Complexity of products of modal logics

Abstract

A relatively new way of combining modal logics is to consider their products. The main application of these product logics lies in the description of parallel computing processes. Axiomatics and decidability of the validity problem have been rather extensively investigated and many logics behave well in these respects. In this paper we look at the product construction from a computational complexity point of view. We show that in many cases there is a drastic increase in complexity, e.g., all products containing the finite S5×S5 products as models have an nexptime-hard satisfaction problem. Products with a functional modality however do not lead to an increase in complexity. For the products K× S5 and S5× S5, we provide a matching upper bound. Combining (modal) logics is a very active area, witness e.g., [4] and the book [1]. A rather special way of combining two modal logics is to consider their products. This approach started with [20], and has recently been developed in great detail in [5]. In temporal logic, products of two logics have been used to describe the temporal logic of intervals (cf. the “product treatment” of the system HS from [8] in [15]: Chapter 4 and the references therein). Almost all products of temporal logics are undecidable, sometimes the validities are not even recursively enumerable [8, 22]. Products of modal (and modal and temporal) logics have applications in the theory of parallel computing [18]. Here we are concerned with the general mathematical theory of products of modal logics, in particular the complexity of several natural decision problems, like the validity problem. With respect to (Hilbert style) axiomatizability a lot of general results are obtained in [5], cf e.g., Theorem 5.7. That paper also contains decidability results for a large number of cases. The general trend for these results is that they are rather hard to prove, but become a lot easier if one of the logics is S5, though even then the filtration arguments are rather involved, and lead to models whose size is in general double exponential in the length of the formula which is to be satisfied. The upper bounds we obtain from these proofs are very bad, in the general case (when none of the logics is S5), the decision-algorithm is non-elementary, and when one of the logics is S5 we only obtain a non-deterministic double exponential time upper bound for the satisfaction problem. Questions concerning computational complexity, have hitherto not been addressed, and we will make a start here. The overall trend is that these logics have a very bad complexity for the satisfaction problem: in many simple cases it is nexptime-hard. Also, even if the satisfaction problem is decidable, the problem whether for a formula φ there exists a model The author is supported by UK EPSRC grant No. GR/K54946.