://www.msnbc.msn.com/id/23012160/ U.S. Al-Qaida shifting to Pakistan]
*[http://news.bbc.co.uk/2/hi/south_asia/7228864.stm Changing ways of Pakistan's militants]
*[http://www.washingtonpost.com/wp-dyn/content/article/2008/02/05/AR2008020502979.html Pakistan Is Threatened]


{{War on Terrorism}}
* Axt, P. (1959) On a Subrecursive Hierarchy and Primitive Recursive Degrees, ''Transactions of the American Mathematical Society'' 92, pp. 85-105
*{{citation|author1-link=Andreas Blass|first1=Andreas|last1=Blass|author2-link=Yuri Gurevich|first2=Yuri|last2=Gurevich|year=2003|url=http://research.microsoft.com/~gurevich/Opera/164.pdf|title=Algorithms: A Quest for Absolute Definitions|journal= Bulletin of European Association for Theoretical Computer Science|volume= 81}}. Includes an excellent bibliography of 56 references.
*{{citation|last1=Boolos|first1= George|last2=Jeffrey|first2= Richard|author1-link=George Boolos|author2-link=Richard Jeffrey|title=Computability and Logic|edition=4th|publisher=Cambridge University Press, London|year=1974, 1999|date=1974, 1980, 1989, 1999|isbn=0-521-20402-X}}: cf. Chapter 3 ''Turing machines'' where they discuss "certain enumerable sets not effectively (mechanically) enumerable".
* Burgin, M. ''Super-recursive algorithms'', Monographs in computer science, Springer, 2005. ISBN 0387955690
* Campagnolo, M.L., Moore, C., and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In ''Proc. of the 4th Conference on Real Numbers and Computers'', Odense University, pp. 91-109
*{{cite journal|last=Church|first=Alonzo|authorlink=Alonzo Church|title=An Unsolvable Problem of Elementary Number Theory|journal=The American Journal of Mathematics|volume=58|pages= 345–363|year=1936a|doi=10.2307/2371045}} Reprinted in ''The Undecidable'', p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (''The Undecidable'') where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc.
*{{cite journal|last=Church|first=Alonzo|authorlink=Alonzo Church|title=A Note on the Entscheidungsproblem|journal=Journal of Symbolic Logic|volume=1 no. 1 and volume 1 no. 3|year=1936b}} Reprinted in ''The Undecidable'', p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.
*{{cite book|last=Daffa'|first=Ali Abdullah al-|title=The Muslim contribution to mathematics|year=1977|publisher=Croom Helm|location=London|id=ISBN 0-85664-464-1}}
*{{cite book|last=Davis|first=Martin|authorlink=Martin Davis|title=The Undecidable: Basic Papers On Undecidable Propostions, Unsolvable Problems and Computable Functions|publisher=Raven Press|location=New York|year=1965}} Davis gives commentary before each article. Papers of [[Gödel]], [[Alonzo Church]], [[Alan Turing|Turing]], [[J. Barkley Rosser|Rosser]], [[Kleene]], and [[Emil Post]] are included; those cited in the article are listed here by author's name.
*{{cite book|last=Davis|first=Martin|authorlink=Martin Davis|title=Engines of Logic: Mathematicians and the Origin of the Computer|publisher=W. W. Nortion|location=New York|year=2000}} Davis offers concise biographies of [[Leibniz]], [[Boole]], [[Frege]], [[Georg Cantor|Cantor]], [[Hilbert]], [[Gödel]] and [[Alan Turing|Turing]] with [[von Neumann]] as the show-stealing villain. Very brief bios of [[Joseph-Marie Jacquard]], [[Babbage]], [[Ada Lovelace]], [[Claude Shannon]], [[Howard Aiken]], etc.
* {{DADS|algorithm|algorithm}}
*{{cite book|last=Dennett|first=Daniel|authorlink=Daniel Dennett|title=Darwin's Dangerous Idea|publisher=Touchstone/Simon & Schuster|location=New York|year=1995}}
*[[Yuri Gurevich]], [http://research.microsoft.com/~gurevich/Opera/141.pdf ''Sequential Abstract State Machines Capture Sequential Algorithms''], ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pages 77–111. Includes bibliography of 33 sources.
*{{cite journal|last=Kleene C.|first=Stephen|authorlink=Stephen Kleene |title=General Recursive Functions of Natural Numbers|journal=Mathematische Annalen|volume=Band 112, Heft 5|pages=727–742|year=1936
| doi = 10.1007/BF01565439 <!--Retrieved from Yahoo! by DOI bot-->}} Presented to the American Mathematical Society, September 1935. Reprinted in ''The Undecidable'', p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper ''An Unsolvable Problem of Elementary Number Theory'' that proved the "decision problem" to be "undecidable" (i.e., a negative result).
*{{cite journal|last=Kleene C.|first=Stephen|authorlink=Stephen Kleene |title= Recursive Predicates and Quantifiers|journal=American Mathematical Society Transactions|volume= Volume 54, No. 1 |pages=41–73|year=1943 |doi= 10.2307/1990131}} Reprinted in ''The Undecidable'', p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the [[Church Thesis]]).
*{{cite book|last=Kleene|first=Stephen C.|authorlink=Kleene|title=Introduction to Metamathematics|edition=Tenth Edition 1991|publisher=North-Holland Publishing Company|year=First Edition 1952}} Excellent — accessible, readable — reference source for mathematical "foundations".
*{{cite book|last=Knuth|first=Donald|authorlink=Donald Knuth|title=Fundamental Algorithms, Third Edition|publisher=Addison-Wesley|location=Reading, Massachusetts|year=1997|ISBN 0-201-89683-4}}
* Kosovsky, N. K. ''Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms'', LSU Publ., Leningrad, 1981
*{{cite journal|last=Kowalski|first=Robert|authorlink=Robert Kowalski|title=Algorithm=Logic+Control|journal=Communications of the ACM|volume=22|issue=7|pages=424–436|publisher= ACM Press|year=1979|id=ISSN 0001-0782|doi=10.1145/359131.359136}}
*[[A. A. Markov]] (1954) ''Theory of algorithms''. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]
*{{cite book|last=Minsky|first=Marvin|authorlink=Marvin Minsky|title=Computation: Finite and Infinite Machines|edition=First|publisher=Prentice-Hall, Englewood Cliffs, NJ|year=1967}} Minsky expands his "...idea of an algorithm — an effective procedure..." in chapter 5.1 ''Computability, Effective Procedues and Algorithms. Infinite machines."
*{{cite journal|last=Post|first=Emil|authorlink=Emil Post|title=Finite Combinatory Processes, Formulation I|journal=The Journal of Symbolic Logic|volume=1|year=1936|pages=pp.103–105|doi=10.2307/2269031}} Reprinted in ''The Undecidable'', p. 289ff. Post defines a simple algorithmic-like process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the so-called [[Church-Turing thesis]].
*{{cite journal|last=Rosser|first=J.B.|authorlink=J.B. Rosser|title=An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem|journal=Journal of Symbolic Logic|volume= 4 |year=1939}} Reprinted in ''The Undecidable'', p. 223ff. Herein is Rosser's famous definition of "effective method": "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer" (p. 225–226, ''The Undecidable'')
*{{cite book|last=Sipser|first=Michael|title=Introduction to the Theory of Computation|publisher=PWS Publishing Company|year=2006}}
*{{cite book|last=Stone|first=Harold S.|title=Introduction to Computer Organization and Data Structures|edition=1972|publisher=McGraw-Hill, New York}} Cf in particular the first chapter titled: ''Algorithms, Turing Machines, and Programs''. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an ''algorithm''" (p. 4).
*{{cite journal|last=Turing|first=Alan M.|authorlink=A. M. Turing|title=On Computable Numbers, With An Application to the Entscheidungsproblem|journal=Proceedings of the London Mathematical Society|volume= series 2, volume 42|pages= 230–265 |year=1936-7|doi=10.1112/plms/s2-42.1.230 }}. Corrections, ibid, vol. 43(1937) pp.544-546. Reprinted in ''The Undecidable'', p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.
*{{cite journal|last=Turing|first=Alan M.|authorlink=A. M. Turing|title=Systems of Logic Based on Ordinals|journal=Proceedings of the London Mathematical Society|volume= series 2, volume 45|pages=161–228|year=1939|doi=10.1112/plms/s2-45.1.161}} Reprinted in ''The Undecidable'', p. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton USA.
*[[United States Patent and Trademark Office]] (2006), [http://www.uspto.gov/web/offices/pac/mpep/documents/2100_2106_02.htm ''2106.02 **>Mathematical Algorithms< - 2100 Patentability''], Manual of Patent Examining Procedure (MPEP). Latest revision August 2006
=== Secondary references ===
*{{cite book|last=Bolter|first=David J.|authorlink=Bolter|title=Turing's Man: Western Culture in the Computer Age|edition=(1984)|publisher=The University of North Carolina Press, Chapel Hill NC}}, ISBN 0-8078-4108-0 pbk.
*{{cite book|last=Dilson|first=Jesse|authorlink=Dilson|title=The Abacus|edition=(1968,1994)|publisher=St. Martin's Press, NY}}, ISBN 0-312-10409-X (pbk.)
*{{cite book|last=van Heijenoort|first=Jean|authorlink=van Heijenoort|title=From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931|edition=(1967)|publisher=Harvard University Press, Cambridge, MA}}, 3rd edition 1976[?], ISBN 0-674-32449-8 (pbk.)
*{{cite book|last=Hodges|first=Andrew|authorlink=Hodges|title=Alan Turing: The Enigma|edition=(1983)|publisher=Simon and Schuster, New York}}, ISBN 0-671-49207-1. Cf Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.


