### Decidable and undecidable logics with a binary modality

�gnes Kurucz, Istv�n N�meti, Ildik� Sain, Andr�s Simon
1995 Journal of Logic, Language and Information
We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic. We investigate here decidability problems concerning logics having an extra binary connective "o" beside the Boolean ones. These logics are strongly related to ordinary first-order logic, see Henkin et al. (1985, ch.
more » ... 3) on this connection in an algebraic setting. Our most important aims are to give a transparent overview of the results and to stress the crucial points and ideas of the proofs, especially when the extra binary connective is associative 9 The emphasis is on those parts of the proof methods which have been well known for the specialists. (The reason for this is the didactic character of the present paper 9 For the details of those ideas which are our own contributions we give references to more technical papers. Since associativity of"o" corresponds to commutativity of first-order existential quantifiers in some sense (cf. op cit), our results and techniques can help to find decidable fragments of first-order logic as well (for such results see N6meti (1985, 1987, 1992)) 9 Preliminaries We say that/2 is a logic with a binary rnodality iff (1-3) below hold. 1. The language of 12 includes the usual Boolean connectives and a binary connective "o" "e" " " def 9 denotes the dual of o , i.e. (~ 9 ~b) = -,(--,~ o --,~b).