On goal-directed provability in classical logic

One of the key features of logic programming is the notion of goal-directed provability. In intuitionistic logic, the notion of uniform proof has been used as a proof-theoretic characterization of this property. Whilst the connections between intuitionistic logic and computation are well known, there is no reason per se why a similar notion cannot be given in classical logic. In this paper we show that there are two notions of goal-directed proof in classical logic, both of which are suitably weaker than that for intuitionistic logic. We show the completeness of this class of proofs for certain fragments, which thus form logic programming languages. As there are more possible variations on the notion of goal-directed provability in classical logic, there is a greater diversity of classical logic programming languages than intuitionistic ones. In particular, we show how logic programs may contain disjunctions in this setting. This provides a proof-theoretic basis for disjunctive logic programs, as well as characterising the “disjunctive” nature of answer substitutions for such programs in terms of the provability properties of the classical connectives Λ and Λ.     

A non-interactive deniable authentication scheme based on designated verifier proofs

A deniable authentication protocol enables a receiver to identify the source of the given messages but unable to prove to a third party the identity of the sender. In recent years, several non-interactive deniable authentication schemes have been proposed in order to enhance efficiency. In this paper, we propose a security model for non-interactive deniable authentication schemes. Then a non-interactive deniable authentication scheme is presented based on designated verifier proofs. Furthermore, we prove the security of our scheme under the DDH assumption.     

Equational tree transformations

We define an equational relation as the union of some components of the least solution of a system of equations of tree transformations in a pair of algebras. We focus on equational tree transformations which are equational relations obtained by considering the least solutions of such systems in pairs of term algebras. We characterize equational tree transformations in terms of tree transformations defined by different bimorphisms. To demonstrate the robustness of equational tree transformations, we give equational definitions of some well-known tree transformation classes for which bimorphism characterizations also exist. These are the class of alphabetic tree transformations, the class of linear and nondeleting extended top-down tree transformations, and the class of bottom-up tree transformations and its linear and linear and nondeleting subclasses. Finally, we prove that a relation is equational if and only if it is the morphic image of an equational tree transformation.     

Light Dialectica Program Extraction from a Classical Fibonacci Proof

We demonstrate program extraction by the Light Dialectica Interpretation (LDI) on a minimal logic proof of the classical existence of Fibonacci numbers. This semi-classical proof is available in MinLog's library of examples. The term of Gödel's T extracted by the LDI is, after strong normalization, exactly the usual recursive algorithm which defines the Fibonacci numbers (in pairs). This outcome of the Light Dialectica meta-algorithm is much better than the T-program extracted by means of the pure Gödel Dialectica Interpretation. It is also strictly less complex than the result obtained by means of the refined A-translation technique of Berger, Buchholz and Schwichtenberg on an artificially distorted variant of the input proof, but otherwise it is identical with the term yielded by Berger's Kripke-style refined A-translation. Although syntactically different, it also has the same computational complexity as the original program yielded by the refined A-translation from the undistorted input classical Fibonacci proof.     

Formalizing Strong Normalization Proofs of Explicit Substitution Calculi in ALF

The past decade has given rise to a number of explicit substitution calculi. An important question of explicit substitution calculi is that of the termination of the underlying calculus of substitution. Proofs of termination of substitutions fall in two categories: those that are easy because a decreasing measure can be established and those that are difficult because such a decreasing measure is not easy to establish. This paper considers two styles of explicit substitution: and s, for which different termination proof methods apply. The termination of s is guaranteed by a decreasing weight, while a decreasing weight for showing the termination of has not yet been found. These termination methods for and s are formalized in the proof checker ALF. During our process of formally checking the termination of and s we comment on what is needed to make a proof formally checkable.     

Reference Constructions in the Single-conclusion Proof Logic

We propose an extension of the propositional proof logic languageby the second-order variables denoting the reference constructors(like ‘the formula which is proven by x’). The prooflogic in this weak second-order language is axiomatized viathe calculus _ref, the (Functional)Logic of Proofs with References. It is supplied with the formalarithmetical semantics: we prove that _ref is sound with respect to arithmetical interpretationsand is a conservative extension of propositional single-conclusionproof logic . Finally,we demonstrate how the language of _ref can be used as a scheme language for arithmetic and providethe corresponding proof conversion method.     

Machine-Checkable Correctness Proofs for Intra-procedural Dataflow Analyses

This paper describes our experience using the interactive theorem prover Athena for proving the correctness of abstract interpretation-based dataflow analyses. For each analysis, our methodology requires the analysis designer to formally specify the property lattice, the transfer functions, and the desired modeling relation between the concrete program states and the results computed by the analysis. The goal of the correctness proof is to prove that the desired modeling relation holds. The proof allows the analysis clients to rely on the modeling relation for their own correctness. To reduce the complexity of the proofs, we separate the proof of each dataflow analysis into two parts: a generic part, proven once, independent of any specific analysis; and several analysis-specific conditions proven in Athena.     

Resolution Proofs of Matching Principles

This paper discusses current techniques for proving lower bounds on the size of resolution proofs from sets of clauses expressing assertions about matchings in bipartite graphs (a class including the well-known pigeon-hole clauses). The techniques are illustrated by demonstrating an improved lower bound for some examples discussed by Kleine Büning. The final section discusses some problems where current techniques appear inadequate, and new ideas seem to be required.