{{Military of Pakistan}}
== External links ==
* {{MathWorld | urlname=Algorithm | title=Algorithm}}
* [http://everydaymath.uchicago.edu/educators/Algorithms_final.pdf Algorithms in Everyday Mathematics]
* {{dmoz|Computers/Algorithms/|Algorithms}}
* [http://sortieralgorithmen.de/ Sortier- und Suchalgorithmen (German)]


{{DEFAULTSORT:Waziristan}}
[[Category:Algorithms|*]]
[[Category:Arabic words and phrases]]
[[Category:Al-Qaeda]]
[[Category:Discrete mathematics]]
[[Category:Al-Qaeda activities]]
[[Category:Civil wars involving the states and peoples of Asia]]
[[Category:Mathematical logic]]
[[Category:Theoretical computer science]]
[[Category:Religion-based civil wars]]
[[Category:Articles with example pseudocode]]
[[Category:Guerrilla wars]]
[[Category:Taliban]]
[[Category:Waziristan]]
[[Category:Wars involving Pakistan]]
[[Category:Wars involving the Taliban]]
[[Category:War on Terror]]


[[cy:Rhyfel Wasiristan]]
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Revision as of 12:47, 10 October 2008

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War in North-West Pakistan
Part of the War on Terrorism

District map of NWFP and FATA.
DateMarch 2004–ongoing
Location
Federally Administered Tribal Areas and North-West Frontier Province in Pakistan
Result ongoing
Belligerents
 Pakistan,
 United States		 Afghanistan Tehrik-i-Taliban Pakistan,
Tehreek-e-Nafaz-e-Shariat-e-Mohammadi,
Lashkar-e-Islam,
al-Qaeda,
Islamic Movement of Uzbekistan (until 2007)
Commanders and leaders
Pakistan Asif Ali Zardari,
Pakistan General Ashfaq Kayani,
Pakistan Lt Gen Masood Aslam,
Pakistan Maj Gen Javed Iqbal,
Pakistan Maj Gen Tariq Khan,
Pakistan Maj Gen Nasser Janjua,
Pakistan Maj Gen Khalid Rabbani
Former commanders
Pakistan Gen Pervez Musharraf,
Pakistan Lt Gen Safdar Hussain,
Pakistan Maj Gen Alam Khattak,
Pakistan Maj Gen Ahmad S Pasha		 Afghanistan Baitullah Mehsud,
Afghanistan Hafiz Gul Bahadur,
Afghanistan Faqir Mohammed,
Maulana Fazlullah,
Mangal Bagh
Fugitives
Osama bin Laden,
Ayman al Zawahiri,
Afghanistan Mullah Mohammed Omar,
Afghanistan Jalaluddin Haqqani,
Tohir Yo‘ldosh
Strength
80,000 Pakistani troops[1],
~80,000 Frontier Corps		 Unknown
Casualties and losses
1,370 soldiers and policemen killed (official)[2]

Other estimates:
1,815 soldiers and policemen killed
112 tribesmen killed[3],
2,259 soldiers wounded[3],
810 soldiers and policemen missing or captured[4]		 4,594+ militants killed[5][6]
(488 foreign fighters [7])
5,320 civilians killed,
25 foreigners killed
(11 Frenchmen, 9 Americans, 4 Chinese, 1 Czech, 1 Dane, 1 Turk and 1 Vietnamese)[8][9][10]

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Template:FixBunching Template:Campaignbox War on Terrorism Template:FixBunching

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The War in North-West Pakistan is an armed conflict between the Pakistani Army and Islamist militants made up by local tribesmen, the Taliban and foreign extremists.[11][12][13] It began in 2004 when tensions rooted in the Pakistani Army's search for al-Qaeda members in Pakistan's mountainous Waziristan area (in the Federally Administered Tribal Areas) escalated into armed resistance by local tribesmen.

