## Modal logic and attribute value structures

P. Blackburn. Modal logic and attribute value structures. In M. de Rijke, editors, *Diamonds and Defaults*, Synthese Language Library, pp. 19–65, Kluwer Academic Publishers, Dordrecht, 1993.

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#### Abstract:

This paper aims to show that attribute-value (AV) formalisms can be regarded as more or less standard modal logics. Basically, in AV formalisms there exist two kinds of primitives, called attributes and atoms. Standardly, an atom is not thought of as a statement. Instead, a (basic) statement is a pair $\pi=\langle\rm ATT,\rm val\rangle$, where ATT is an attribute and val an atom. In $\pi$, val is called the value of the attribute ATT. Complex AV structures can be built from basic ones using primitive operations that vary from formalism to formalism. Virtually all formalisms allow AV structures as values of an attribute, whereby some recursion is built into the language, and they have an operation corresponding to conjunction. Many formalisms use disjunction and negation as well. There are two ways to reformulate this in standard logic. One is to assimilate attributes to functions, and atoms to objects. The pair $\pi$ is then read as "the value of the function ATT equals val". This is the approach of M. Johnson [Attribute-value logic and the theory of grammar, Univ. Chicago Press, Chicago, IL, 1988; per bibl.]. The other road is to take atoms as basic propositions. The attributes are then simply modal operators. Thus $\pi$ is now read as "it is possible following an ATT-arc that val". The author argues in favour of the second interpretation on the grounds that modal logic arises naturally when one considers AV structures not as syntactic objects but only as descriptions of syntactic objects. Moreover, it removes the distinction between AV structures and atoms, which is somewhat artificial given that both can appear as values of an attribute. Finally, modal logic provides enough expressive power for the applications in natural language, so that the power of predicate logic, which causes undecidability in the general case, is not really needed. As a case in point, the decidability of a modal logic that models the so-called Kasper-Rounds logic is shown.

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@INCOLLECTION{blackburn93:_modal, author = {P. Blackburn}, title = {Modal logic and attribute value structures}, booktitle = {Diamonds and Defaults}, publisher = {Kluwer Academic Publishers}, year = {1993}, editor = {M. de Rijke}, series = {Synthese Language Library}, pages = {19-65}, address = {Dordrecht}, abstract = { This paper aims to show that attribute-value (AV) formalisms can be regarded as more or less standard modal logics. Basically, in AV formalisms there exist two kinds of primitives, called attributes and atoms. Standardly, an atom is not thought of as a statement. Instead, a (basic) statement is a pair $\pi=\langle{\rm ATT},{\rm val}\rangle$, where ATT is an attribute and val an atom. In $\pi$, val is called the value of the attribute ATT. Complex AV structures can be built from basic ones using primitive operations that vary from formalism to formalism. Virtually all formalisms allow AV structures as values of an attribute, whereby some recursion is built into the language, and they have an operation corresponding to conjunction. Many formalisms use disjunction and negation as well. There are two ways to reformulate this in standard logic. One is to assimilate attributes to functions, and atoms to objects. The pair $\pi$ is then read as "the value of the function ATT equals val". This is the approach of M. Johnson [Attribute-value logic and the theory of grammar, Univ. Chicago Press, Chicago, IL, 1988; per bibl.]. The other road is to take atoms as basic propositions. The attributes are then simply modal operators. Thus $\pi$ is now read as "it is possible following an ATT-arc that val". The author argues in favour of the second interpretation on the grounds that modal logic arises naturally when one considers AV structures not as syntactic objects but only as descriptions of syntactic objects. Moreover, it removes the distinction between AV structures and atoms, which is somewhat artificial given that both can appear as values of an attribute. Finally, modal logic provides enough expressive power for the applications in natural language, so that the power of predicate logic, which causes undecidability in the general case, is not really needed. As a case in point, the decidability of a modal logic that models the so-called Kasper-Rounds logic is shown. } }

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