A Controlled Quantification in Parsing of Montague Grammar

Montague grammar [5] provides a good framework for a direct translation from a natural language sentence into its logical formulas. This is enabled by the fact that every syntactic rule has its corresponding translation rule. As soon as a subphrase is recognized by a syntactic rule, its corresponding translation rule translates the recognized subphrase into its logical formula. M-grammar of Landsbergen [3] and semantic equivalence parsing (SEP) of Warren and Friedman [7] are parsing algorithms that are based on the above framework of Montague grammar. In this paper, only a version of Montague grammar is discussed, that is, the grammar for the English fragments in "The proper treatment of quantification in ordinary English" (PTQ) of Montague [4]. This version is peculiar but most well known. Syntactic rules of Montague grammar consist of context-free rules and noncontext-free rules. The context-free rules scan the given input sentence and recognize its syntactic structure. The noncontext-free rules are the quantification rules whose roles are to treat semantic problems as follows [2]: (1) They bind anaphoric pronouns to their coreferential noun phrases (e.g., "a woman walks and she talks"). (2) They handle the de re~de dicto (or specific/ nonspecific) ambiguities in opaque contexts (e.g., "John seeks a unicorn"). (3) They handle the scope ambiguities which are caused by more than one quantifier in a sentence (e.g., "every man loves a woman"). However, quantification rules can be applied to any phrase which does not need any quantification. At every parsing step, it cannot be determined whether a quantification rule should be applied or not. This nondeterminism causes wrong applications of quantification rules, or redundant parsing paths. SEP [7] is a nondeterministic program such that every parsing path is tried. If an application of a quantification rule results in a wrong parse, SEP backtracks and one of the other alternative rules is tried. On the other hand, if distinct parsing paths translate an input sentence into equivalent logical formulas, only the first one is printed. Here, the equivalency of logical formulas is determined by an equivalency test. However, it is an undecidable problem to test the equivalence of intensional logical formulas [7]. (The intensional logic is the underlying logic of Montague grammar.) Hence, SEP uses the following weaker definition of equivalence: "logical formulas are equivalent if they are identical within change of bound variable, after logical reduction and extensionalization." By this definition only, the equivalence of intensional logical formulas cannot be completely tested. In this paper, a parsing algorithm is proposed,