Clashes erupted between the Pakistani troops and al-Qaeda's and other militants joined by local rebels and pro-Taliban forces. The Pakistani actions were presented as a part of the War on Terrorism, and had connections to the war and Taliban insurgency in Afghanistan.[14][15]

After a truce in September 2006, clashes escalated again in 2007, with local tribesmen ousting militant Uzbeks and attacking the Pakistani security forces again in July 2007.

Since the conflict began, Pakistan has lost more than twice the number of its soldiers compared to the number of US troops killed in Afghanistan. However, as of September 2008, the number of militants killed by the Pakistan Army reached the 4,500 mark.[16]

Background

July 2002: Initial troop movements

In July 2002, Pakistani troops entered the Tirah Valley in the Khyber Agency for the first time since Pakistan independence in 1947. They proceeded to move into the Shawal Valley of North Waziristan, and later South Waziristan. This was made possible after long negotiations with various tribes, who reluctantly agreed to allow the military's presence on the assurance that it would bring in funds and development work.

However, once the military action started in South Waziristan a number of Waziri sub-tribes viewed the action as an attempt to subjugate them. As attempts to persuade them to hand over the foreign militants failed, and missteps by the authorities increased feelings of ill-will, the security campaign against suspected al-Qaeda militants turned into an undeclared war in 2004 between the Pakistani military and the rebel tribesmen.

December 2003: Waziri attempts on President Musharraf's life

In December 2003, two assassination attempts against President Pervez Musharraf were traced to Waziristan. The government responded by intensifying military pressure on the area, however the fighting was costly and government forces would sustain heavy casualties throughout 2004 and into early 2005 when the government switched to a tactic of negotiation instead of direct conflict.[17]

Timeline

March 2004–September 2006 phase

2004: Fighting breaks out

Main article: Battle of Wana

In March 2004, heavy fighting broke out at Azam Warsak, near the South Waziristan town of Wana. Pakistani troops faced an estimated 400 militants holed up in several fortified settlements. It was speculated at the time that Osama bin Laden's deputy Ayman al-Zawahiri was among those trapped by the Pakistani Army, but he either escaped or was never among these fighters.

Peace deals with Taliban

In April 2004 the Government of Pakistan signed the first of three peace agreements with militants in South Waziristan. It was signed with Taliban commander Nek Muhammad Wazir, but was immediately abrogated once Nek Muhammad was killed by American Hellfire missile in June 2004. The second was signed in February 2005 with Nek's successor Baitullah Mehsud, which brought relative calm in the South Waziristan region. This deal will be later mimicked in the neighboring North Waziristan territory in September 2006 as the third and final truce between the government and the militants. However, all of these truces would not have a substantial effect in reducing bloodshed.[17] The later two deals were officially broken in August 2007 after the Lal Masjid episode raising the suicide attacks on forces and civilians ten-fold throughout the country.

2005

On May 4, 2005, Pakistani commandos captured Abu Faraj al-Libbi after a raid outside the town of Mardan, 30 miles north of Peshwar. Abu Farraj al-Libbi was a high ranking al-Qaeda official, rumored to be third after Osama bin Laden and Ayman al-Zawahiri. Al-Libbi replaced Khalid Shaikh Mohammed after his arrest in March 2003 in connection with the September 11th attacks. The Pakistani government arrested al-Libbi and held him on charges in relation to being a chief planner in two assassination attempts on the life of President Pervez Musharraf in December 2003.[18]

2006

On January 13, 2006, the U.S. launched an airstrike on the village of Damadola. The attack occurred in the Bajaur tribal area, about 7 km (4.5 miles) from the Afghan border, and killed at least 18 people. The attack again targeted Ayman al-Zawahiri, but later evidence suggests he was not there. DNA analysis showed that at least five high-ranking al-Qaeda foreigners were killed, including Midhat Mursi, a bomb building expert, for whom a $5 million award was offered.[citation needed]

On March 4, 2006, Pakistani forces started a massive assault on pro-Taliban elements in the region. Pakistani officials said 46 militants and five soldiers died after fighting erupted, although some reports put the death toll at over 70.[citation needed]

On June 21, 2006, pro-Taliban militants in the Bannu region of North Waziristan stated they shot down a military helicopter that was reported to have crashed. The government denied missile fire as the cause, stating it was due to technical faults. On the same day militants killed an inspector and two constables on a road connecting Bannu and the main town of Miranshah; the police officers were reportedly ambushed and killed while praying alongside their vehicle.[19]

June 2006: Ceasefire

Also on June 21, 2006 the military head of the Taliban in Waziristan, Sirajuddin Haqqani, issued a decree that it was no longer Taliban policy to fight the Pakistan Army. This marked the end of significant fighting in South Waziristan, however the Taliban intentionally did not circulate the decree in North Waziristan thereby keeping pressure on the Government as the terms for a comprehensive accord were worked out.[20]

On June 26, 2006, a suicide car bomber killed nine Pakistani soldiers. Officials say that the explosives-laden vehicle detonated about six kilometres (four miles) east of Miranshah, the main town in the North Waziristan region.[21]

September 2006: Waziristan peace accord signed

On September 5 2006, the Waziristan Accord, an agreement between tribal leaders and the Pakistani government was signed in Miranshah, North Waziristan.[22] to end all fighting. The agreement includes the following provisions:[23]

  • The Pakistani Army will help reconstruct infrastructure in tribal areas of North and South Waziristan.
  • The Pakistani Military will not tolerate any assistance to intruders in North Waziristan, and will monitor actions in the region.
  • The Pakistan government is to compensate tribal leaders for the loss of life and property of innocent tribesmen.
  • “Foreigners” (informally understood to be foreign jihadists) are not allowed to use Pakistani territory for any terrorist activity anywhere in the world.
  • 2,500 foreigners who were originally held on suspicion of having links to al-Qaeda or the Taliban[24] were to be detained for necessary action against them.

