Knowledge representation with logic

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Knowledge representation with logic is a type of knowledge representation that is based on formal logic . To build knowledge-based systems, objects in the real world must be represented in a language that a computer understands so that it can deal with this knowledge. So z. B. an expert formalize his knowledge about his subject area and make it usable for many others (so-called expert system). A logical system is a formal language suitable for this.

Knowledge representation with logic

Knowledge based system

If one uses classical logic as a representation of a knowledge-based system , one speaks of a classical-logical system .

Classic-logical system

Such a logical system can be divided into two basic components. The first of the two components of a classical-logical system is the so-called inference relation to model human reasoning . The second component consists of a representation language in which the knowledge base , the core of the knowledge-based system, can be formulated. The task of this representation language is taken over in the classical-logical system by the classical logics. This means that the existing knowledge is encoded in predicate logic formulas . The most important potential of the classical-logical system defined on these two components, however, consists in the inference of knowledge itself. If the inference relation is defined according to the logical representation language, there is the possibility of inferring knowledge , that is, from what is already available in the system Know how to derive new knowledge about the inference relation.

Inference in the classical-logical system

From the division of the inference into deduction, induction and abduction it follows that induction as well as abduction as inference mechanisms are not necessarily correct. Thus, deduction is the only sure method in the end (cf. 3-division of the inference according to Peirce (1839–1914)), because the knowledge derived in this case is always true. Because of this property, logical systems use the deductive instance of inference to model the logical inference operator. A correct sequence of steps of this inference procedure, in the course of which new knowledge B is derived from existing knowledge W, is also referred to as proof.

With the use of such a deductive inference component, however, it becomes impossible to revise conclusions that have once been derived as true. Therefore, in classical-logical systems that are based on purely deductive methods, it is impossible to correctly model non- monotonic reasoning as a fundamental characteristic of human inference.

One possible approach to solving this defect in the inference model of the logical system is the gradual gradation of the correctness of derivations. This gradation can be achieved in two ways. On the one hand, it does this by using probabilities. By quantifying derivatives using percentages, one obtains a probabilistic logic in which gradual gradations are entirely possible. On the other hand, the gradual grading of conclusions in fuzzy logic is made possible by the use of degrees to describe vague predicates.

Building a logical system

Basic components

• ${\displaystyle \Sigma }$ = Signature
• Int ( ) = set of all interpretations about the signature ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$
• For ( ) = set of all formulas above the signature ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$
• ${\displaystyle \models _{\Sigma }}$ = Fulfillment relation

The signature Σ

The set theoretical intuition behind the concept of the signature is a set of names and concepts through which all elements of a knowledge base W to be represented are formalized. More precisely, the elements of a signature are names that are classified according to predicates and functors and differentiated according to their arity.

Propositional signature

In propositional logic, signatures contain zero-character names or identifiers, which are also known as propositional variables.

Example: ${\displaystyle \Sigma _{AL}=\{Schnee,schneit,Sonne,kalt\}}$

Predicate logic signature

Signatures in first level predicate logic contain zero and multi-digit functors and predicates. Thus, a signature can be viewed as a tuple in predicate logic, where git:

${\displaystyle \Sigma }$= (Func, Pred)

With

• Func = set of zero or multiple-place functors Zero-place functors are called constants
• Pred = set of zero or multi-digit predicates

Due to the fact that propositional logic is a true subset of first level predicate logic, the set of all predicate logic signatures also contains the set of all propositional signatures as a true subset. It follows that the propositional variables can be modeled by zero-digit predicates. These can represent atomic formulas, i.e. the atoms of propositional logic.

Example: ${\displaystyle \Sigma _{PL1}=\{Schneewegschaufeln(x,y),Mann,Gehweg,Kinder,Einfahrt\}}$

The set of interpretations Int (Σ)

The most important property of an interpretation within the logical system is that, together with the fulfillment relation, it establishes the connection between the syntax (in the form of the signature ) of the representation language and the semantics of statements by assigning the names of the signature to objects of a knowledge base W assigns. ${\displaystyle \Sigma }$

Interpretation in propositional logic

When interpreting a propositional signature, each propositional variable from the signature is assigned a truth value. This assignment is made through an interpretation for which the following applies: ${\displaystyle \Sigma }$ ${\displaystyle \sigma }$

${\displaystyle \sigma \in Int(\Sigma )=\Sigma \longrightarrow \{0,1\}}$

The amount Int (called ) the set of all functions of a given signature after . This interpretation of a signature is also called assignment, because this function assigns a truth value to each proposition variable. ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$ ${\displaystyle \{0,1\}}$ ${\displaystyle \sigma }$ ${\displaystyle \Sigma }$

Interpretation in predicate logic

In PL1, the structure of an interpretation can be described as follows:

${\displaystyle I=(U_{I},Func_{I},Pred_{I})}$

where:

• ${\displaystyle U_{I}}$= non-empty carrier set with all objects of an interpretation
• ${\displaystyle Func_{I}}$ = Functional amount:

${\displaystyle \{f_{I}:\underbrace {U_{I}\times \dots \times U_{I}} _{n-mal}\rightarrow U_{I}|f\in Func\;mit\;Stelligkeit\;n\in N\}}$

• ${\displaystyle Pred_{I}}$ = Set of relations:

