The global consequence relation of a normal modal logic \ is formulated as a global sequent calculus which extends the local sequent theory of \ with global sequent rules. All global sequent calculi of normal modal logics admits global cut elimination. This property is utilized to show that decidability is preserved from the local to global sequent theories of any normal modal logic over \. The preservation of Craig interpolation property from local to global sequent theories of any normal modal (...) logic is shown by proof-theoretic method. (shrink)

While specifications and verifications of concurrent systems employ Linear Temporal Logic , it is increasingly likely that logical consequence in image will be used in the description of computations and parallel reasoning. Our paper considers logical consequence in the standard image with temporal operations image and image . The prime result is an algorithm recognizing consecutions admissible in image, so we prove that image is decidable w.r.t. admissible inference rules. As a consequence we obtain algorithms verifying the validity of consecutions (...) in image and solving the satisfiability problem. We start by a simple reduction of logical consecutions of image to equivalent ones in the reduced normal form . Then we apply a semantic technique based on image-Kripke structures with formula definable subsets. This yields necessary and sufficient conditions for a consecution to be not admissible in image. These conditions lead to an algorithm which recognizes consecutions admissible in image by verifying the validity of consecutions in special finite Kripke structures of size square polynomial in reduced normal forms of the consecutions. As a consequence, this also solves the satisfiability problem for image. (shrink)

The paper studies the logic TL(NBox+-wC) – logic of discrete linear time with current time point clusters. Its language uses modalities Diamond+ (possible in future) and Diamond- (possible in past) and special temporal operations, – Box+w (weakly necessary in future) and Box-w (weakly necessary in past). We proceed by developing an algorithm recognizing theorems of TL(NBox+-wC), so we prove that TL(NBox+-wC) is decidable. The algorithm is based on reduction of formulas to inference rules and converting the rules in special reduced (...) normal form, and, then, on checking validity of such rules in models of singleexponential size in the rules. Also we consider the admissibility problem for TL(NBox+-wC) and show how to reduce the problem for admissibility to the decidability of TL(NBox+-wC) itself using definable universal modalities. (shrink)