The agreement, dubbed the Waziristan accord, has been viewed by some political commentators as a success for Pakistan.[25] Further details of the agreement, as well as comments on the agreement made by US, Pakistani, and Taliban spokesmen is available in the Waziristan accord article.

October 2006: The madrassa air strike

On October 30, 2006, the Pakistani army conducted an air strike, with possible U.S. involvement, against a madrassa in the Bajaur region bordering Afghanistan. The madrassa was destroyed killing 70 to 80 people.

In retaliation for the attack the militants conducted a suicide bombing on an army camp on November 8, 2006, killing 42 Pakistani soldiers and wounding 20.

March–April 2007 phase

In March, Pakistan signed a peace treaty with Faqir Mohammed, the Taliban leader in Bajaur.[26][27] The Taliban now held three districts in the Federally Administered Tribal Areas: South Waziristan, North Waziristan and Bajaur Agency.[28]

Waziri-Uzbek tensions

Reportedly, the fighting sparked by the killing of Saiful Adil, an al-Qaeda-linked Arab, blamed on the Uzbeks by Maulavi Nazir, described as a top pro-Taliban militant commander in the region. According to the other version, fighting started after Mullah Nazir, whom the government says has come over to its side, ordered the Uzbek followers of Tohir Yo‘ldosh, formerly a close confidant of Osama bin Laden, to disarm. It was also preceded by the clashes between the IMU and a pro-government tribal leader in Azam Warsak, in which 17 to 19 people died before a ceasefire was announced.[29]

Defeat of the Islamic Movement of Uzbekistan

Sirajuddin Haqqani, the son of Taliban leader Jalaluddin Haqqani, tried to stop the fighting but failed. Local Taliban militants allied to the tribesmen were reported attacking and seizing the IMU's private jail in Azam Warsak. The Pakistan Army said did not intend to step in, but witnesses say government artillery fired on the Uzbek bunkers they set up to fight the tribesmen.

Heavy fighting resumed on March 29, ending a week-long ceasefire between tribal fighters and foreign militants. According to initial reports, tribesmen attacked a checkpoint manned by Uzbek militants and captured two of them. The clashes also left one tribal fighter dead and three wounded. The following day, a senior Pakistani official announced that 52 people were killed during the past two days, 45 of them Uzbeks and the rest tribesmen. One of Maulvi Nazir's aides put the death toll at 35 Uzbeks and 10 tribal fighters. However, residents in the area said that the death toll on both sides was inflated.

The conflict further escalated on April 2 when a council of elders declared jihad against foreign militants and started to raise an army of tribesmen. According to Pakistani security officials, heavy fighting concentrated in the village of Doza Ghundai left more than 60 people dead, including 50 foreigners, 10 tribal fighters and one Pakistani soldier. He also said that "dozens" of Uzbeks had surrendered to tribal forces and that many bunkers used by militants were seized or destroyed.

On April 12 2007 the army general in charge of South Waziristan said that tribal fighters had cleared the Uzbeks out of the valleys surrounding Wana and the foreign militants had been pushed back into the mountains on the Afghan border.[30] Four days later, the local tribesmen has urged Islamabad to resume control of law and order in the area.[31]

July 2007–February 2008 phase

Main article: 2007 timeline of the War in Waziristan

Lal Masjid siege

On July 3, 2007, the militant supporters of Lal Masjid and Pakistani security forces clashed in Islamabad after the students from the mosque attacked a nearby government ministry building. The Pakistani security forces immediately put up a siege around the mosque complex which lasted until July 11 and resulted in 108 deaths. This represented the main catalyst for the conflict and eventual breakdown of the truce that existed between Pakistan and the Taliban in the northwest. Already during the siege there were several attacks in Waziristan in retaliation for the siege.

Truce in Waziristan broken

As the siege in Islamabad ensued, several attacks on Pakistani troops in Waziristan were reported. On July 14, 2007, a suicide bomber attacked a Pakistani Army convoy killing 26 soldiers and wounding 54. On July 15, 2007, two suicide bombers attacked another Pakistani Army convoy killing 16 soldiers and 5 civilians and wounding another 47 people. And in a separate incident a fourth suicide bomber attacked a police headquarters killing 28 police officers and recruits and wounding 35 people.[32][33]

The assault on the Red Mosque prompted pro-Taliban rebels along the border with Afghanistan to scrap the controversial Waziristan Accord with the government. [34]

The new war in Waziristan

The Army moved large concentration of troops into Waziristan and engaged in fierce clashes with militants in which at least 100 militants were killed including wanted terrorist and former Guantanamo Bay detainee, Abdullah Mehsud.[35] The militants also struck back by attacking Army convoys, security check points and sending suicide bombers killing dozens of soldiers and police and over 100 civilians. In one month of fighting during the period from July 24 to August 24, 2007, 250 militants and 60 soldiers were killed.

On September 2, 2007, just a few dozen militants led by Baitullah Mehsud managed to ambush a 17-vehicle army convoy and captured an estimated 247 soldiers in it, without a shot being fired; an event that shocked the nation.[36] Several officers were among the captured.

After the army returned to Waziristan, they garrisoned the areas and set up check-points, but the militants hit hard. In mid-September Taliban forces attacked a number of Pakistani army outposts all across North and South Waziristan. This resulted in some of the heaviest fighting of the war. On September 12, 2007, the first outpost was attacked and overrun by the Taliban resulting in the capture of 12 Pakistani soldiers. The next day on September 13, 2007, a suicide bomber in Ghazi Tarbela attacked a Pakistani army base, destroying the main mess hall and killing 20 members of the Karar commando group; Pakistan's most elite army unit. Another 29 soldiers were wounded. A series of attacks ensued and by September 20, 2007 a total of five Pakistani Army military outposts had been overrun and more than 25 soldiers captured. More than 65 soldiers were either killed or captured and almost 100 wounded.

A little over two weeks later, the Army responded with helicopter gunships, jet fighters and ground troops. They hit militant positions near the town of Mir Ali. In heavy fighting over four days between October 7 and October 10 2007, 257 people were killed, including 175 militants, 47 soldiers and 35 civilians.

File:FATA (4).PNG
With the fall of the Swat Valley, the fighting spread to the whole FATA.

