Positive Literals Research Articles

The longest letter-duplicated subsequence and related problems

Motivated by computing duplication patterns in sequences, a new problem called the longest letter-duplicated subsequence (LLDS) is proposed. Given a sequence S of length n, a letter-duplicated subsequence is a subsequence of S in the form of x1d1x2d2…xkdk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_1^{d_1}x_2^{d_2}\\ldots x_k^{d_k}$$\\end{document} with xi∈Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_i\\in \\Sigma $$\\end{document}, xj≠xj+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_j\ e x_{j+1}$$\\end{document} and di≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_i\\ge 2$$\\end{document} for all i in [k] and j in [k-1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[k-1]$$\\end{document}. A linear time algorithm for computing a longest letter-duplicated subsequence (LLDS) of S can be easily obtained. In this paper, we focus on two variants of this problem: (1) ‘all-appearance’ version, i.e., all letters in Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} must appear in the solution, and (2) the weighted version. For the former, we obtain dichotomous results: We prove that, when each letter appears in S at least 4 times, the problem and a relaxed version on feasibility testing (FT) are both NP-hard. The reduction is from (3+,1,2-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3^+,1,2^-)$$\\end{document}-SAT, where all 3-clauses (i.e., containing 3 lals) are monotone (i.e., containing only positive literals) and all 2-clauses contain only negative literals. We then show that when each letter appears in S at most 3 times, then the problem admits an O(n) time algorithm. Finally, we consider the weighted version, where the weight of a block xidi(di≥2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_i^{d_i} (d_i\\ge 2)$$\\end{document} could be any positive function which might not grow with di\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_i$$\\end{document}. We give a non-trivial O(n2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^2)$$\\end{document} time dynamic programming algorithm for this version, i.e., computing an LD-subsequence of S whose weight is maximized.

MTS-PRO2SAT: Hybrid Mutation Tabu Search Algorithm in Optimizing Probabilistic 2 Satisfiability in Discrete Hopfield Neural Network

The primary objective of introducing metaheuristic algorithms into traditional systematic logic is to minimize the cost function. However, there is a lack of research on the impact of introducing metaheuristic algorithms on the cost function under different proportions of positive literals. In order to fill in this gap and improve the efficiency of the metaheuristic algorithm in systematic logic, we proposed a metaheuristic algorithm based on mutation tabu search and embedded it in probabilistic satisfiability logic in discrete Hopfield neural networks. Based on the traditional tabu search algorithm, the mutation operators of the genetic algorithm were combined to improve its global search ability during the learning phase and ensure that the cost function of the systematic logic converged to zero at different proportions of positive literals. Additionally, further optimization was carried out in the retrieval phase to enhance the diversity of solutions. Compared with nine other metaheuristic algorithms and exhaustive search algorithms, the proposed algorithm was superior to other algorithms in terms of time complexity and global convergence, and showed higher efficiency in the search solutions at the binary search space, consolidated the efficiency of systematic logic in the learning phase, and significantly improved the diversity of the global solution in the retrieval phase of systematic logic.

Conditional random <i>k</i> satisfiability modeling for <i>k</i> = 1, 2 (CRAN2SAT) with non-monotonic Smish activation function in discrete Hopfield neural network

<abstract> <p>The current development of logic satisfiability in discrete Hopfield neural networks (DHNN)has been segregated into systematic logic and non-systematic logic. Most of the research tends to improve non-systematic logical rules to various extents, such as introducing the ratio of a negative literal and a flexible hybrid logical structure that combines systematic and non-systematic structures. However, the existing non-systematic logical rule exhibited a drawback concerning the impact of negative literal within the logical structure. Therefore, this paper presented a novel class of non-systematic logic called conditional random <italic>k</italic> satisfiability for <italic>k</italic> = 1, 2 while intentionally disregarding both positive literals in second-order clauses. The proposed logic was embedded into the discrete Hopfield neural network with the ultimate goal of minimizing the cost function. Moreover, a novel non-monotonic Smish activation function has been introduced with the aim of enhancing the quality of the final neuronal state. The performance of the proposed logic with new activation function was compared with other state of the art logical rules in conjunction with five different types of activation functions. Based on the findings, the proposed logic has obtained a lower learning error, with the highest total neuron variation <italic>TV</italic> = 857 and lowest average of Jaccard index, <italic>JSI</italic> = 0.5802. On top of that, the Smish activation function highlights its capability in the DHNN based on the result ratio of improvement <italic>Zm</italic> and <italic>TV</italic>. The ratio of improvement for Smish is consistently the highest throughout all the types of activation function, showing that Smish outperforms other types of activation functions in terms of <italic>Zm</italic> and <italic>TV.</italic> This new development of logical rule with the non-monotonic Smish activation function presents an alternative strategy to the logic mining technique. This finding will be of particular interest especially to the research areas of artificial neural network, logic satisfiability in DHNN and activation function.</p> </abstract>

