The Complexity of Model Checking Higher-Order Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed \lambda-calculus and the modal \lambda-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal \lambda-calculus. This paper provides complexity results for its model checking problem. In particular we consider those fragments of HFL built by using only types of bounded order k and arity m. We establish k-fold exponential time completeness for model checking each such fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singly-exponential in the size of the formula and k-fold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence, we obtain an exponential time upper bound on the expression complexity of each such fragment. The lower bound is established by a reduction from the word problem for alternating (k-1)-fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are already hard with respect to k-fold exponential time, we obtain, as a corollary, k-fold exponential time completeness for the data complexity of our fragments of HFL, provided m exceeds 3. This also yields a hierarchy result in expressive power.Comment: 33 pages, 2 figures, to be published in Logical Methods in Computer Scienc

Applications of Intuitionistic Logic in Answer Set Programming

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.Comment: 30 pages, Under consideration for publication in Theory and Practice of Logic Programmin

Deciding Full Branching Time Logic by Program Transformation

We present a method based on logic program transformation, for verifying Computation Tree Logic (CTL*) properties of finite state reactive systems. The finite state systems and the CTL* properties we want to verify, are encoded as logic programs on infinite lists. Our verification method consists of two steps. In the first step we transform the logic program that encodes the given system and the given property, into a monadic ω -program, that is, a stratified program defining nullary or unary predicates on infinite lists. This transformation is performed by applying unfold/fold rules that preserve the perfect model of the initial program. In the second step we verify the property of interest by using a proof method for monadic ω-program

Proving theorems by program transformation

In this paper we present an overview of the unfold/fold proof method, a method for proving theorems about programs, based on program transformation. As a metalanguage for specifying programs and program properties we adopt constraint logic programming (CLP), and we present a set of transformation rules (including the familiar unfolding and folding rules) which preserve the semantics of CLP programs. Then, we show how program transformation strategies can be used, similarly to theorem proving tactics, for guiding the application of the transformation rules and inferring the properties to be proved. We work out three examples: (i) the proof of predicate equivalences, applied to the verification of equality between CCS processes, (ii) the proof of first order formulas via an extension of the quantifier elimination method, and (iii) the proof of temporal properties of infinite state concurrent systems, by using a transformation strategy that performs program specialization

Principles and Implementation of Deductive Parsing

We present a system for generating parsers based directly on the metaphor of parsing as deduction. Parsing algorithms can be represented directly as deduction systems, and a single deduction engine can interpret such deduction systems so as to implement the corresponding parser. The method generalizes easily to parsers for augmented phrase structure formalisms, such as definite-clause grammars and other logic grammar formalisms, and has been used for rapid prototyping of parsing algorithms for a variety of formalisms including variants of tree-adjoining grammars, categorial grammars, and lexicalized context-free grammars.Comment: 69 pages, includes full Prolog cod

Transformation-Based Bottom-Up Computation of the Well-Founded Model

We present a framework for expressing bottom-up algorithms to compute the well-founded model of non-disjunctive logic programs. Our method is based on the notion of conditional facts and elementary program transformations studied by Brass and Dix for disjunctive programs. However, even if we restrict their framework to nondisjunctive programs, their residual program can grow to exponential size, whereas for function-free programs our program remainder is always polynomial in the size of the extensional database (EDB). We show that particular orderings of our transformations (we call them strategies) correspond to well-known computational methods like the alternating fixpoint approach, the well-founded magic sets method and the magic alternating fixpoint procedure. However, due to the confluence of our calculi, we come up with computations of the well-founded model that are provably better than these methods. In contrast to other approaches, our transformation method treats magic set transformed programs correctly, i.e. it always computes a relevant part of the well-founded model of the original program.Comment: 43 pages, 3 figure