Battle of Swat Valley

By the end of October fighting erupted in the Swat district of the North-West Frontier Province, with a large Taliban force, under the command of Maulana Fazlullah, trying to impose Sharia law. Around 3,000 paramilitary soldiers were sent to confront them. After almost a week of heavy fighting the battle came to a standstill with both sides suffering heavy casualties. Then on November 1 and November 3, 220 paramilitary soldiers and policemen surrendered or deserted after a military position on a hill-top and two police stations were overrun. This left the Taliban in control of most of the Swat district.

The fighting in Swat is the first serious insurgent threat from pro-Taliban forces in what is known as a settled area of Pakistan. Forces loyal to Maulana Fazlullah, including some foreign fighters, after taking control of a series of small towns and villages, tried to implement strict Islamic law in November 2007. In mid-November the regular army was deployed with the help of helicopter gunships to crush the uprising. The Pakistan Army deployed over 2,500 men. By the beginning of December the fighting had ended and the Army recaptured Swat. Almost 400 pro-Taliban fighters were dead along with 15 Pakistani soldiers and 20 civilians in the military offensive.[37]

The Rawalpindi attacks

The city of Rawalpindi, which is the military headquarters of the Pakistan Armed Forces, was an attractive target for the militants and they were planning to hit the city.[citation needed] On September 3, two suicide bombers targeted a military intelligence (ISI) bus and a line of cars carrying ISI officers. The bus attack killed a large number of Defence Ministry workers and the other attack killed an Army colonel. In all 31 people, 19 soldiers and 12 civilians, were killed.

Over two months later on November 24, in a similar fashion a pair of suicide bombers struck. Again, one of the targets was a military intelligence bus. Almost everyone on the bus was killed. The other bomber blew up at a military checkpoint. 35 people were killed, almost all military.[citation needed]

State of emergency

The 2007 Pakistani state of emergency was declared by Pervez Musharraf on 2007-11-03 and lasted until 2007-12-15. During this time the constitution of the country was suspended.[38][39] This action and its responses are generally related to the controversies surrounding the re-election of Musharraf during the presidential election that had occurred on 2007-10-06, and also was claimed by the government to be the reaction to the actions by Islamic militants in Waziristan.[40]

Benazir Bhutto's assassination

Main article: Benazir Bhutto assassination

On 27 December 2007, Pakistani opposition leader Benazir Bhutto was killed upon leaving a political rally for the Pakistan Peoples Party (PPP) in Rawalpindi, Pakistan.[41][42] A suicidal assassin reportedly fired shots in Bhutto's direction just prior to detonating an explosive pellet-laden vest, killing approximately 24 people and wounding many more.[43][44][45] Musharraf and the army blamed the attack on Al-Qaida, but the following day a statement by Commander Baitullah Mehsud was sent to the media saying that he and Al-Qaida had no involvement in the murder of the former Prime Minister, he briefed that these were the crimes of Musharraf and the army. The killing was followed by a wave of violence across the country that left 58 people dead, including four police officers. Most of the violence was directed at Musharraf and the pro-Musharraf political party, Pakistan Muslim League (Q). The public chanted slogans against the army and Musharraf: "Musharraf Dog", "General is a murderer", "uniform (army) wearing murderers", etc. Bhutto had previously survived an assassination attempt made on her life during her homecoming which left 139 people dead and hundreds wounded.[46]

More fighting in South Waziristan

In January 2008, pro-Taliban militants overran Sararogha Fort, and may have overrun a fort in Ladah as well. Both forts are in South Waziristan, and were held by the Pakistani army.[47] Gunmen fired on a children's aid group in in the North West Frontier Province February 25, 2008, leaving four staffers dead. On the same day as the children's aid group attack, another suicide bomber struck in the garrison-town of Rawalpindi killing Pakistani Lt. Gen. Mushtaq Baig along with two more soldiers and five civilians. Baig was the army's surgeon general and the highest-level military official to be assassinated since Pakistan joined the U.S.-led war on terror.[48]

At least 45 people died and 82 were wounded in a suicide attack on the funeral February 29 2008 of a district superintendent of police – killed earlier in the day in a separate attack – in northwest Pakistan's Swat province.[49]

Peace agreement

On 7 February, 2008, a leading militant group led by Baitullah Mehsud offered a truce and peace negotiations resulting in a suspension of violence.[50]

On May 21, 2008 Pakistan signed a peace agreement with Taliban fighters. Pakistan's government promised to "gradually" pull out troops from the northwestern valley of Swat. In return the Taliban were due to shut down training camps, hand over foreign fighters and halt suicide attacks on government installations and security forces under the 15-point pact.[51]

June 2008 - Present

Main article: 2007 timeline of the War in Waziristan

Despite the agreement sporadic fighting continued until late June and escalated with the takeover of the town of Jandola on June 24, by the militants. 22 pro-government tribal fighters were captured and executed by the Taliban at that time.[52]

There had been growing concern about threats to Peshawar from Taliban fighters. In early June, a Taliban force from Khyber entered the city and seized 16 Christians, before later releasing them. Mahmood Shah, a former security chief in the tribal regions, said: "The situation is such that [the Taliban] are all around Peshawar. They are on our doorstep."[53]

On June 28, 2008, Pakistan's Army started an offensive against Taliban fighters in Khyber. The military took control of a key town and demolished an insurgent group's building. 1 militant was reportedly killed while 2 soldiers died in Swat valley.[54] The operation was halted in early July.

On July 6, 2008, a suicide bomber attacked a police station in Islamabad killing 12 policemen and seven civilians. Another 53 people were wounded. The police were part of a protection cordon set up by the government for a rally which was attended by about 12,000 people who were marking the first anniversary of government forces storming Islamabad's Red Mosque, or Lal Masjid.[55]

On July 12, 2008, militants ambushed a military convoy killing 17 soldiers.[citation needed]

On July 19, 2008, clashes erupted between the Taliban and a rival faction of alleged pro-government Taliban militants. 10-15 of the pro-government fighters were killed and another 120 were captured. Among the captured were two commanders who were tried under "Islamic" law by the Taliban and then executed.

