Showing papers on "Disjunctive normal form published in 2006"

Clause/term resolution and learning in the evaluation of quantified Boolean formulas

Abstract: Resolution is the rule of inference at the basis of most procedures for automated reasoning. In these procedures, the input formula is first translated into an equisatisfiable formula in conjunctive normal form (CNF) and then represented as a set of clauses. Deduction starts by inferring new clauses by resolution, and goes on until the empty clause is generated or satisfiability of the set of clauses is proven, e.g., because no new clauses can be generated. In this paper, we restrict our attention to the problem of evaluating Quantified Boolean Formulas (QBFs). In this setting, the above outlined deduction process is known to be sound and complete if given a formula in CNF and if a form of resolution, called "Q-resolution", is used. We introduce Q-resolution on terms, to be used for formulas in disjunctive normal form. We show that the computation performed by most of the available procedures for QBFs -based on the Davis-Logemann-Loveland procedure (DLL) for propositional satisfiability- corresponds to a tree in which Q-resolution on terms and clauses alternate. This poses the theoretical bases for the introduction of learning, corresponding to recording Q-resolution formulas associated with the nodes of the tree. We discuss the problems related to the introduction of learning in DLL based procedures, and present solutions extending state-of-the-art proposals coming from the literature on propositional satisfiability. Finally, we show that our DLL based solver extended with learning, performs significantly better on benchmarks used in the 2003 QBF solvers comparative evaluation.

Multiobjective evolutionary induction of subgroup discovery fuzzy rules: a case study in marketing

Abstract: This paper presents a multiobjective genetic algorithm which obtains fuzzy rules for subgroup discovery in disjunctive normal form. This kind of fuzzy rules lets us represent knowledge about patterns of interest in an explanatory and understandable form which can be used by the expert. The evolutionary algorithm follows a multiobjective approach in order to optimize in a suitable way the different quality measures used in this kind of problems. Experimental evaluation of the algorithm, applying it to a market problem studied in the University of Mondragon (Spain), shows the validity of the proposal. The application of the proposal to this problem allows us to obtain novel and valuable knowledge for the experts.

QBF modeling: exploiting player symmetry for simplicity and efficiency

Abstract: Quantified Boolean Formulas (QBFs) present the next big challenge for automated propositional reasoning. Not surprisingly, most of the present day QBF solvers are extensions of successful propositional satisfiability algorithms (SAT solvers). They directly integrate the lessons learned from SAT research, thus avoiding re-inventing the wheel. In particular, they use the standard conjunctive normal form (CNF) augmented with layers of variable quantification for modeling tasks as QBF. We argue that while CNF is well suited to “existential reasoning” as demonstrated by the success of modern SAT solvers, it is far from ideal for “universal reasoning” needed by QBF. The CNF restriction imposes an inherent asymmetry in QBF and artificially creates issues that have led to complex solutions, which, in retrospect, were unnecessary and sub-optimal. We take a step back and propose a new approach to QBF modeling based on a game-theoretic view of problems and on a dual CNF-DNF (disjunctive normal form) representation that treats the existential and universal parts of a problem symmetrically. It has several advantages: (1) it is generic, compact, and simpler, (2) unlike fully non-clausal encodings, it preserves the benefits of pure CNF and leverages the support for DNF already present in many QBF solvers, (3) it doesn't use the so-called indicator variables for conversion into CNF, thus circumventing the associated illegal search space issue, and (4) our QBF solver based on the dual encoding (Duaffle) consistently outperforms the best solvers by two orders of magnitude on a hard class of benchmarks, even without using standard learning techniques.

