Negative 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.

Unsupervised logic mining with a binary clonal selection algorithm in multi-unit discrete Hopfield neural networks via weighted systematic 2 satisfiability

<p>Evaluating behavioral patterns through logic mining within a given dataset has become a primary focus in current research. Unfortunately, there are several weaknesses in the research regarding the logic mining models, including an uncertainty of the attribute selected in the model, random distribution of negative literals in a logical structure, non-optimal computation of the best logic, and the generation of overfitting solutions. Motivated by these limitations, a novel logic mining model incorporating the mechanism to control the negative literal in the systematic Satisfiability, namely Weighted Systematic 2 Satisfiability in Discrete Hopfield Neural Network, is proposed as a logical structure to represent the behavior of the dataset. For the proposed logic mining models, we used ratio of <italic>r</italic> to control the distribution of the negative literals in the logical structures to prevent overfitting solutions and optimize synaptic weight values. A new computational approach of the best logic by considering both true and false classification values of the learning system was applied in this work to preserve the significant behavior of the dataset. Additionally, unsupervised learning techniques such as Topological Data Analysis were proposed to ensure the reliability of the selected attributes in the model. The comparative experiments of the logic mining models by utilizing 20 repository real-life datasets were conducted from repositories to assess their efficiency. Following the results, the proposed logic mining model dominated in all the metrics for the average rank. The average ranks for each metric were Accuracy (7.95), Sensitivity (7.55), Specificity (7.93), Negative Predictive Value (7.50), and Mathews Correlation Coefficient (7.85). Numerical results and in-depth analysis demonstrated that the proposed logic mining model consistently produced optimal induced logic that best represented the real-life dataset for all the performance metrics used in this study.</p>

Synergizing intelligence and knowledge discovery: Hybrid black hole algorithm for optimizing discrete Hopfield neural network with negative based systematic satisfiability

<p>The current systematic logical rules in the Discrete Hopfield Neural Network encounter significant challenges, including repetitive final neuron states that lead to the issue of overfitting. Furthermore, the systematic logical rules neglect the impact on the appearance of negative literals within the logical structure, and most recent efforts have primarily focused on improving the learning capabilities of the network, which could potentially limit its overall efficiency. To tackle the limitation, we introduced a Negative Based Higher Order Systematic Logic to the network, imposing restriction on the appearance of negative literals within the clauses. Additionally, a Hybrid Black Hole Algorithm was proposed in the retrieval phase to optimize the final neuron states. This ensured that the optimized states achieved maximum diversity and reach global minima solutions with the lowest similarity index, thereby enhancing the overall performance of the network. The results illustrated that the proposed model can achieve up to 10,000 diversified and global solutions with an average similarity index of 0.09. The findings indicated that the optimized final neuron states are in optimal configurations. Based on the findings, the development of the new systematic SAT and the implementation of the Hybrid Black Hole algorithm to optimize the retrieval capabilities of DHNN to achieve multi-objective functions result in updated final neuron states with high diversity, high attainment of global minima solutions, and produces states with a low similarity index. Consequently, this proposed model could be extended for logic mining applications to tackle classification tasks. The optimized final neuron states will enhance the retrieval of high-quality induced logic, which is effective for classification and knowledge extraction.</p>

(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.

S-Type Random k Satisfiability Logic in Discrete Hopfield Neural Network Using Probability Distribution: Performance Optimization and Analysis

Recently, a variety of non-systematic satisfiability studies on Discrete Hopfield Neural Networks have been introduced to overcome a lack of interpretation. Although a flexible structure was established to assist in the generation of a wide range of spatial solutions that converge on global minima, the fundamental problem is that the existing logic completely ignores the probability dataset’s distribution and features, as well as the literal status distribution. Thus, this study considers a new type of non-systematic logic termed S-type Random k Satisfiability, which employs a creative layer of a Discrete Hopfield Neural Network, and which plays a significant role in the identification of the prevailing attribute likelihood of a binomial distribution dataset. The goal of the probability logic phase is to establish the logical structure and assign negative literals based on two given statistical parameters. The performance of the proposed logic structure was investigated using the comparison of a proposed metric to current state-of-the-art logical rules; consequently, was found that the models have a high value in two parameters that efficiently introduce a logical structure in the probability logic phase. Additionally, by implementing a Discrete Hopfield Neural Network, it has been observed that the cost function experiences a reduction. A new form of synaptic weight assessment via statistical methods was applied to investigate the effect of the two proposed parameters in the logic structure. Overall, the investigation demonstrated that controlling the two proposed parameters has a good effect on synaptic weight management and the generation of global minima solutions.

