Deciding regular grammar logics with converse through first-order logic

TL;DR

This paper presents a translation of regular grammar logics with converse into GF2, a fragment of first-order logic, establishing EXPTIME decidability and extending previous results to include various modal logics.

Contribution

It introduces a simple translation method of regular grammar logics with converse into GF2, demonstrating their satisfiability problem is in EXPTIME and encompassing other modal logics.

Findings

Satisfiability for regular grammar logics with converse is in EXPTIME.

Translation into GF2 captures various modal logics without explicit frame conditions.

GF2 is the natural first-order fragment for these logics.

Abstract

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.