QBF modeling : Exploiting player symmetry for simplicity and efficiency

Abstract: Quantified Boolean Formulas (QBFs) present the next big challenge for automated propositional reasoning. Not surprisingly, most of the present day QBF solvers are extensions of successful propositional satisfiability algorithms (SAT solvers). They directly integrate the lessons learned from SAT research, thus avoiding re-inventing the wheel. In particular, they use the standard conjunctive normal form (CNF) augmented with layers of variable quantification for modeling tasks as QBF. We argue that while CNF is well suited to existential reasoning as demonstrated by the success of modern SAT solvers, it is far from ideal for universal reasoning needed by QBF. The CNF restriction imposes an inherent asymmetry in QBF and artificially creates issues that have led to complex solutions, which, in retrospect, were unnecessary and sub-optimal. We take a step back and propose a new approach to QBF modeling based on a game-theoretic view of problems and on a dual CNF-DNF (disjunctive normal form) representation that treats the existential and universal parts of a problem symmetrically. It has several advantages: (1) it is generic, compact, and simpler, (2) unlike fully non-clausal encodings, it preserves the benefits of pure CNF and leverages the support for DNF already present in many QBF solvers, (3) it doesn't use the so-called indicator variables for conversion into CNF, thus circumventing the associated illegal search space issue, and (4) our QBF solver based on the dual encoding (Duaffle) consistently outperforms the best solvers by two orders of magnitude on a hard class of benchmarks, even without using standard learning techniques.

Solving QBF with combined conjunctive and disjunctive normal form

Abstract: Similar to most state-of-the-art Boolean Satisfiabilily (SAT) solvers, all contemporary Quantified Boolean Formula (QBF) solvers require inputs to be in the Conjunctive Normal Form (CNF). Most of them also store the QBF in CNF internally for reasoning. In order to use these solvers, arbitrary Boolean formulas have to be transformed into equi-satisfiable formulas in Conjunctive Normal Form by introducing additional variables. In this paper, we point out an inherent limitation of this approach, namely the asymmetric treatment of satisfactions and conflicts. This deficiency leads to artificial increase of search space for QBF solving. To overcome the limitation, we propose to transform a Boolean formula into a combination of an equisatisfiable CNF formula and an equi-tautological DNF formula for QBF solving. QBF solvers based on this approach treat satisfactions and conflicts symmetrically, thus avoiding the exploration of unnecessary search space. A QBF solver called IQTest is implemented based on this idea. Exrerimental results show that it significantly outperforms existing QBF solvers.

Minimizing DNF formulas and AC/sup 0//sub d/ circuits given a truth table

Abstract: For circuit classes R, the fundamental computational problem Min-R asks for the minimum R-size of a Boolean function presented as a truth table. Prominent examples of this problem include Min-DNF, which asks whether a given Boolean function presented as a truth table has a k-term DNF, and Min-Circuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that Min-DNF is NP-complete. It is significantly simpler than the known reduction of Masek (1979), which is from Circuit-SAT. We then give a more complex reduction, yielding the result that Min-DNF cannot be approximated to within a factor smaller than (log N)/sup /spl Upsi//, for some constant /spl Upsi/ > 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for set cover is often used in practice to approximate Min-DNF. The question of whether Min-DNF can be approximated to within a factor of o(log N) remains open, but we construct an instance of Min-DNF on which the solution produced by the greedy algorithm is /spl Omega/(log N) larger than optimal. Finally, we extend known hardness results for Min-TC/sup 0//sub d/ to obtain new hardness results for Min-AC/sup 0//sub d/, under cryptographic assumptions.

An interactive tool for manipulating logical formulae

Abstract: Logic is constructive in nature, and in a course on logic a student learns how to manipulate logical formulas For example, a student has to learn how to simplify a logical formula, how to transform a logical formula into disjunctive normal form (DNF), and how to prove equivalences of logical formulae Solving logical exercises is often done with pen and paper, but e-learning tools offer great possibilities In particular for a distance learning university such as the Dutch Open University it is important to support the interactive construction of solutions to logical exercises Currently all exercises and solutions can be found in our lecture notes for the courses that teach logic

Analyzing and Extending MUMCUT for Fault-based Testing of General Boolean Expressions

Abstract: Boolean expressions are widely used to model decisions or conditions of a specification or source program. The MUMCUT, which is designed to detect seven common faults where Boolean expressions under test are assumed to be in Irredundant Disjunctive Normal Form (IDNF), is an efficient fault-based test case selection strategy in terms of the fault-detection capacity and the size of selected test suite. Following up our previous work that reported the fault-detection capacity of the MUMCUT when it is applied to general form Boolean expressions, in this paper we present the characteristic of the types of single faults committed in general Boolean expressions that a MUMCUT test suite fails to detect, analyze the certainty why a MUMCUT test suite fails to detect these types of undetected faults, and provide some extensions to enhance the detection capacity of the MUMCUT for these types of undetected faults.

