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Parsing and generation as Datalog query evaluation
, 2011
"... Parsing and generation (or surface realization) are two of the most important tasks in the processing of natural language by humans and by computers. In this paper, we study both tasks in the style of formal language theory. We show that the problems of parsing and surface realization for grammar fo ..."
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Parsing and generation (or surface realization) are two of the most important tasks in the processing of natural language by humans and by computers. In this paper, we study both tasks in the style of formal language theory. We show that the problems of parsing and surface realization for grammar formalisms with “contextfree ” derivations, coupled with a kind of Montague semantics (satisfying a certain restriction) can be reduced in a uniform way to Datalog query evaluation. This makes it possible to apply to parsing and surface realization known efficient evaluation methods for Datalog. Moreover, the reduction has the following complexitytheoretic consequences for all such formalisms: (i) the decision problem of recognizing grammaticality (surface realizability) of an input string (logical form) is in LOGCFL; and (ii) the search problem of computing all derivation trees (in the form of shared forest) from an input string or input logical form is in functional LOGCFL. The reduction is carried out by way of “contextfree ” grammars on typed λterms, a relaxation of the secondorder fragment of de Groote’s abstract categorial grammar.
Tools and Techniques for Formalising Structural Proof Theory
, 2009
"... Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the l ..."
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Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the literature. Two intermediate results can be used to drastically reduce the amount of effort needed in proofs of Cut admissibility: Weakening and Invertibility. Indeed, whereas there are proofs of Cut admissibility which do not use Invertibility, Weakening is almost always necessary. Use of these results simply shifts the bureaucracy, however; Weakening and Invertibility, whilst more easy to prove, are still not trivial. We give a framework under which sequent calculi can be codified and analysed, which then allows us to prove various results: for a calculus to admit Weakening and for a rule to be invertible in a calculus. For the latter, even though many calculi are investigated, the general condition is simple and easily verified. The results have been applied to G3ip, G3cp, G3c, G3s, G3LC and G4ip. Invertibility is important in another respect; that of proofsearch. Should all rules in a calculus be invertible, then terminating rootfirst proof search gives a decision procedure
The StringMeaning Relations Definable by Lambek Grammars and ContextFree Grammars
"... Abstract. We show that the class of stringmeaning relations definable by the following two types of grammars coincides: (i) Lambek grammars where each lexical item is assigned a (suitably typed) lambda term as a representation of its meaning, and the meaning of a sentence is computed according to t ..."
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Abstract. We show that the class of stringmeaning relations definable by the following two types of grammars coincides: (i) Lambek grammars where each lexical item is assigned a (suitably typed) lambda term as a representation of its meaning, and the meaning of a sentence is computed according to the lambdaterm corresponding to its derivation; and (ii) cyclefree contextfree grammars that do not generate the empty string where each rule is associated with a (suitably typed) lambda term that specifies how the meaning of a phrase is determined by the meanings of its immediate constituents. 1