On July 21, 2008, heavy fighting in Baluchistan killed 32 militants, 9 soldiers and 2 civilians. More than two dozen militants were captured and a large weapons cache was found.[56]

Between July 28 and August 4 2008, heavy fighting flared up in the northwestern Swat valley leaving 94 militants and 22 soldiers and policemen dead. Another 28 civilians were also killed.[57]

Heavy fighting erupted on August 6 2008, in the Loisam area of Bajaur district. Loisam lies on the strategically important road leading towards the main northwestern city of Peshawar. The fighting started when hundreds of militants poured into the area attacking government forces. After four days of fighting on August 10 the military withdrew from the area. 100 militants and 9 soldiers were confirmed killed and another 55 soldiers were missing, at least three dozen of them captured by the militants.[58][59] While the fighting was going on in Bajaur, in the Buner area of North West Frontier Province militants killed at least nine policemen in an attack on a check post.[60] The checkpoint was then abandoned, and the local Pakistani forces withdrew to Khar, the main town of Bajaur Agency. There were reports that the town of Khar was then besieged by tribal militants. [61]

On 23 August 2008, at least 15 people were killed in a suicide attack at a police check post in Char Bagh area of Swat. Also, three persons including two kids were killed in a bomb blast in Abuha.[62]

Tribesmen declare war against the Taliban

By the beginning of September 2008, Pakistani tribal elders began organising a private army of approximately 30,000 tribesmen to fight the taliban. A lashkar, or private army, comprised of Pakistani tribesmen, began torching the houses of Taliban commanders in Bajaur, near the Afghan border, vowing to fight them until they are expelled.

A local jirga decided to form the lashkar in the wake of the increasing presence of the local Taliban in the area. The lashkar began torching houses, including the house of a local Taliban commander named Naimatullah, who had occupied several government schools and converted them into seminaries.

A tribal elder named Malik Munsib Khan, who heads the lashkar, said that tribesmen would continue their struggle until the Taliban were expelled from the area, adding that anyone found sheltering Taliban militants would be fined one million rupees and their houses will be torched. The tribesmen also torched two important centres of the Taliban in the area and gained control of most of the tehsil.

The main reasons for this was that the operations that were taking place in the Federally Administered Tribal Areas had displaced some 300,000 people while dozens of citizens have been killed in clashes between the militants and military. This showcases why the Taliban have become highly unpopular with the tribal elders and their tribesmen.[63][64][65]

US support and aid for Pakistani tribesmen

Recent American military proposals outlines an intensified effort to enlist tribal leaders in the frontier areas of Pakistan in the fight against Al Qaeda and the Taliban, as part of a broader effort to bolster Pakistani forces against militancy in the region.

The proposal is modeled in part on a similar effort by American forces in Iraq that has been hailed as a great success in fighting foreign insurgents there. But it raises the question of whether such partnerships can be forged without a significant American military presence in Pakistan. And it is unclear whether enough support can be found among the tribes. Small numbers of United States military personnel have served as advisers to the Pakistani Army in the tribal areas, giving planning advice and helping to integrate American intelligence. Under this new approach, the number of advisers would increase.

American officials said these security improvements complemented a package of assistance from the Agency for International Development and the State Department for the seven districts of the tribal areas that amounted to $750 million over five years, and would involve work in education, health and other sectors. The State Department's Bureau of International Narcotics and Law Enforcement Affairs is also assisting the Frontier Corps with financing for counternarcotics work.[66][67][68]

Islamabad Marriott Hotel Bombing

On 23 September 2008, the Pakistani Army, backed by helicopter gunships and artillery killed more than 60 insurgents in northwest Pakistan in offensives as the response to the Islamabad Marriott Hotel bombing over the weekend at the Marriott hotel in the capital Islamabad that killed 53 people. In the nearby Bajur tribal region, the Army killed at least 10 militants during an ongoing offensive there, according to government officials.[69]

The Bajur operations, which the army says has left more than 700 suspected militants dead, has won praise from U.S. officials who are worried about rising violence in Pakistan and Afghanistan.[70]

Renewed Bajaur offensive

Pakistani President Asif Ali Zardari publicly vowed revenge in response to the Islamabad Marriott Hotel bombing. By 26 September 2008, Pakistani troops had successfully conducted and completed a major offensive in the Bajaur and the Tang Khata regions of the Federally Administered Tribal Areas. Pakistani troops had killed over 1,000 militants in a huge offensive, a day after President Asif Ali Zardari lashed out at US forces over a clash on the Afghan border.

Tariq Khan, Inspector General of the Paramilitary Frontier Corps, mentioned to journalists that since the beginning of the Bajaur operations, there were up to 2,000 militant fighters including hundreds of foreigners who were fighting with the soldiers and the security forces. The overall death toll was over 1,000 militants and also adding that 27 Pakistani soldiers had also been killed with 111 soldiers seriously wounded.[71][72]

Five top Al-Qaeda and Taliban commanders were among those killed in a month-long operation in Bajaur. Of the five militant commanders killed, four appeared to be foreigners: Egyptian Abu Saeed Al-Masri; Abu Suleiman, an Arab; an Uzbek commander named Mullah Mansoor; and an Afghan commander called Manaras. The fifth was a Pakistani commander named only Abdullah, a son of ageing hardline leader Maulvi Faqir Mohammad who is based in Bajaur and has close ties to Al-Qaeda second-in-command Ayman al-Zawahiri.[73][74]

On 27 September 2008, Pakistani troops killed at least 16 militants after coming under attack in a tribal region near the Afghan border. Militants attacked three military posts near Khar, the main town in the restive Bajaur tribal district but soldiers repulsed them with artillery and mortar fire. There were no reports of millitary casualties.[75][76]

On 3 October 2008, Pakistani troops, backed by artillery, killed 25 Taliban militants after militants attacked security checkposts in four villages in the restive region. Separately, the local government ordered Afghan refugees in Bajaur to leave the area within three days. In other violence, a suicide bomber blew himself up as he tried to storm the house of Asfandyar Wali Khan, head of the Awami National Party and who is also a member of Pakistan's ruling coalition in the western town of Charsadda, killing four people but missing the politician.[77][78]

Intensified US. strikes

Since the end of August, the United States has stepped up its attacks in the Federally Administered Tribal Areas, following the change of the country's President.[79] On September 3 a commando attack took place in a village near the Afghan border in South Waziristan, and there have been 3 strikes from unmanned drones in North Waziristan, culminating on the morning of 8 September 2008, when a United States Air Force drone aircraft fired a number of missiles at a madrassa (religious school). The airstrike, which unsuccessfully targeted Jalaluddin Haqqani, killed 23 people.[80]

On 25 September 2008, following exchanges of gunfire between US and Pakistani forces on the frontier on Thursday, President Zardari told the United Nations that Pakistan would not tolerate violations of its sovereignty, even by its allies. The incident happened after two US military helicopters came under fire from the Pakistani side, a US military spokesman said, insisting that they had been about a mile and a half inside Afghanistan.