Rational Acceptance and Conjunctive/Disjunctive Absorption

Abstract: A bounded formula is a pair consisting of a propositional formula ? in the first coordinate and a real number within the unit interval in the second coordinate, interpreted to express the lower-bound probability of ?. Converting conjunctive/disjunctive combinations of bounded formulas to a single bounded formula consisting of the conjunction/disjunction of the propositions occurring in the collection along with a newly calculated lower probability is called absorption. This paper introduces two inference rules for effecting conjunctive and disjunctive absorption and compares the resulting logical system, called System Y, to axiom System P. Finally, we demonstrate how absorption resolves the lottery paradox and the paradox of the preference.

A Genetic-Programming-Based Approach for the Learning of Compact Fuzzy Rule-Based Classification Systems

Abstract: In the design of an interpretable fuzzy rule-based classification system (FRBCS) the precision as much as the simplicity of the extracted knowledge must be considered as objectives. In any inductive learning algorithm, when we deal with problems with a large number of features, the exponential growth of the fuzzy rule search space makes the learning process more difficult. Moreover it leads to an FRBCS with a rule base with a high cardinality. In this paper, we propose a genetic-programming-based method for the learning of an FRBCS, where disjunctive normal form (DNF) rules compete and cooperate among themselves in order to obtain an understandable and compact set of fuzzy rules, which presents a good classification performance with high dimensionality problems. This proposal uses a token competition mechanism to maintain the diversity of the population. The good results obtained with several classification problems support our proposal.

A Genetic-Programming-Based Approach for the Learning of Compact Fuzzy Rule-Based Classification Systems

Abstract: In the design of an interpretable fuzzy rule-based classification system (FRBCS) the precision as much as the simplicity of the extracted knowledge must be considered as objectives. In any inductive learning algorithm, when we deal with problems with a large number of features, the exponential growth of the fuzzy rule search space makes the learning process more difficult. Moreover it leads to an FRBCS with a rule base with a high cardinality. In this paper, we propose a genetic-programming-based method for the learning of an FRBCS, where disjunctive normal form (DNF) rules compete and cooperate among themselves in order to obtain an understandable and compact set of fuzzy rules, which presents a good classification performance with high dimensionality problems. This proposal uses a token competition mechanism to maintain the diversity of the population. The good results obtained with several classification problems support our proposal.

Handling of satisfaction and conflicts in a quantified Boolean formula solver

Abstract: In order to provide for more efficient QBF satisfiability determination, the formula to be checked is transformed into one formula which is equi-satisfiable, and one which is equi-tautological. The conjunction or disjunction of these two formulas, then, is used to determine satisfiability, with the result being that a determination of satisfiability is more easily achieved. A conjunctive normal form transformation of the initial formula yields a group of clauses, only one of which must be unsatisfiable for the formula to be unsatisfiable. A disjunctive normal form transformation of the initial formula yields a group of cubes, only one of which must be satisfiable in order for the formula to be determined to be satisfiable.

Disjunctive Normal Form Generation Algorithm for Discernibility Function in Rough Set Theory

Abstract: The automatic generation algorithm of discernibility function ’s disjunctive normal form(DNF) is studied based on rough set theory. The tansformation method form discernibility matrix to conjunction matrix is presented. And the mathimetic model of the transformation from conjunction matrix to disjunction matrix is esteblished for the attribute reduct, then the program flow graph is also put forward. Based on the model, direct search method is presented which can save the computing space and CPU occupation time, enhance the efficiency of decision rules generation. Finally, by the factual process for UCI database, the validity of direct serach method is proved.

