On natural deduction in fixpoint logics

In the current paper we present a powerful technique of obtaining natural deduction (or, in other words, Gentzen-like) proof systems for first-order fixpoint logics. The term "fixpoint logics" refers collectively to a class of logics consisting of modal logics with modalities definable at meta-level by fixpoint equations on formulas. The class was found very interesting as it contains most logics of programs with e.g. dynamic logic, temporal logic and, of course, mu-calculus among them. Fixpoint logics were intensively studied during the last decade. In this paper we are going to present some results concerning deductive systems for first-order fixpoint logics. In particular we shall present some powerful and general technique for obtaining natural deduction (Gentzen-like) systems for fixpoint logics. As those logics are usually totally undecidable, we show how to obtain complete (but infinitary) proof systems as well as relatively complete (finitistic) ones. More precisely, given fixpoint equations on formulas defining nonclassical connectives of a logic, we automatically derive Gentzen-like proof systems for the logic. The discussion of implementation problems is also provided

Searching for Invariants using Temporal Resolution

Abstract. In this paper, we show how the clausal temporal resolution technique developed for temporal logic provides an effective method for searching for invariants, and so is suitable for mechanising a wide class of temporal problems. We demonstrate that this scheme of searching for invariants can be also applied to a class of multi-predicate induction problems represented by mutually recursive definitions. Completeness of the approach, examples of the application of the scheme, and overview of the implementation are described.

On correspondence between modal and classical logic: automated approach

The current paper is devoted to automated techniques in correspondence theory. The theory we deal with concerns the problem of finding classical first-order axioms corresponding to propositional modal schemas. Given a modal schema and a semantics based method of translating propositional modal formulas into classical first-order ones, we try to derive automatically classical first-order formula characterizing precisely the class of frames validating the schema. The technique we consider can, in many cases, be easily applied even without any computer support. Although we mainly concentrate on Kripke semantics, the technique we apply is much more general, as it is based on elimination of second-order quantifiers from formulas. We show many examples of application of the method. Those can also serve as new, automated proofs of considered correspondences. We essentially strengthen the considered elimination technique. Thus, as a side-effect of this paper we get a stronger elimination based method for proving a subset of second-order logic

Annotation theories over finite graphs

In the current paper we consider theories with vocabulary containing a num- ber of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph’s nodes. Such theories, which we call annotation theories , can be used in many applications, including the formalization of argumentation, approxim ate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPTime -or co-NPTime -complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of anno- tation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21]