Complexity of Strict Implication

The aim of the present paper is to analyze the complexity of strict implication (together with falsum, conjunction and disjunction). We prove that Ladner’s Theorem remains valid when we restrict the language to the strict implication fragment, and that the same holds for Hemaspaandra’s Theorem. As a consequence we have that the validity problem for most standard normal modal logics is the same one than the validity problem for its strict implication fragment. We also prove that the validity problems for Visser’s basic propositional logic and Visser’s formal propositional logic are PSPACE complete. Finally, a polynomial reduction from most standard normal modal logics into its strict implication fragment is presented. Strict implication is defined in the modal language as φ0→φ1 := (φ0⊃φ1) strict implication, where ⊃ refers to material implication. Strict implication was already considered by Lewis in the birth of modal logic. Nevertheless, there is almost no complexity result in the literature for proper fragments of the modal language that are based on strict implication. One of the few exceptions is intuitionistic propositional logic [22]. On the other hand, there are a lot of complexity results for logics in the modal language. Two of the most outstanding results are Ladner’s Theorem [15, Theorem 3.1] and Hemaspaandra’s Theorem [21, 13]. They state, respectively, that all modal subsystems of S4 have a PSPACE hard validity problem and that all modal extensions of S4.3 have a co-NP complete validity problem. In this paper we analyze the complexity of the language based on strict implication (→) together with falsum (⊥), conjunction (∧) and disjunction (∨). We notice that true (>) is easily definable in this language, while neither material implication (⊃) nor classical negation (∼) are definable. The main results of the paper say that Hemaspaandra’s Theorem and Ladner’s Theorem also hold when we consider the restriction to the strict implication fragment. Hence, the complexity of the validity problem for most standard Advances in Modal Logic, Volume 5. c © 2005, Félix Bou.