Negative Ordered Hyper-Resolution as a Proof Procedure for Disjunctive Logic Programming

Abstract: We prove that negative hyper-resolution using any liftable and well-founded ordering refinement is a sound and complete procedure for answering queries in disjunctive logic programs. In our formulation, answers of queries are defined using disjunctive substitutions, which are more flexible than answer literals used in theorem proving systems.

Molecular learning of wDNF formulae

Abstract: We introduce a class of generalized DNF formulae called wDNF or weighted disjunctive normal form, and present a molecular algorithm that learns a wDNF formula from training examples. Realized in DNA molecules, the wDNF machines have a natural probabilistic semantics, allowing for their application beyond the pure Boolean logical structure of the standard DNF to real-life problems with uncertainty. The potential of the molecular wDNF machines is evaluated on real-life genomics data in simulation. Our empirical results suggest the possibility of building error-resilient molecular computers that are able to learn from data, potentially from wet DNA data.

Upper and Lower Bounds on the Number of Disjunctive Forms

Abstract: We evaluate the upper and lower bounds on the number of disjunctive (normal) forms of an n-variable Boolean function (for our purpose it is sufficient to take the constant 1 function which always takes the value 1). We use a one-to-one correspondence between the disjunctive forms and the antichains in the ternary n-cube which is isomorphic to the partially ordered set formed by all terms of the given function. For the upper bound we use a newly invented decomposition of the partially ordered set into chains (we introduce trees which span the cube). For the lower bounds, we evaluate the number of anticains in the cube by analyzing the dependency among the three consecutive layers instead of two. Put DF(1) the number of different disjunctive forms for the constant 1 function. We obtain newly improved upper and lower bounds.

The Representational Power of Conjunctive Normal Form.

Abstract: There is continuing research interest in comparison of the complexity of problems within the class NP-Complete. This paper examines the representational power of conjunctive normal form Boolean expressions to establish a proper hierarchy for finite languages, where the language of an expression is defined to be the set of bit strings corresponding to its satisfying assignments. The increasing representational complexity parallels and perhaps explains the increasing complexity of k-SAT algorithms for higher values of k. The hierarchy provides a general framework for comparisons of satisfiability algorithms and leads to the conclusion that logical resolution is not a good algorithm for satisfiabiltiy testing.

A study of fuzzy and many-valued logics in cellular automata

Abstract: In this paper we provide an analytical study of the theory of multi-valued and fuzzy cellular automata where the fuzziness appears as the result of the application of an underlying multi-valued or continuous logic as opposed to standard logic as used conventionally. Using the disjunctive normal form of any one of the 255 ECA's so defined, we modify the underlying logic structure and redefine the ECA within the framework of this new logic. The idea here is to show that the evolution of space-time diagrams of ECA's under even a probabilistic logic can exhibit non-chaotic behavior. This is looked at specifically for Probabilistic Rule 110, in contrast with Boolean Rule 110 which is known to be capable of universal computation.

Chapter 7 – EQUIVALENCES IN TWO-VALUED LOGIC

Abstract: Two-valued logic schema can be constructed either on the two valued set descriptions or on the fuzzy-valued set descriptions. This chapter reviews the two-valued set and logic formulas for the combination of two linguistic concepts A and B. In this theory, the descriptive membership values A(x) is equal to A and B(x) is equal to b, a, b for all {0,1}, corresponding to the set symbols A and B and the truth assignments {T,F} of the verity. The (canonical) normal form expressions is derived in general and in particular for “A OR B” with normal form derivation algorithm; the first step is to derive the disjunctive normal form (DNF) expression. The second step is to derive the conjunctive normal form (CNF) expression. To demonstrate the equivalence of the DNF and CNF, direct fuzzification of DNF and CNF expressions is stated that directly fuzzifies DNF and CNF and in turn, fuzzy disjunctive and fuzzy conjunctive canonical forms, are not equivalent to each other, even though they are equivalent in form only to their two valued descriptive set versions, that is, DNF and CNF, respectively.