(1+1) genetic programming with functionally complete instruction sets can evolve Boolean conjunctions and disjunctions with arbitrarily small error

Recently it has been proven that simple GP systems can efficiently evolve a conjunction of n variables if they are equipped with the minimal required components. In this paper, we make a considerable step forward by analysing the behaviour and performance of a GP system for evolving a Boolean conjunction or disjunction of n variables using a complete function set that allows the expression of any Boolean function of up to n variables. First we rigorously prove that a GP system using the complete truth table to evaluate the program quality, and equipped with both the AND and OR operators and positive literals, evolves the exact target function in O(ℓnlog2⁡n) iterations in expectation, where ℓ≥n is a limit on the size of any accepted tree. Additionally, we show that when a polynomial sample of possible inputs is used to evaluate the solution quality, conjunctions or disjunctions with any polynomially small generalisation error can be evolved with probability 1−O(log2⁡(n)/n). The latter result also holds if GP uses AND, OR and positive and negated literals, thus has the power to express any Boolean function of n distinct variables. To prove our results we introduce a super-multiplicative drift theorem that gives significantly stronger runtime bounds when the expected progress is only slightly super-linear in the distance from the optimum.

PRO2SAT: Systematic Probabilistic Satisfiability logic in Discrete Hopfield Neural Network

Satisfiability is prominent in the field of computer science and mathematics because SAT provides an alternative to represent the knowledge of any datasets. Fueled by this nature, recent paradigm tends to converge towards modelling Artificial Neural Network (ANN) through SAT. Despite extensive implementation of SAT in ANN, there are severely limited strategy to control the distribution of negative and positive literals in the logical rule. One of the most feasible approaches in controlling the behavior of the literal is by employing probabilistic behavior to each neuron in the ANN. In this paper, a novel logical rule namely Probabilistic 2 Satisfiability was proposed by implementing the probability to each variable in the 2 Satisfiability clause. In this context, the negativity of each variable will be determined using the probability which leads to higher search space. The proposed Probabilistic 2 Satisfiability was implemented into the special single layered Discrete Hopfield Neural Network where the cost function of each variable was derived by minimizing the inconsistency of the logic. The behavior of the proposed Probabilistic 2 Satisfiability was assessed based on various performance metrics including several newly introduced metrics. According to the experimental results, the proposed model has a probability of at least 81.8% in outperforming the existing method. Interestingly, the proposed model was reported to have the largest solution space when the ratio of positive was within [0.1, 0.4]. The comparison of experimental results with other state of the art logical rule demonstrates that the proposed model is promising in retrieving global neuron state.

The Phase Transition Analysis for the Random Regular Exact 2-(d, k)-SAT Problem

In a regular (d,k)-CNF formula, each clause has length k and each variable appears d times. A regular structure such as this is symmetric, and the satisfiability problem of this symmetric structure is called the (d,k)-SAT problem for short. The regular exact 2-(d,k)-SAT problem is that for a (d,k)-CNF formula F, if there is a truth assignment T, then exactly two literals of each clause in F are true. If the formula F contains only positive or negative literals, then there is a satisfiable assignment T with a size of 2n/k such that F is 2-exactly satisfiable. This paper introduces the (d,k)-SAT instance generation model, constructs the solution space, and employs the method of the first and second moments to present the phase transition point d* of the 2-(d,k)-SAT instance with only positive literals. When d<d*, the 2-(d,k)-SAT instance can be satisfied with high probability. When d>d*, the 2-(d,k)-SAT instance can not be satisfied with high probability. Finally, the verification results demonstrate that the theoretical results are consistent with the experimental results.

On Signings and the Well-Founded Semantics

AbstractIn this note, we use Kunen’s notion of a signing to establish two theorems about the well-founded semantics of logic programs, in the case where we are interested in only (say) the positive literals of a predicate p that are consequences of the program. The first theorem identifies a class of programs for which the well-founded and Fitting semantics coincide for the positive part of p. The second theorem shows that if a program has a signing, then computing the positive part of p under the well-founded semantics requires the computation of only one part of each predicate. This theorem suggests an analysis for query answering under the well-founded semantics. In the process of proving these results, we use an alternative formulation of the well-founded semantics of logic programs, which might be of independent interest.

Compression with Wildcards: From CNFs to Orthogonal DNFs by Imposing the Clauses One-by-One

Abstract We present a novel technique for converting a Boolean conjunctive normal form (CNF) into an orthogonal disjunctive normal form (DNF), aka exclusive sum of products. Our method (which will be pitted against a hardwired command from Mathematica) zooms in on the models of the CNF by imposing its clauses one by one. Clausal imposition invites parallelization, and wildcards beyond the common don’t-care symbol compress the output. The method is most efficient for few but large clauses. Generalizing clauses, one can in fact impose superclauses. By definition, superclauses are obtained from clauses by substituting each positive literal by an arbitrary conjunction of positive literals.

