Computing Interpolants without Proofs

Our terminology and notation is the same as that of [1], of which this note is a sequel.We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

Space Bounds for Resolution

On the hydra number of disconnected graphs

AI systems often need to deal with inconsistent information. For this reason since the early 2000s some AI researchers have developed ways to measure the amount of inconsistency in a knowledge base. By now there is a substantial amount of research about various aspects of inconsistency measuring. The problem is that most of this work applies only to knowledge bases formulated as sets of formulas in propositional logic. Hence this work is not really applicable to the way that information is actually stored. The purpose of this paper is to extend inconsistency measuring to real world information. We first define the concept of general information space which encompasses various types of databases and scenarios in AI systems. Then, we show how to transform any general information space to an inconsistency equivalent propositional knowledge base, and finally apply propositional inconsistency measures to find the inconsistency of the general information space. Our method allows for the direct comparison of the inconsistency of different information spaces, even though the data is presented in different ways. We demonstrate the transformation on three general information spaces: a relational database, a graph database, and a Blocks world scenario, where we apply several inconsistency measures after performing the transformation.

Cut formulas in propositional logic

The topic of this thesis is the development of SAT-based techniques and tools for solving industrial combinatorial problems. First, it describes the architecture of state-of-the-art SAT and SMT Solvers based on the classical DPLL procedure. These systems can be used as black boxes for solving combinatorial problems. However, sometimes we can increase their efficiency with slight modifications of the basic algorithm. Therefore, the study and development of techniques for adjusting SAT Solvers to specific combinatorial problems is the first goal of this thesis. Namely, SAT Solvers can only deal with propositional logic. For solving general combinatorial problems, two different approaches are possible: - Reducing the complex constraints into propositional clauses. - Enriching the SAT Solver language. The first approach corresponds to encoding the constraint into SAT. The second one corresponds to using propagators, the basis for SMT Solvers. Regarding the first approach, in this document we improve the encoding of two of the most important combinatorial constraints: cardinality constraints and pseudo-Boolean constraints. After that, we present a new mixed approach, called lazy decomposition, which combines the advantages of encodings and propagators. The other part of the thesis uses these theoretical improvements in industrial combinatorial problems. We give a method for efficiently scheduling some professional sport leagues with SAT. The results are promising and show that a SAT approach is valid for these problems. However, the chaotical behavior of CDCL-based SAT Solvers due to VSIDS heuristics makes it difficult to obtain a similar solution for two similar problems. This may be inconvenient in real-world problems, since a user expects similar solutions when it makes slight modifications to the problem specification. In order to overcome this limitation, we have studied and solved the close solution problem, i.e., the problem of quickly finding a close solution when a similar problem is considered.

This dissertation is concerned with the application of notions and methods from the field of information theory to the field of communication complexity. It consists of two main parts. In the first part of the dissertation, we prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond to variables, and the internal nodes are labeled by binary connectives. Under certain assumptions, this representation is unique. Thus, one can define the depth of a formula as the depth of the tree that represents it. The complexity of the evaluation of general read-once formulae has attracted interest mainly in the decision tree model. In the communication complexity model many interesting results deal with specific read-once formulae, such as disjointness and tribes. In this dissertation we use information theory methods to prove lower bounds that hold for any read-once formula. Our lower bounds are of the form n(f )/c^d(f ) , where n(f ) is the number of variables and d(f ) is the depth of the formula, and they are optimal up to the constant in the base of the denominator. In the second part of the dissertation, we explore the applicability of the information-theoretic method in the number-on-the-forehead model. The work of Bar-Yossef, Jayram, Kumar & Sivakumar [BYJKS04] revealed a beautiful connection between Hellinger distance and two-party randomized communication protocols. Inspired by their work and motivated by the open questions in the number-on-the-forehead model, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party protocols. We provide a small toolbox that allows one to manipulate several Hellinger volume terms and also to lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. Finally, we show how to apply the new tools to obtain a lower bound on the informational complexity of the ANDk function.

This paper shows that idempotents in finite rings of integers can act as Generalized Boolean Assignments (GBA's) by providing a completeness theorem. We introduce the notion of a generic Generalized Boolean Assignment. The mere propagation of such an assignment reveals feasibility (existence of a solution) of a formula in propositional logic. Then, we demystify this general concept by formulating the process on the bit-level: It turns out that propagation of a GBA only simulates bitwise (non-communicating) parallel computing. We capitalize on this by modifying the state-of-the-art local search Sat solver UnitWalk accordingly. This modification involves a more complicated parallelism.

In this paper we describe a teaching tool for proving equivalences between propositional logic formulae, using rewrite rules such as De Morgan’s laws and double negation. This tool is based on an earlier tool for rewriting logical formulae into disjunctive normal form (DNF). Both tools make use of a rewrite strategy, which specifies how an exercise can be solved stepwise. Different types of feedback can be calculated automatically from such a strategy specification. We describe a strategy for constructing expert-like equivalence proofs, and present two techniques for improving the proofs that are generated by the strategy.Keywordspropositional logicequivalencese-learningfeedback

We prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond to variables, and the internal nodes are labeled by binary connectives. Under certain assumptions, this representation is unique. Thus, one can define the depth of a formula as the depth of the tree that represents it. The complexity of the evaluation of general read-once formulae, has attracted interest mainly in the decision tree model. In the communication complexity model, many interesting results deal with specific read-once formulae, such as DISJOINTNESS and TRIBES. In this paper we use information theory methods to prove lower bounds that hold for any read-once formula. Our lower bounds are of the form n(f)/c^{d(f)}, where n(f) is the number of variables and d(f) the depth of the formula, and they are optimal up to the constant c in the denominator.

