The Two-Variable Guarded Fragment with Transitive Guards Is 2EXPTIME-Hard

Abstract

We prove that the satisfiability problem for the two-variable guarded fragment with transitive guards GF ²+TG 2EXPTIME-hard. This result closes the open questions left in [4],[17].In fact,we show 2EXPTIME-hardness of minGF ² +TG a fragment of GF ² +TG with- out equality and with just one transitive relation .,which is the only non-unary symbol allowed. Our lower bound for minGF ² +TG the upper bound for the whole guarded fragment with transitive guards and the unrestricted number of variables GF +TG by Szwast and Tendera [17 ],so in fact GF ²+TG 2EXPTIME-complete. It is surpris- ing that the complexity of minGF ² +TG higher then the complexity of the variant with one-way transitive guards GF ² +TG [9 ]. The latter logic appears naturally as a counterpart of temporal logics without past operators.

Supported by KBN grant 8 T11C 043 19

By the first order logic we mean in this paper the first order logic without constants and function symbols.

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