Augmenting Negation Normal Form With Irrelevant Variables

Irrelevant variables are always omitted in knowledge compilation languages since their assignments do not change the satisfiability of sentences. In order to identify new knowledge compilation languages and reduce the scale of compiling result of d-DNNF, we augment NNF with irrelevant variables in this paper. The NNF PNI , NNF PI , and NNF NI are proposed based on different combinations of positive literals, negative literals, and irrelevant variables. Each sentence in NNF, NNF PI , NNF NI , or NNF PNI can be translated to an equivalent sentence in another language in polynomial time. We also define d-DNNF NI , d-DNNF PI , and d-DNNF PNI based on decomposability and determinism in NNF, which are subclasses of NNF PI , NNF NI , and NNF PNI , respectively. A number of querying and transforming methods for d-DNNF PNI , d-DNNF PI , and d-DNNF PI are designed to solve relevant reasoning problems in knowledge compilation map. Overall, d-DNNF PI and d-DNNF NI do not reduce the tractability of d-DNNF, so we propose a compressing method for d-DNNF based on d-DNNF PI and d-DNNF NI . The experimentally, the compiling results of the d-DNNF PI and d-DNNF NI are better with respect to d-DNNF for most instances, and our compressing method is significantly effective for all instances.

On the Expressive Power of Sub-Propositional Fragments of Modal Logic

Modal logic is a paradigm for several useful and applicable formal systems in computer science. It generally retains the low complexity of classical propositional logic, but notable exceptions exist in the domains of description, temporal, and spatial logic, where the most expressive formalisms have a very high complexity or are even undecidable. In search of computationally well-behaved fragments, clausal forms and other sub-propositional restrictions of temporal and description logics have been recently studied. This renewed interest on sub-propositional logics, which mainly focus on the complexity of the various fragments, raise natural questions on their the relative expressive power, which we try to answer here for the basic multi-modal logic Kn. We consider the Horn and the Krom restrictions, as well as the combined restriction (known as the core fragment) of modal logic, and, orthogonally, the fragments that emerge by disallowing boxes or diamonds from positive literals. We study the problem in a very general setting, to ease transferring our results to other meaningful cases.

On the performance of MaxSAT and MinSAT solvers on 2SAT-MaxOnes

We analyze and compare two solvers for Boolean optimization problems: WMaxSatz, a solver for Partial MaxSAT, and MinSatz, a solver for Partial MinSAT. Both MaxSAT and MinSAT are similar, but previous results indicate that when solving optimization problems using both solvers, the performance is quite different on some cases. For getting insights about the differences in the performance of the two solvers, we analyze their behaviour when solving 2SAT-MaxOnes problem instances, given that 2SAT-MaxOnes is probably the most simple, but NP-hard, optimization problem we can solve with them. The analysis is based first on the study of the bounds computed by both algorithms on some particular 2SAT-MaxOnes instances, characterized by the presence of certain particular structures. We find that the fraction of positive literals in the clauses is an important factor regarding the quality of the bounds computed by the algorithms. Then, we also study the importance of this factor on the typical case complexity of Random-p 2SAT-MaxOnes, a variant of the problem where instances are randomly generated with a probability p of having positive literals in the clauses. For the case p=0, the performance results indicate a clear advantage of MinSatz with respect to WMaxSatz, but as we consider positive values of p WMaxSatz starts to show a better performance, although at the same time the typical complexity of Random-p 2SAT-MaxOnes decreases as p increases. We also study the typical value of the bound computed by the two algorithms on these sets of instances, showing that the behaviour is consistent with our analysis of the bounds computed on the particular instances we studied first.

Finding a Periodic Attractor of a Boolean Network

In this paper, we study the problem of finding a periodic attractor of a Boolean network (BN), which arises in computational systems biology and is known to be NP-hard. Since a general case is quite hard to solve, we consider special but biologically important subclasses of BNs. For finding an attractor of period 2 of a BN consisting of n OR functions of positive literals, we present a polynomial time algorithm. For finding an attractor of period 2 of a BN consisting of n AND/OR functions of literals, we present an O(1:985(n)) time algorithm. For finding an attractor of a fixed period of a BN consisting of n nested canalyzing functions and having constant treewidth w, we present an O(n(2p(w+1))poly(n)) time algorithm.

Coincidence of the sets of minimal and irreducible join graphs over primary structure of algebraic Bayesian networks

An algebraic Bayesian network (ABN) is a probabilistic-logic graphical model of bases of knowledge patterns with uncertainty. A primary structure of an ABN is a set of knowledge patterns, that are ideals of conjunctions of positive literals except the empty conjunction endowed with scalar or interval probability estimates. A secondary ABN structure is represented by a graph constructed over the primary structure, which is called a join graph. From the point of view of learning of a global ABN structure, of interest are join graphs with the minimum number of edges and irreducible join graphs.

