Deciding Regular Grammar Logics with Converse Through First-Order Logic

Abstract

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF 2 , 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 GF 2 , 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 GF 2 without extra machinery such as fixed point-operators.

Modal Logics with Hard Diamond-Free Fragments

“…As we mention in the Introduction, all the logics we consider can be seen as regular grammar logics with converse ( [7]), for which the satisfiability problem is in EXP. This already gives an upper bound for the satisfiability of D 2 ⊕ ⊆ K4 (and for the general case of (N, ⊂, F ) from Section 4).…”

“…In this subsection we present the basic definitions about regular grammar logics with converse and we sketch an argument why (N, ⊂, F ) is (or can be reduced to) a regular grammar logic with converse. Definitions and most of our arguments come from [7]. 7 For every agent i ∈ N , let i be a new agent, i := i, N := {i | i ∈ N }, and in every frame (W, (R i ) i∈N ∪N ), R i = R…”

“…Definitions and most of our arguments come from [7]. 7 For every agent i ∈ N , let i be a new agent, i := i, N := {i | i ∈ N }, and in every frame (W, (R i ) i∈N ∪N ), R i = R…”

“…The reader is encouraged to see [7] and [19] for a more complete presentation of regular grammar modal logics with converse.…”

“…Demri and De Nivelle have shown in [7] through a translation into a fragment of first-order logic that the satisfiability problem for the whole family is in EXP (see also [6]). Then, Nguyen and Sza las in [19] gave a tableau procedure for the general satisfiability problem (where the logic itself is given as input in the form of a finite automaton) and determined that it is also in EXP.…”