President Zardari told the United Nations, "Just as we will not let Pakistani's territory to be used by terrorists for attacks against our people and our neighbours, we cannot allow our territory and our sovereignty to be violated by our friends," he said, without citing the United States or the border flareup.[81]

Casualties

Casualties of the War in Pakistan (According to news reports)
Period Pakistani forces Militants Civilians
December 2001–March 2004
(prelude) 		29 killed Unknown 120+
March 2004–September 2006 700 killed,
11 captured (all released) 		1,000 1,000
October 2006–April 2007 78 killed 100-300 N/A
July 2007–June 2008 882 killed,
40 missing,
559 captured (439 released) 		1,928 1,827
June 2008–present 296 killed,
200 captured (116 released) 		1,327 462
Sources:[citation needed]

There has been no conclusive reports on the casualties of the war, though some authors, especially Pakistani writers, have estimated that the total casualties on both sides to be more than 1,000 by 2006.[82]

On September 13 2007, then-U.S. Deputy Secretary of State John Negroponte noted during a visit to Islamabad that Pakistan lost 1,000 soldiers fighting the militants since joining the war on terrorism.[83]

On November 14, 2007, senior Pakistan Army officials told at a news conference that a total of 28 suicide attacks killed some 600 Pakistani security men, in addition to 1,300 civilians in the period after the Lal Masjid siege. It also said that from 2001 till November 14, 2007, at least 966 military men were killed and 2,259 others were injured; 488 foreign extremists were killed, 24 others were arrested and 324 foreign extremists were injured[7]

Some have speculated that the unofficial number of Pakistani soldiers killed in action to be somewhere around 3,000 by the late 2006.[84][85] A Pakistan writer, Ayaz Amir states that the army's "Casualties were high, perhaps unsustainable, although we’ll never know the exact figures, the Pakistan army not given to embarrassing disclosures."[86]. It should be noted however that Pakistani military provides substantial benefits to families of fallen soldiers[87], and thus it is unlikely that higher end numbers are correct since it would be difficult to hide fatalities from families. Also Ayaz Amirs and Bill Roggio's claims are spurious to say the least; their assertion that Pakistan Army hides casualties is belied by the fact that the PR arm of the Pakistani military gives a daily update on the operations, complete with casualty figures; broken down by units.[88]

United States role

Pakistan received about $4 billion from the United States for the logistical support it provided for the counter-terrorism operations from 2002 to 2006, and for its own military operation mainly in Waziristan and other tribal areas along the Durand line, according to a report of the Asian Development Bank. The Bush administration also offered a $3 billion five-year aid package to Pakistan for becoming a frontline ally in its 'war on terror'. Annual instalments of $600 million each split evenly between military and economic aid, began in 2005.[89]

In his autobiography, President Musharraf wrote that the United States had paid millions of dollars to the Pakistan government as bounty money for capturing al-Qaeda operators from tribal areas bordering Afghanistan. About 359 of them were handed over to the US for prosecution. [89]