The fraction of large random trees representing a given Boolean function in implicational logic

AbstractWe consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this system. Then we show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f. We also prove that most expressions computing any given function in this system are “simple”, in a sense that we make precise. The probability of all read‐once functions of a given complexity is also evaluated in this model. At last, using the same techniques, the relation between the probability of a function and its complexity is also obtained when random expressions are drawn according to a critical branching process. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

Branching Process Approach for 2-Sat Thresholds

It is well known that, as n tends to ∞, the probability of satisfiability for a random 2-SAT formula on n variables, where each clause occurs independently with probability α / 2n, exhibits a sharp threshold at α = 1. We study a more general 2-SAT model in which each clause occurs independently but with probability α i / 2n, where i ∈ {0, 1, 2} is the number of positive literals in that clause. We generalize the branching process arguments used by Verhoeven (1999) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.

Branching Process Approach for 2-Sat Thresholds

It is well known that, as n tends to ∞, the probability of satisfiability for a random 2-SAT formula on n variables, where each clause occurs independently with probability α / 2n, exhibits a sharp threshold at α = 1. We study a more general 2-SAT model in which each clause occurs independently but with probability αi / 2n, where i ∈ {0, 1, 2} is the number of positive literals in that clause. We generalize the branching process arguments used by Verhoeven (1999) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.

A program-level approach to revising logic programs under the answer set semantics

AbstractAn approach to the revision of logic programs under the answer set semantics is presented. For programs P and Q, the goal is to determine the answer sets that correspond to the revision of P by Q, denoted P * Q. A fundamental principle of classical (AGM) revision, and the one that guides the approach here, is the success postulate. In AGM revision, this stipulates that α ∈ K * α. By analogy with the success postulate, for programs P and Q, this means that the answer sets of Q will in some sense be contained in those of P * Q. The essential idea is that for P * Q, a three-valued answer set for Q, consisting of positive and negative literals, is first determined. The positive literals constitute a regular answer set, while the negated literals make up a minimal set of naf literals required to produce the answer set from Q. These literals are propagated to the program P, along with those rules of Q that are not decided by these literals. The approach differs from work in update logic programs in two main respects. First, we ensure that the revising logic program has higher priority, and so we satisfy the success postulate; second, for the preference implicit in a revision P * Q, the program Q as a whole takes precedence over P, unlike update logic programs, since answer sets of Q are propagated to P. We show that a core group of the AGM postulates are satisfied, as are the postulates that have been proposed for update logic programs.

Level Mapping Induced Loop Formulas for Weight Constraint and Aggregate Logic Programs

Level mapping and loop formulas are two different means to justify and characterize answer sets for normal logic programs. Both of them specify conditions under which a supported model is an answer set. Though serving a similar purpose, in the past the two have been studied largely in isolation with each other. In this paper, we study level mapping and loop formulas for weight constraint and aggregate (logic) programs. We show that, for these classes of programs, loop formulas can be devised from level mapping characterizations. First, we formulate a level mapping characterization of stable models and show that it leads to a new formulation of loop formulas for arbitrary weight constraint programs, without using any new atoms. This extends a previous result on loop formulas for weight constraint programs, where weight constraints contain only positive literals. Second, since aggregate programs are closely related to weight constraint programs, we further use level mapping to characterize the underlying answer set semantics based on which we formulate loop formulas for aggregate programs. The main result is that for aggregate programs not involving the inequality comparison operator, the dependency graphs can be built in polynomial time. This compares to the previously known exponential time method.

Index conditions of resolution

In this paper, the following results are proved: (1) Using both deletion strategy and lock strategy, resolution is complete for a clause set where literals with the same predicate or proposition symbol have the same index. (2) Using deletion strategy, both positive unit lock resolution and input lock resolution are complete for a Horn set where the indexes of positive literals are greater than those of negative literals. (3) Using deletion strategy, input half-lock resolution is complete for a Horn set.

Reasoning with ordered binary decision diagrams

We consider problems of reasoning with a knowledge-base, which is represented by an ordered binary decision diagram, for two cases of general and Horn knowledge-bases. Our main results say that both finding a model of a knowledge-base and deducing from a knowledge-base can be done in linear time for a general knowledge-base, but that abduction is NP-complete even for a Horn knowledge-base. Then, we consider abduction when its assumption set consists of all propositional literals (i.e., an answer for a given query is allowed to include any positive literals), and show that it can be done in polynomial time if the knowledge-base is Horn, while it remains NP-complete for the general case. Some other solvable cases are also discussed.