An AC 1-complete model checking problem for intuitionistic logic

Abstract

Abstract. We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC 1 . The basic tool we use is the connection between intuitionistic logic and Heyting algebras, investigating its complexity theoretical aspects. For superintuitionistic logics with one variable, we obtain NC 1 -completeness for the model checking problem.Key words. complexity, intuitionistic logic, model checking, AC 1 , Heyting algebra AMS subject classifications. 03B20, 06D20, 68Q171. Introduction. Intuitionistic logic (see e.g. [10,28]) is a part of classical logic that can be proven using constructive proofs-e.g. by proofs that do not use reductio ad absurdum. For example, the law of the excluded middle a ∨ ¬a and the weak law of the excluded middle ¬a ∨ ¬¬a do not have constructive proofs and are not valid in intuitionistic logic. Not surprisingly, constructivism has its costs. Whereas the validity problem is coNP-complete for classical propositional logic [6], for intuitionistic propositional logic it is PSPACE-complete [24,25]. The computational hardness of intuitionistic logic is already reached with the fragment that has only formulas with two variables: the validity problem for this fragment is already PSPACE-complete [22]. Recall that every fragment of classical propositional logic with a fixed number of variables has an NC 1 -complete validity problem (follows from [2]).The most common semantics for intuitionistic logic are Heyting semantics [11] and Kripke semantics [14, 13]-see also [23, Chap. 2]. The Heyting semantics bases on Heyting algebras, and Kripke semantics bases on directed graphs that can straightforwardly be adapted to model state-transition systems. Therefore it is used as the standard semantics for modal and hybrid logics. Model checking as we do with Kripke models is not suited for Heyting algebras. Therefore we also use Kripke semantics for intuitionistic logic. All our and all mentioned complexity results below refer to Kripke semantics.In this paper, we consider the complexity of intuitionistic propositional logic IPC with one variable. The model checking problem-i.e. the problem to determine whether a given formula is satisfied by a given intuitionistic Kripke model-for IPC is P-complete [17], even for the fragment with two variables only [18]. More surprisingly, for the fragment with one variable IPC 1 we show the model checking problem to be AC 1 -complete. To our knowledge, this is the first "natural" AC 1 -complete problem, whereas formerly known AC 1 -complete problems (see e.g. [1]) have some explicit logarithmic bound in the problem definition. A basic ingredient for the AC 1 -completeness lies in normal forms for models and formulas as found by Nishimura [20], that we

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