The decision problem of provability logic with only one atom

The decision problem for provability logic remains PSPACE -complete even if the number of propositional atoms is restricted to one. In some cases the set of all tautologies of a modal logic is in coNP. An example of a logic like that is the well-known S5. However, most of the traditional modal systems, including S4 and T, have PSPACE -complete decision problem. So one can say that adding modalities to the language of classical propositional logic does increase algorithmic complexity — not a surprising paradigm. The methods for constructing a polynomial space decision procedure and for proving PSPACE -completeness of a modal logic can be learnt from R. Ladner’s paper [Lad77]. Provability logic GL is not mentioned in [Lad77], but it is not difficult to verify that GL has PSPACE -complete decision problem as well. In this paper we go farther and use Ladner’s methods to show that the decision problem of GL is PSPACE -complete even if the number of propositional atoms used to build modal formulas is restricted to one. This fact can be interpreted as saying that, in case of provability logic, allowing more than one atom does not increase the expressive power of the language. The structure of the present paper is similar to that of our [Šve03] where an alternative simple proof of R. Statman’s result concerning PSPACE -completeness of intuitionistic propositional logic is presented. Modal formulas are built up from propositional atoms and the symbol ⊥ for falsity using logical connectives and a unary symbol 2 for necessity. We use 3A ∗This paper was supported by grant 401/01/0218 of the Grant Agency of the Czech Republic.