${\displaystyle \{p_{I}\subseteq \underbrace {U_{I}\times \dots \times U_{I}} _{n-mal}\rightarrow U_{I}|p\in Pred\;mit\;Stelligkeit\;n\in N\}}$

An interpretation I in PL1 maps functors and predicates of the signature to objects of the world to be represented above the universe U according to the following table:

Zero-place functors	Elements from U
One or more digit functors	Functions
Zero-digit predicates	Allocation with truth value
Single-digit predicates	Subset of U
Multi-digit predicates	Relations R ${\displaystyle \subseteq \underbrace {U\times \dots \times U} _{Stelligkeit\;n}}$

Example:

Let signature with = p, with arity i and interpretation ${\displaystyle \Sigma =(\{eins^{0},plus^{2}\},\{gleich^{2}\})}$ ${\displaystyle p^{i}}$

${\displaystyle (N,\{+\in N\times N\rightarrow N\},\{\{(n,m)\in N\times N|n=m\}\})}$

given. The following applies:

I (one)	${\displaystyle 1\in N=U}$
I (plus)	${\displaystyle +\in N\times N\rightarrow N}$
I (same)	${\displaystyle \{(n,m)\in N\times N|n=m\}=\{(0,0),(1,1),\dots \}}$

The set of formulas For (Σ)

The set of formulas via a signature is an essential part of a logical system. Formulas form the syntactic representation of objects of the world W to be represented, of statements about these objects, as well as of facts with which the world W is described. An essential property of the formulas of a logic system is its well-formulated unit (English well-formed formula ). For ( ) contains all formulas that can be formed from the elements of the signature according to the given grammar for formulas . It is precisely for these formulas that the property of being well formulated applies. ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$

Formulas in propositional logic

If the signature is a purely propositional signature, i. H. the signature only contains propositional variables (= zero-digit predicates), so these themselves already form atomic propositional formulas, the so-called literals. In the case of a propositional signature, the set For ( ) thus includes the signature itself and all more complex forms that can be formed by logical links in accordance with the grammar for formulas. ${\displaystyle \Sigma }$

Example:

Let a signature = {Mon, Tue, Wed, Thu, Fri, Sat, Sun} be given. So z. B. the following formulas can be formed: ${\displaystyle \Sigma }$

${\displaystyle \neg Mo\Rightarrow Di}$ ${\displaystyle \neg Mo\wedge \neg Di\wedge \neg Mi\wedge \neg Do\wedge \neg Fr\Rightarrow Sa\wedge So}$ ${\displaystyle Mo\vee Di\vee Mi\vee Do\vee Fr\Rightarrow Sa\vee So}$

Formulas in predicate logic

In addition to the propositional formulas For ( ) listed in the previous section , formulas in the predicate logic formula set For ( ) can also contain variables and quantifications using these variables. If a signature contains the single-digit predicate P (x), the formula set For ( ) contains the predicate itself, as well as existential and universal quantification of the statement P about the individual variable x ${\displaystyle \Sigma _{AL}}$ ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$

Example:

Let a signature example: = {father (x, y), grandfather (x, y)} and let x, y, z be variables, the following formula can be derived from this: Father (x, y) father (y, z) grandfather (x, z) ${\displaystyle \forall x.\forall y.\forall z:}$ ${\displaystyle \wedge }$ ${\displaystyle \Rightarrow }$

Furthermore, let a signature = {loves (x, y)} be given, where x, y denote variables for people. The following sentences can be formulated using predicate logic formulas using this signature: ${\displaystyle \Sigma }$

Everybody loves somebody	${\displaystyle \forall x.\exists y:loves(x,y)}$
Somebody loves somebody	${\displaystyle \exists x.\exists y:loves(x,y)}$
Everybody loves everybody	${\displaystyle \forall x.\forall y:loves(x,y)}$
Nobody loves everybody	${\displaystyle \neg (\exists x.\forall y:loves(x,y))}$
Somebody loves nobody	${\displaystyle \exists x.\forall y:\neg loves(x,y)}$

The fulfillment relation

Together with the interpretation of a signature, the fulfillment relation represents the connection between the objects of a world W syntactically represented by formulas and their semantics in W. A fulfillment relation indicates when a formula is valid in an interpretation and whether a formula in an interpretation is true or false is. Since this relation is one of the basic components of the logical system, every logical system provides such a satisfaction relation:

${\displaystyle \models _{\Sigma }\subseteq Int(\Sigma )\times For(\Sigma )}$

Example:

Let be an interpretation, A a literal and let (A) = 1, then . ${\displaystyle I_{1}}$ ${\displaystyle I_{1}}$ ${\displaystyle I_{1}\models _{\Sigma _{AL}}A}$

If you transfer the fulfillment relation to a relation between formulas, you get the logical conclusion:

${\displaystyle \models _{\Sigma }\subseteq For(\Sigma )\times For(\Sigma )}$ ${\displaystyle F\models _{\Sigma }G\Leftrightarrow \forall I\in \Sigma \longrightarrow \{0,1\}:I\models _{\Sigma }F\Rightarrow I\models _{\Sigma }G}$

It is read as "from F logically follows G" or "G logically follows from F". ${\displaystyle F\models _{\Sigma }G}$