Tautology testing with a generalized matrix reduction method

AbstractA formalization of the tautology problem in terms of matrices is given. From that a generalized matrix reduction method is derived. Its application to a couple of selected examples indicates a relatively efficient behaviour in testing the validity of a given formula in propositional logic—not only for machines but also for humans. A further result from that formalization is a reduction of the tautology problem to a part of Presburger arithmetic which involves formulas of the ∀∃∃⋯∃-type where all quantifiers have finite range

Comments on “tautology testing with a generalized matrix reduction method”

AbstractBibel [1] has given a proof system for the propositional calculus called (generalized) matrix reduction. When matrix splitting is restricted to one literal at a time the system is the same as Galil's system [2] of enumeration dags. In fact the relation is even closer. The matrices produced by the reduction on a set of literals {I} are exactly the set of clauses appearing on a dag after |I| consecutive branches with substitute for the same literals. The clauses M1 (which do not appear in the matrices Mc) are exactly the clauses whose branches close with the empty clause Λ. Thus the saving in space is at most by a factor of |I|, but |I| is bounded from above by log2 M to ‘guarantee polynomial behaviour’. Hence Galil's system polynomially simulates matrix reduction and thus matrix reduction is also an exponential proof procedure

Circuit complexity, proof complexity, and polynomial identity testing

We introduce a new algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP is not equal to VP). As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complexity. More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size circuits of depth d - a notoriously open question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity

A theory of resolution

We review the fundamental resolution-based methods for first-order theorem proving and present them in a uniform framework. We show that these calculi can be viewed as specializations of non-clausal resolution with simplification. Simplification techniques are justified with the help of a rather general notion of redundancy for inferences. As simplification and other techniques for the elimination of redundancy are indispensable for an acceptable behaviour of any practical theorem prover this work is the first uniform treatment of resolution-like techniques in which the avoidance of redundant computations attains the attention it deserves. In many cases our presentation of a resolution method will indicate new ways of how to improve the method over what was known previously. We also give answers to several open problems in the area

KNOWLEDGE REPRESENTATION AND INFERENCE FOR ANALYSIS AND DESIGN OF DATABASES AND TABULAR RULE-BASED SYSTEMS

Rulc-based Systems constitute a powerful tool for speciftcation of knowledge in design and implementation of knowledge-based Systems. They provide also a universal programming paradigm for domains such as intelligent control, decision support, situation classification and opcrational knowledge encoding. In order to assure safe and reliable performance, such Systems should satisfy certain format reąuirements, including completeness and consistency. This paper addresses the issue of analysis and verification of selected properties of a class of such Systems in a systematic way. A uniform, tabular scheme of single-levcl rule-bascd Systems is considered. Such systcms can be applied as a generalized form of databases for speciftcation of data pattems (unconditional knowledge), or can be used for deftning attributive decision tables (conditional knowledge in form of rules). They can also serve as lower-level componcnts of a hierarchical, multi-lcvcl control and decision support knowledge-based systcms. An algebraic knowledge rcprescntation paradigm using extcnded tabular rcprcsentation, similar to relational databasc tables is prcsentcd and algebraic bascs for system analysis, vcrification and design support arc outlined

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Quantum Algorithm for Variant Maximum Satisfiability

In this paper, we proposed a novel quantum algorithm for the maximum satisfiability problem. Satisfiability (SAT) is to find the set of assignment values of input variables for the given Boolean function that evaluates this function as TRUE or prove that such satisfying values do not exist. For a POS SAT problem, we proposed a novel quantum algorithm for the maximum satisfiability (MAX-SAT), which returns the maximum number of OR terms that are satisfied for the SAT-unsatisfiable function, providing us with information on how far the given Boolean function is from the SAT satisfaction. We used Grover’s algorithm with a new block called quantum counter in the oracle circuit. The proposed circuit can be adapted for various forms of satisfiability expressions and several satisfiability-like problems. Using the quantum counter and mirrors for SAT terms reduces the need for ancilla qubits and realizes a large Toffoli gate that is then not needed. Our circuit reduces the number of ancilla qubits for the terms T of the Boolean function from T of ancilla qubits to ≈⌈log2⁡T⌉+1. We analyzed and compared the quantum cost of the traditional oracle design with our design which gives a low quantum cost