Computing Simple Quantifiers
In this Chapter I introduce the idea of semantic automata—simple computational devices corresponding to basic quantifiers in natural language. In line with a procedural approach to semantics, given a quantified sentence and a finite model, a semantic automaton computes the truth-value of this sentence in that model. In order to build the semantic automata theory, I first show how to encode finite models as strings of symbols, translating between generalized quantifier theory and formal language theory. With the help of this encoding I show what kind of automata correspond to particular quantifiers. This leads to a number of characterization results, for instance, a classic theorem of Van Benthem establishing equivalence between quantifiers definable in first-order logic (e.g., ‘more than 5’) and quantifiers recognizable by finite-automata. Quantifier ‘most’, which is not definable in first-order logic, will require a recognition device with some sort of unbounded working memory, e.g., a push-down automaton. The question arises: are these logical characterizations cognitively plausible? In the next chapter, I will argue that the answer is positive.
KeywordsMonadic generalized quantifiers Logical definability Finite model-theory First-order quantifiers Proportional quantifiers Automata theory Chomsky’s hierarchy Computations Finite-automata Push-down automata
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