Journal of Symbolic Logic

Small substructures and decidability issues for first-order logic with two variables

Emanuel Kieroński and Martin Otto

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Abstract

We study first-order logic with two variables FO2 and establish a small substructure property. Similar to the small model property for FO2 we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO2 under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO2 has the finite model property and is complete for non-deterministic exponential time, just as for plain FO2. With two equivalence relations, FO2 does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO2 is undecidable.

Article information

Source
J. Symbolic Logic Volume 77, Issue 3 (2012), 729-765.

Dates
First available in Project Euclid: 13 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1344862160

Digital Object Identifier
doi:10.2178/jsl/1344862160

Mathematical Reviews number (MathSciNet)
MR2987136

Zentralblatt MATH identifier
1284.03136

Citation

Kieroński, Emanuel; Otto, Martin. Small substructures and decidability issues for first-order logic with two variables. J. Symbolic Logic 77 (2012), no. 3, 729--765. doi:10.2178/jsl/1344862160. https://projecteuclid.org/euclid.jsl/1344862160


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