Tautology testing with a generalized matrix reduction method

A catalog of Boolean concepts

DIMACS Series in Discrete Mathematicsand Theoretical Computer ScienceVolume00, 19xxA Branch and Cut Algorithm for MAX-SAT and WeightedMAX-SATSteveJoy, John Mitchell, and Brian BorchersAbstract.We describ e a branch and cut algorithm for b oth MAX-SAT andweighted MAX-SAT. This algorithm uses the GSAT pro cedure as a primalheuristic. At eachnode we solve a linear programming (LP) relaxation of theproblem. Twostyles of separating cuts are added: resolution cuts and o ddcycle inequalities.We compare our algorithm to an extension of the Davis Putnam Love-land (EDPL) algorithm. Our algorithm is more e ective than EDPL on someproblems, notably MAX-2-SAT. EDPL is more e ective on some other classesof problems.1. Intro ductionThe satis ability problem (SAT) is a problem in prop ositional logic. A logicformula consists of the conjunction of clauses. Each clause consists of a disjunctionof literals. Each literal is a variable or its negation. The SAT problem seeks to ndan assignment to the variables which satis es the logic formula, or an indicationthat no such assignment exists. The satis ability problem is NP-complete [5].There are a numb er of exact algorithms for the satis ability problem. Theseinclude Davis-Putnam-Loveland [4,23], resolution [26], and integer programmingapproaches [1,16, 18, 20,21]. A numb er of heuristics that use randomizationalso exist; the rst randomized lo cal search algorithm for satis abilitywas due toGu [9, 10, 11, 12]. Other algorithms include the GSAT heuristic [29, 30] and theGRASP heuristic [25]. For surveys of algorithms for SAT problems see [13,14].In this pap er weinvestigate the related MAX-SAT problem. Given a collectionof clauses, we seek a variable assignment that maximizes the numb er of satis edclauses. The weighted MAX-SAT problem assigns a wt to each clause, and seeksan assignment that maximizes the sum of the weights of the satis ed clauses. Bothof these problems are NP-hard. It is p ossible to approximate MAX-SAT within afactor of 1.325 in p olynomial time [6].Most SAT heuristics have b een extended to MAX-SAT. Several heuristics forMAX-SAT are summarized in Hansen and Jaumard [15]. The GSAT heuristic hasalso b een extended to weighted MAX-SAT[22].1991Mathematics Subject Classi cation.03B05, 49M35, 65K05, 90C10.Research supp orted by ONR Grant N00014{94{1{0391 to Rensselaer Polytechnic Institute.c0000 American Mathematical So ciety1052-1798/00 $1.00 + $.25 p er page1

The article considers non-standard semantics for syllogistic systems. In these systems, the formulas of propositional logic are the meanings of general terms, and the values of categorical statements are determined by the logical relationship between these formulas. If formula A is the meaning of term S, and if formula B is the meaning of term P, then the formula “SaP” is valid if and only if A entails B. Otherwise, the formula of the form “SoP” is valid. The “SeP” formula is valid if and only if A entails ¬B, otherwise the “SiP” formula is valid. This semantics is based on the intensional approach to the construction of syllogistic theories that goes back to Leibniz’s ideas. Firstly, the entailment relation by which categorical statements are interpreted allows, as Voishvillo has shown, an intensional interpretation: A entails B if the logical content of B is a part of the logical content of A. Secondly, propositional formulas can be considered as abbreviations for quantifier-free first-degree formulas with a single free variable, and such formulas, according to Voishvillo, state the contents of concepts. Therefore, in this semantics, the value of a syllogistic formula is determined not by the relationship between the extensions of the subject and the predicate, but by the relationship between their intensions. This article shows how different types of entailment affect the class of universally valid syllogistic formulas. Theories that use the relations of classical entailment and FDE-entailment have already been constructed. The classical entailment can be defined on the set of Carnap-style (consistent and complete) state-descriptions, and the first-degree relevant entailment can be defined on the set of generalized state-descriptions (the latter can be either contradictory or incomplete). In the article, semantics using two other relations: the Hao Wang-style entailment logic (defined on the set of all consistent state-descriptions) and its dual logic entailment (defined on the set of all complete state-descriptions) were built. The deductive capabilities of four syllogistic systems that use different types of logical entailments were compared.

Reaction systems are a qualitative formalism for modeling systems of biochemical reactions characterized by the non-permanency of the elements: molecules disappear if not produced by any enabled reaction. Reaction systems execute in an environment that provides new molecules at each step. Brijder, Ehrenfeucht and Rozemberg introduced the idea of predictors. A predictor of a molecule s, for a given n, is the set of molecules to be observed in the environment to determine whether s is produced or not at step n by the system. We introduced the notion of formula based predictor, that is a propositional logic formula that precisely characterizes environments that lead to the production of s after n steps. In this paper we revise the notion of formula based predictor by defining a specialized version that assumes the environment to provide molecules according to what expressed by a temporal logic formula. As an application, we use specialized formula based predictors to give theoretical grounds to previously obtained results on a model of gene regulation.