Attribute-value matrix

An attribute-value matrix (AWM), also known as a feature-value structure , is a formal structure that is mainly used in linguistics , especially in the field of unification grammars and the head-driven phrase structure grammar . In this context, feature structures are modeled with attribute-value matrices . Together with the mechanisms of subsumption , typification and unification , it offers the possibility of formally describing linguistic structures.

An AWM is a two-column matrix. Each row of this matrix represents a characteristic . This is divided into the name of the characteristic, which can be found in the left column, and the value of the characteristic in the right column. The order of the characteristics is not important, but there must be no characteristics with the same name and different values. So a simple AWM modeling a dog (specifically a four-year-old dachshund named Waldi) would be

{\ displaystyle A = {\ begin {bmatrix} \ mathrm {NAME} & waldi \\\ mathrm {BREED} & dachshund \\\ mathrm {AGE} & 4 \ end {bmatrix}}}

In this case all assigned values are atomic , which means that they cannot be broken down further. But it is also possible to enter complex values. These are stored as new attribute-value matrices as a value within the matrix. So if you wanted to add more information about the color and texture of the fur you could expand the matrix like this:

{\ displaystyle A '= {\ begin {bmatrix} \ mathrm {NAME} & waldi \\\ mathrm {BREED} & dachshund \\\ mathrm {AGE} & 4 \\\ mathrm {FELL} & {\ begin {bmatrix} \ mathrm {COLOR} & brown \\\ mathrm {ART} & rough \ end {bmatrix}} \ end {bmatrix}}}

The characteristic FELL refers to a value that is itself an AWM. This AWM now specifies the individual properties of the fur: COLOR and TYPE. So Waldi is a brown wire-haired dachshund.

Relations and Operations

Subsumption

Subsumption is a relation that compares two attribute-value matrices for their information content. If an AWM B is at least as informative as an AWM A, then: (A subsumes B). B must therefore contain at least all the information that A contains, but can also provide additional information. The following applies to the two AWM listed above , because A 'contains the information on the fur in addition to the information from A. ${\ displaystyle A \ sqsubseteq B}$ ${\ displaystyle A \ sqsubseteq A '}$

The subsumption applies if and only if ${\ displaystyle A \ sqsubseteq B}$

All atomic features from A with the same value are included in B and
All complex features from A are subsumed by the corresponding complex features from B.

Conversely, the subsumption is invalid if ${\ displaystyle A \ sqsubseteq B}$

An atomic feature from A has a different value than an atomic feature from B or
An atomic feature is contained in A, but not in B, or
A complex characteristic from A does not subsume or the corresponding complex characteristic from B
A complex feature is contained in A, but not in B.

The attribute-value matrix is subsumed because both match in the attribute “RACE”, but the second also contains the attribute “NAME”, so it is more special. ${\ displaystyle {\ begin {bmatrix} \ mathrm {BREED} & dachshund \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} \ mathrm {NAME} & fiffi \\\ mathrm {BREED} & dachshund \ end {bmatrix}}}$

The two AWM and are not subsumed in any direction, as their NAME feature, which contains two different values "waldi" and "fiffi", is not compatible. ${\ displaystyle {\ begin {bmatrix} \ mathrm {NAME} & waldi \\\ mathrm {BREED} & dachshund \\\ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} \ mathrm {NAME} & fiffi \\\ mathrm {BREED} & dachshund \\\ end {bmatrix}}}$

The most general AWM is the empty attribute-value matrix that subsumes all other AWMs because it does not contain any information at all.

Unification

Unification is a binary operation that attempts to merge two attribute-value matrices into one result AWM. This operation is comparable to the union of sets , but has to be carried out recursively due to the recursive structure of attribute-value matrices.

Two attribute-value matrices A and B are unified to form an AWM C (notation:) by ${\ displaystyle A \ sqcup B = C}$

the atomic features of both output matrices are stored in C.
the corresponding complex values of both output matrices are unified and stored in C.

If the case occurs within this recursive process that two characteristics with the same name but different values are to be stored in the result matrix, then the unification fails. In this case, the result of the operation is the specially defined 'impossible' AWM . ${\ displaystyle \ bot}$

example 1

{\ displaystyle {\ begin {bmatrix} \ mathrm {NAME} & waldi \\\ mathrm {AGE} & 4 \\\ mathrm {FELL} & {\ begin {bmatrix} \ mathrm {COLOR} & brown \ end {bmatrix}} \ \\ end {bmatrix}} \ sqcup {\ begin {bmatrix} \ mathrm {NAME} & waldi \\\ mathrm {BREED} & dachshund \\\ mathrm {FELL} & {\ begin {bmatrix} \ mathrm {ART} & rough \ end {bmatrix}} \\\ end {bmatrix}} = {\ begin {bmatrix} \ mathrm {NAME} & waldi \\\ mathrm {BREED} & dachshund \\\ mathrm {AGE} & 4 \\\ mathrm {FELL} & {\ begin {bmatrix} \ mathrm {COLOR} & brown \\\ mathrm {ART} & rough \ end {bmatrix}} \\\ end {bmatrix}}}

This unification is successful: Each atomic value either only occurs in a starting matrix, or the values are the same ("waldi"), and the subordinate AWM for FELL is also unifiable.

Example 2

{\ displaystyle {\ begin {bmatrix} \ mathrm {NAME} & waldi \\\ mathrm {RACE} & dachshund \ end {bmatrix}} \ sqcup {\ begin {bmatrix} \ mathrm {NAME} & fiffi \\\ mathrm {RACE} & dackel \ end {bmatrix}} = \ bot}

The unification fails here because the NAME attribute has different values ("waldi" or "fiffi").

Web links

Attribute-value matrices as a feature structure, introduction (Postscript document)