Showing papers on "Conjunctive normal form published in 1987"

Learning Decision Lists

Abstract: This paper introduces a new representation for Boolean functions, called decision lists, and shows that they are efficiently learnable from examples. More precisely, this result is established for k-;DL – the set of decision lists with conjunctive clauses of size k at each decision. Since k-DL properly includes other well-known techniques for representing Boolean functions such as k-CNF (formulae in conjunctive normal form with at most k literals per clause), k-DNF (formulae in disjunctive normal form with at most k literals per term), and decision trees of depth k, our result strictly increases the set of functions that are known to be polynomially learnable, in the sense of Valiant (1984). Our proof is constructive: we present an algorithm that can efficiently construct an element of k-DL consistent with a given set of examples, if one exists.

Superlinear Speedup for Parallel Backtracking

Abstract: We have implemented a backtracking strategy for the satisfiability problem on a ring of processors and we observed a superlinear speedup in the average. In this paper we describe a model from which this superlinear speedup can be deduced. The model is based on the fact that in the average the solutions are distributed nonuniformly in the case of the satisfiability problem. To our knowledge this phenomenon was not used before in the analysis of algorithms.

Preserving consistency across abstraction mappings

Abstract: An abstraction mapping over clausal form theories in first-order predicate calculus is presented that involves the renaming of predicate symbols. This renaming is not 1-1, in the sense that several predicate symbols Ri,..., Rn from the original theory are all replaced by a single symbol R in the abstract theory. In order to preserve consistency, however, the clauses that distinguish the Ri's must be discarded in the abstract theory. This leads to a simple semantics; the union of the extensions of each of the Ri's in any model of the original theory forms the extension of R in a model of the abstract theory.

On the Computational Complexity of Quantified Horn Clauses

Abstract: A polynomial time algorithm is presented for the evaluation problem for quantified propositional Horn clauses. This answers an open problem posed by Itai and Makowski in (IM 87).

Link inheritance in abstract clause graphs

Abstract: Clause graphs, as they were defined in the 1970s, are graphs representing first order formulas in conjunctive normal form: the nodes are labelled with literals and the edges (links) connect complementary unifiable literals, i.e. they provide an explicit representation of the resolution possibilities. This report describes a generalization of this concept, called abstract clause graphs. The nodes of abstract clause graphs are still labelled with literals, the links however connect literals that are ‘unifiable’ relative to a given relation between literals. This relation is not explicitely defined, only certain abstract properties are required. For instance the existence of a special purpose unification algorithm is assumed, which computes substitutions, the application of which makes the relation hold for two literals. When instances of already existing literals are added to the graph (e.g. due to resolution or factoring), the links to the new literals are derived from the links of their ancestors. An inheritance mechanism for such links is presented which operates only on the attached substitutions and does not have to unify the literals. The definition of abstract clause graphs and the theory about link inheritance is general enough to provide a framework so that as new ideas are proposed for graph based theorem provers, the methodology for both implementing links and proving their properties will be readily available.

An Extended Disjunctive Normal Form Approach for Optimizing Recursive Logic Queries in Loosely Coupled Environments

Abstract: We present an approach to processing logic queries in loosely coupled environments. We emphasize the importance of the loose coupling technique as a practkal solution to provide deductive capabiities to existing DBMS+especially when an efficient access to a very large database is required in the. process of inferencing. We propose the Extended Disjunctive Normal Form (EDNF) as the basis of our approach. The EDNF is an extension of the disjunctive normal form of relational algebra expressions so as to include recursion. The EDNF is well suited for a loosely coupled environment, where an existing DBMS and optimiition can be fully exploited. It also serves as a clear, graphical characterixation of various recursions that can occur in logic queries. We first present the basic form of the EDNF and then use it as a building block to process a more general class of queries. We extend valid usage of Clark% negation-as-failure evaluation technique to incorporate negation for most practical situations. We also propose new criteria for safety and termination in the presence of negation. To the extent of the authors’ knowledge, optimixation in loosely coupled environments has not been seriously addressed in previous research. We believe our technique provides significant progress in this dhection.

Prolog and expert systems

Abstract: The first part of the thesis provides an introduction to the logic programming language Prolog and some areas of current research The use of compilation to make Prolog faster and more efficient is studied and a modified representation for complex structures is presented Two programming tools are also presented The second part of the thesis focuses on one problem which arises when implementing an Expert System using Prolog A practical three-valued Prolog implementation is described An interpreter accepts three-valued formulae and converts these into a Prolog representation Formulae are in clausal form which allows disjunctive conclusions to rules True and false formulae are stated explicitly and therefore the interpreter is able to perform useful consistency checks when information is added to the data base

On the Average Case Complexity of Backtracking for the Exact-Satisfiability Problem

Abstract: We analyse the average case performance of a simple backtracking algorithm for determining all exact-satisfying truth assignments of boolean formulas in conjunctive normal form with r clauses over n variables. A truth assignment exact-satisfies a formula, if in each clause exactly one literal is set to true. We show: If formulas are chosen by generating clauses independently, where each variable occurs in a clause either unnegated with probability p or negated with probability q or none of both with probability 1-p-q (p,q>0, p+q≦1), then the average number of nodes in the backtracking trees of formulas from these classes is bounded by a constant, for all neN, if r≧1n2/(pq) is chosen. (In case of p=q=1/3 the result holds for all r≧6.)

The Logical Work of Mordchaj Wajsberg

Abstract: In this paper Mordchaj Wajsberg’s life and research work in logic are described, and an attempt is made to situate the latter among the accomplishments of the rest of the Polish school of logic.

Algorithms for Propositional Updates

Abstract: We have introduced and analysed propositional updating. It turns out that there are a number of different solutions to that problem and the selection of a method may depend on the pragmatics of the application. As had to be expected computional problems are NP-hard. Heuristics may not be applied in this situation so we have developed algorithms having typical applications in mind. This will allow us to test the method for acceptance before refining the algorithms.

On Certain Bandwidth Restricted Versions of the Satisfaiability Problem of Propositional CNF Formulas

Abstract: We consider the complexity of certain restricted versions of the satisfiability problem of propositional formulas in conjunctive normal form. We define the corresponding languages in terms of kernel constructibility, where log-bandwidth violation is allowed in the corresponding kernels, but the constructing relations obey the bandwidth restriction