Towards Correctness of Program Transformations Through Unification and Critical Pair Computation

Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.Comment: In Proceedings UNIF 2010, arXiv:1012.455

A Logic-Style Version of Interactive Proofs

Interactive proofs are defined in terms of a conversation between two Turing machines, the prover and the verifier. We define an equivalent kind of proof in terms more usual for logic: our proofs are derivations from axioms by rules of inference. Namely, we consider proofs in formal arithmetic extended by some additional rule that uses random numbers. Such a proof can be considerably shorter than any proof of the same formula in arithmetic (however, a proof in the randomized system is allowed to have a small probability of error). This kind of proof is a special case of Arthur-Merlin proofs: Merlin generates a proof, Arthur supplies random numbers and checks the proof. On the other hand, we show that if a language L belongs to PSPACE then membership in L has polynomially long proofs in our system. This result, together with the well-known equality IP =PSPACE, shows equivalence in power between interactive proofs and polynomially long proofs in our extension of arithmetic. y y Copyri..

MAX-SAT for formulas with constant clause density can be solved faster than in O(2 n ) time

Abstract. We give an exact deterministic algorithm for MAX-SAT. On input CNF formulas with constant clause density (the ratio of the number of clauses to the number of variables is a constant), this algorithm runs in O(c n) time where c < 2 and n is the number of variables. Worst-case upper bounds for MAX-SAT less than O(2 n) were previously known only for k-CNF formulas and for CNF formulas with small clause density.

On moderately exponential time for SAT

Abstract. Can sat be solved in "moderately exponential" time, i.e., in time p(|F |) 2 cn for some polynomial p and some constant c < 1, where F is a CNF formula of size |F | over n variables? This challenging question is far from being resolved. In this paper, we relate the question of moderately exponential complexity of sat to the question of moderately exponential complexity of problems defined by existential second-order sentences. Namely, we extend the class SNP (Strict NP) that consists of Boolean queries defined by existential second-order sentences where the first-order part has a universal prefix. The extension is obtained by allowing a ∀ . . . ∀ ∃ . . . ∃ prefix in the first-order part. We prove that if sat can be solved in moderately exponential time then all problems in the extended class can also be solved in moderately exponential time

Complexity of Query Answering in Logic Databases with Complex Values

This paper characterizes the computational complexity of nonrecursive queries in logic databases with complex values. Queries are represented by Horn clause logic programs. Complex values are represented by terms in equational theories (finite sets and multisets are examples of such complex values). We show that the problem of whether a query has a nonempty answer is NEXP-hard for nonrecursive range-restricted queries. We also show that this problem is in NEXP if complex values satisfy the following condition: the solvability problem for equations in the corresponding equational theory is in NP. Since trees, finite sets and multisets satisfy this condition, the query answering problem for logic databases with trees, finite sets and multisets is shown to be NEXP-complete. 2 2 Copyright c fl 1997, 1998 Evgeni Dantsin and Andrei Voronkov. This technical report and other technical reports in this series can be obtained at http://www.csd.uu.se/papers/reports.html or at ftp.csd.uu.se in th..