A Decision Procedure for a Sublanguage of Set Theory Involving Monotone, Additive, and Multiplicative Functions, I: The Two-Level Case

Authors:
Calogero G. Zarba;Domenico Cantone;Jacob T. Schwartz
Affiliations:
LORIA and INRIA, Lorraine, Villers-lès-Nancy Cedex, France 54602;Dipartimento di Matematica e Informatica, Università di Catania, Catania, Italy 95125;Courant Institute of Mathematical Sciences, New York University, New York, USA 10012
Venue:
Journal of Automated Reasoning
Year:
2004

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Abstract

2LS is a decidable many-sorted set-theoretic language involving one sort for elements and one sort for sets of elements. In this paper we extend 2LS with constructs for expressing monotonicity, additivity, and multiplicativity properties of set-to-set functions. We call the resulting language 2LSmf. We prove that 2LSmf is decidable by reducing the problem of determining the satisfiability of its sentences to the problem of determining the satisfiability of sentences of 2LS. Furthermore, we prove that the language 2LSmf is stably infinite with respect to the sort of elements. Therefore, by using a many-sorted version of the Nelson--Oppen combination method, 2LSmf can be combined with other languages modeling the sort of elements.