See also

References

  1. ^ Bloomberg.com: Worldwide
  2. ^ http://ap.google.com/article/ALeqM5hkiMxbHNH0BqgpWA2ZG6VD6wVTmAD93GH45O1
  3. ^ a b http://in.news.yahoo.com/071114/139/6n8m0.html
  4. ^ Over 200 soldiers in captivity of militants By Mushtaq Yusufzai & Sailab Mahsud September 3, 2007 The News International, Pakistan
  5. ^ http://www.thenews.com.pk/top_story_detail.asp?Id=15771
  6. ^ http://news.xinhuanet.com/english/2008-09/26/content_10118035.htm
  7. ^ a b 600 Pakistan security men killed in 28 suicide attacks after Lal Masjid operation | Top News
  8. ^ [http://www.defenselink.mil/news/newsarticle.aspx?id=15292
  9. ^ http://ap.google.com/article/ALeqM5hkiMxbHNH0BqgpWA2ZG6VD6wVTmAD93GH45O1
  10. ^ http://www.gmanews.tv/story/123833/Suicide-attacks-soar-in-Pakistan-since-2007
  11. ^ The War in Pakistan - washingtonpost.com
  12. ^ FEATURE-Pakistan lifts veil on not-so-secret Waziristan war
  13. ^ BBC NEWS | South Asia | Pakistan's undeclared war
  14. ^ David Montero (2006-06-22). "Killing scares media away from Waziristan". Christian Science Monitor. Retrieved 2008-08-25.
  15. ^ "Pakistan attacks Waziristan compound". Al Jazeera. 2006-03-16. Retrieved 2008-08-25.
  16. ^ http://www.timesonline.co.uk/tol/news/world/asia/article4111277.ece
  17. ^ a b Rohde, David (2006-09-10). "Al Qaeda Finds Its Center of Gravity". New York Times. Retrieved 2006-09-12. {{cite news}}: Check date values in: |date= (help)
  18. ^ John Diamond (2005-05-04). "Pakistan reports arrest of Osama bin Laden's operations chief". USA Today.
  19. ^ "'Policemen killed' in Waziristan". BBC News. 2006-06-22. Retrieved 2006-08-07. {{cite news}}: Check date values in: |date= (help)
  20. ^ "Forces, militants heading for truce". Dawn. 2006-06-22.
  21. ^ "'Suicide attack' on Pakistan army". BBC News. 2006-06-26. Retrieved 2006-08-07. {{cite news}}: Check date values in: |date= (help)
  22. ^ Pakistan, Taliban militants sign peace agreement
  23. ^ Waziristan accord signed
  24. ^ US outraged as Pakistan frees Taliban fighters - Daily Telegraph
  25. ^ Some See Pakistan's Truce As a Defeat - Fox news
  26. ^ Roggio B (17 March 2007). "Pakistan signs the Bajaur Accord". The Long War Journal. Retrieved on 14 January 2008.
  27. ^ Stakelbeck E (3 April 2007). "Bajaur: When 'Peace' Yields War". CBN News. Retrieved on 14 January 2008.
  28. ^ Roggio B. "The fall of northwestern Pakistan: An online history". The Long War Journal. Retrieved on 13 January 2008.
  29. ^ "Rival militants clash in Pakistan". BBC. 20 March 2007.[failed verification]
  30. ^ Plett B (12 April 2007). "Tribesmen 'oust' foreign fighters". BBC. Retrieved on 16 January 2008.
  31. ^ "Tribe in Pakistan security plea". BBC. 16 April 2007. Retrieved on 16 January 2008.
  32. ^ FOXNews.com - Two Days of Homicide Attacks Kill 70 in Pakistan - International News | News of the World | Middle East News | Europe News
  33. ^ Suicide Bombers Kill 49 in Pakistan
  34. ^ BBC NEWS | South Asia | Scores killed in Pakistan attacks
  35. ^ http://news.yahoo.com/s/ap/20070724/ap_on_re_as/pakistan_militant_leader
  36. ^ Pakistan crisis 'hits army morale' By Ahmed Rashid 6 September 2007 - BBC
  37. ^ 12-hour curfew clamped on Swat -DAWN - Top Stories; November 17, 2007
  38. ^ "Musharraf imposes emergency rule". Dawn. 2007-11-03. Retrieved 2007-11-03. {{cite web}}: Check date values in: |date= (help)
  39. ^ "Martial law declared in Pakistan". CNN. 2007-11-03. Retrieved 2007-11-03. {{cite web}}: Check date values in: |date= (help)
  40. ^ "Pervez Musharraf spoils for a fight as he declares emergency rule". The Times. 2007-11-04. Retrieved 2007-11-04. {{cite news}}: Check date values in: |date= (help)
  41. ^ "Bhutto killed in suicide attack" (HTML). Al Jazeera English. Retrieved 2007-12-27.
  42. ^ Benazir Bhutto is dead Reuter's video
  43. ^ "Benazir Bhutto Assination NBC News Coverage". NBC. 2007-12-27. Retrieved 2007-12-27.
  44. ^ "Benazir Bhutto Assination CBS News Coverage". CBS. 2007-12-27. Retrieved 2007-12-27.
  45. ^ "Benazir Bhutto Assination ABC News Coverage". ABC. 2007-12-27. Retrieved 2007-12-27.
  46. ^ Gall, Carlotta (2007-10-19). "Bomb Attack Kills Scores in Pakistan as Bhutto Returns". The New York Times. Retrieved 2007-12-27. {{cite news}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  47. ^ "Pakistani troops 'flee border post'". Al Jazeera. 17 January 2008. Retrieved on 18 January 2008.
  48. ^ Pakistan attacks hit aid group, military HQ - CNN.com
  49. ^ 45 killed in Pakistan funeral blast - CNN.com
  50. ^ BBC NEWS | South Asia | Pakistani militants 'call truce'
  51. ^ Al Jazeera English - News - Pakistan Troops To Vacate Swat
  52. ^ Baitullah men storm Jandola -DAWN - Top Stories; June 24, 2008
  53. ^ Al Jazeera English - CENTRAL/S. ASIA - Pakistan forces in Khyber offensive
  54. ^ France 24 | Pakistan broadens anti-militant offensive | France 24
  55. ^ Suicide blast targeting police kills 16 at Pakistan rally - CNN.com
  56. ^ Fighting flares in Pakistan's Baluchistan; 43 killed - Yahoo! News
  57. ^ Ninety-four militants, 14 troops killed in NW Pakistan - Yahoo! News
  58. ^ 100 militants, nine soldiers killed in Pakistan clashes - Yahoo! News
  59. ^ Pakistan fighting ends as troops withdraw - Los Angeles Times
  60. ^ Taliban militants kill nine policemen in Pakistan - Yahoo! India News
  61. ^ BBC NEWS | South Asia | 'Dozens die' in Pakistan clashes
  62. ^ 15 feared dead in Swat suicide bombings
  63. ^ http://www.csmonitor.com/2008/0901/p99s01-duts.html
  64. ^ http://www.mcclatchydc.com/226/story/53151.html
  65. ^ http://www.mcclatchydc.com/226/story/53151.html
  66. ^ http://www.iht.com/articles/2007/11/19/america/19policy.php
  67. ^ http://www.nytimes.com/2007/11/19/washington/19policy.html?_r=1&adxnnl=1&ref=world&adxnnlx=1222717017-fvB/geZ6uJD4OkQF6LCPSg&oref=slogin
  68. ^ http://216.239.59.104/search?q=cache:Fda3_xPQYTkJ:www.state.gov/p/sca/rls/2008/106495.htm+US+support+pakistani+tribes&hl=en&ct=clnk&cd=19&gl=uk
  69. ^ [1]
  70. ^ [2]
  71. ^ http://www.thestandard.com.hk/breaking_news_detail.asp?id=6788&icid=4&d_str=20080926
  72. ^ http://timesofindia.indiatimes.com/World/Pakistan/Pakistan_says_five_top_militants_among_1000_dead_in_offensive/articleshow/3530363.cms
  73. ^ http://afp.google.com/article/ALeqM5iGTgxqoTldD4lBtlbyfU5MSNowjg
  74. ^ http://in.reuters.com/article/southAsiaNews/idINIndia-35671020080926
  75. ^ http://afp.google.com/article/ALeqM5gqbV97erFMfXlFjngT7SYUGUxHdQ
  76. ^ http://www.dawn.com/2008/09/28/top4.htm
  77. ^ http://www.gulf-daily-news.com/Story.asp?Article=230579&Sn=WORL&IssueID=31197
  78. ^ http://afp.google.com/article/ALeqM5iQP3xOQVBIz1uiIMsTcCOVVAMNiA
  79. ^ Suspected US Missile Strike Hits Taliban Commander's House - Voice of America, September 8, 2008
  80. ^ Perlez, J. & Shah, P.Z. 2008, 'US attack on Taliban kills 23 in Pakistan', International Herald Tribune, 9 September. Retrieved on 10 September 2008.
  81. ^ http://afp.google.com/article/ALeqM5iGTgxqoTldD4lBtlbyfU5MSNowjg
  82. ^ Pacifying Waziristan by Muhammad Munir
  83. ^ http://uasos.com/world/fierce-fighting-in-waziristan-as-negroponte-visits-pakistan-reuters.html
  84. ^ The Fall of Waziristan: An Online History
  85. ^ Waziristan: Terror Haven for Jihadists? by Erick Stakelbeck November 15, 2006 CBN news
  86. ^ War and peace, army style - September 15, 2006 Dawn (newspaper)
  87. ^ Year of the Soldier; Strategy Page
  88. ^ Latest Update from Pakistan Military Webpage, contradicting the claims made by Ayaz Amir and Bill Roggio
  89. ^ a b "Pakistan: $1 billion from U.S. to fight terror". Aki/Dawn. 2006-11-14. Retrieved 2006-11-24.

External links

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