Abstract
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.
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- 1.
We assume some familiarity with the basic terminology of automata theory, which we review in Appendix A.2.1.
- 2.
The original proof of Van Benthem makes use of the number triangle representation of first-order quantifiers. The textbook by Partee et al. (1990) discusses algorithms that turn number trees into semantic automata.
- 3.
The proof uses the quantifier elimination for \({{\mathrm{FO}}}(\mathsf D_n)\).
- 4.
Or equivalently a CE-quantifier of type (1,1).
- 5.
At least not without assuming any additional invariance properties for the quantifier in question.
References
van Benthem, J. (1986). Essays in Logical Semantics. Reidel.
Ginsburg, S., & Spanier, E. H. (1966). Semigroups, Presburger formulas, and languages. Pacific Journal of Mathematics, 16, 285–296.
Kanazawa, M. (2013). Monadic quantifiers recognized by deterministic pushdown automata. In M. Aloni, M. Franke, & F. Roelofsen (Eds.), Proceedings of the 19th Amsterdam Colloquium (pp. 139–146).
Mostowski, M. (1998). Computational semantics for monadic quantifiers. Journal of Applied Non-Classical Logics, 8, 107–121.
Partee, B. H., ter Meulen, A. G., & Wall, R. (1990). Mathematical Methods in Linguistics. Studies in Linguistics and Philosophy. Springer.
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Szymanik, J. (2016). Computing Simple Quantifiers. In: Quantifiers and Cognition: Logical and Computational Perspectives. Studies in Linguistics and Philosophy, vol 96. Springer, Cham. https://doi.org/10.1007/978-3-319-28749-2_4
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DOI: https://doi.org/10.1007/978-3-319-28749-2_4
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