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This paper introduces Graded Computation Tree Logic with finite path semantics (GCTL f ⁎ , for short), a variant of Computation Tree Logic CTL⁎, in which path quantifiers are interpreted over finite paths and can count the number of such paths. State formulas of GCTL f ⁎ are interpreted over Kripke structures. The syntax of GCTL f ⁎ has path quantifiers of the form E ≥ g ψ which express that there are at least g many distinct finite paths that satisfy ψ. After defining and justifying the logic GCTL f ⁎ , we solve its model checking problem and establish that its computational complexity is PSPACE-complete. Moreover, we investigate GCTL f ⁎ under the imperfect information setting. Precisely, we introduce GCTLK f ⁎ , an epistemic extension of GCTL f ⁎ and prove that the model checking problem also in this case is PSPACE-complete. [ABSTRACT FROM AUTHOR]

Abstract: Commutative, integral and bounded -algebras form a subvariety of residuated lattices which provides the algebraic semantics of an interesting common fragment of intuitionistic logic and of several fuzzy logics. It is known that both the equational theory and the quasiequational theory of commutative -algebras are decidable (in contrast to the noncommutative case), but their complexity has not been studied yet. In this paper, we prove that both theories are in , and that the quasiequational theory is -hard. [Copyright &y& Elsevier]

Abstract: Periodicity constraints are used in many logical formalisms, in fragments of Presburger LTL, in calendar logics, and in logics for access control, to quote a few examples. In the paper, we introduce the logic , an extension of Linear-Time Temporal Logic LTL with past-time operators whose atomic formulae are defined from a first-order constraint language dealing with periodicity. Although the underlying constraint language is a fragment of Presburger arithmetic shown to admit a PSPACE-complete satisfiability problem, we establish that model-checking and satisfiability problems remain in PSPACE as plain LTL (full Presburger LTL is known to be highly undecidable). This is particularly interesting for dealing with periodicity constraints since the language of has a language more concise than existing languages and the temporalization of our first-order language of periodicity constraints has the same worst case complexity as the underlying constraint language. Finally, we show examples of introduction the quantification in the logical language that provide to , EXPSPACE-complete problems. As another application, we establish that the equivalence problem for extended single-string automata, known to express the equality of time granularities, is PSPACE-complete by designing a reduction from QBF and by using our results for . [Copyright &y& Elsevier]