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On the Computational Complexity of the Numerically Definite Syllogistic and Related Logics

Published online by Cambridge University Press:  15 January 2014

Ian Pratt-Hartmann
Ian Pratt-Hartmann*
Affiliation:
School of Computer Science, University of Manchester, Manchester M13 9PL, UKE-mail: ipratt@cs.man.ac.uk
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Abstract

The numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 1 , March 2008 , pp. 1 - 28
Copyright
Copyright © Association for Symbolic Logic 2008

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