Stratified guarded first-order transition systems

AbstractFirst-order transition systems are a convenient formalism to specify parametric systems such as multi-agent workflows or distributed algorithms. In general, any nontrivial question about such systems is undecidable. Here, we present three subclasses of first-order transition systems where every universal invariant can effectively be decided via fixpoint iteration. These subclasses are defined in terms of syntactical restrictions: negation, stratification and guardedness. While guardedness represents a particular pattern how input predicates control existential quantifiers, stratification limits the information flow between predicates. Guardedness implies that the weakest precondition for every universal invariant is again universal, while the remaining sufficient criteria enforce that either the number of occurring negated literals decreases in every iteration, or the number of required instances of input predicates or the number of first-order variables remains bounded. We argue for each of these three cases that termination of the fixpoint iteration can be guaranteed. We apply these results to identify classes of multi-agent systems, when formalized as first-order transition systems, where noninterference in presence of declassification is decidable for coalitions of attackers of bounded size.

Extending Fuzzy Linguistic Logic Programming with Negation †

Fuzzy linguistic logic programming (FLLP) is a framework for representation and reasoning with linguistically expressed human knowledge. In this paper, we extend FLLP by allowing negative literals to appear in rule bodies, resulting in normal logic programs. We study the stable model semantics and well-founded semantics of such programs and their relation. The two kinds of semantics are adapted from those of classical ones based on the Gelfond–Lifschitz transformation and van Gelder’s alternating fixpoint approach, respectively. To our knowledge, until now, there has been no work on the well-founded semantics of normal programs in any fuzzy logic programming (FLP) framework based on Vojtáš’s FLP. Moreover, the relation between the two kinds of semantics is usually studied using a bilattice setting of the truth domain. However, our truth domains do not possess a complete knowledge-ordering lattice and, thus, do not have a bilattice structure. The two kinds of semantics possess properties similar to those of the classical case. Every stable model contains the well-founded (partial) model, and the well-founded total model coincides with the unique stable model, but not vice versa. Since the well-founded semantics is closely related to the stable model semantics, it can help compute stable models more efficiently.

Weighted Random k Satisfiability for [formula omitted] (r2SAT) in Discrete Hopfield Neural Network

Current studies on non-systematic satisfiability in Discrete Hopfield Neural Network are able to avoid production of repetitive final neuron states which improves the quality of global solutions retrieved. This finding has made it possible to obtain diverse final neuron states without additional modifications in Discrete Hopfield Neural Network. However, non-systematic satisfiability exhibited a drawback for neglecting the distribution of positive and negative literals in the logical structure as a neuron representation. Thus, this paper considers a new class of non-systematic satisfiability logic named Weighted Random k Satisfiability for k=1,2 with an inclusive of weighted ratio of negative literals. A logic phase is proposed as an organized layer to assign the distribution of negative literals unbiasedly by using Genetic Algorithm. The performance of Genetic Algorithm in producing the right structure of Weighted Random k Satisfiability will be compared with exhaustive search. Additionally, the training capability of Discrete Hopfield Neural Network to minimize the cost function will be investigated by embedding state of the art metaheuristics. Overall analysis demonstrated the effect of different ratios of negative literals and advanced training algorithm positively impacted the synaptic weight management and production of global minima solutions.

GRAN3SAT: Creating Flexible Higher-Order Logic Satisfiability in the Discrete Hopfield Neural Network

One of the main problems in representing information in the form of nonsystematic logic is the lack of flexibility, which leads to potential overfitting. Although nonsystematic logic improves the representation of the conventional k Satisfiability, the formulations of the first, second, and third-order logical structures are very predictable. This paper proposed a novel higher-order logical structure, named G-Type Random k Satisfiability, by capitalizing the new random feature of the first, second, and third-order clauses. The proposed logic was implemented into the Discrete Hopfield Neural Network as a symbolic logical rule. The proposed logic in Discrete Hopfield Neural Networks was evaluated using different parameter settings, such as different orders of clauses, different proportions between positive and negative literals, relaxation, and differing numbers of learning trials. Each evaluation utilized various performance metrics, such as learning error, testing error, weight error, energy analysis, and similarity analysis. In addition, the flexibility of the proposed logic was compared with current state-of-the-art logic rules. Based on the simulation, the proposed logic was reported to be more flexible, and produced higher solution diversity.

Non-Systematic Weighted Satisfiability in Discrete Hopfield Neural Network Using Binary Artificial Bee Colony Optimization

Recently, new variants of non-systematic satisfiability logic were proposed to govern Discrete Hopfield Neural Network. This new variant of satisfiability logical rule will provide flexibility and enhance the diversity of the neuron states in the Discrete Hopfield Neural Network. However, there is no systematic method to control and optimize the logical structure of non-systematic satisfiability. Additionally, the role of negative literals was neglected, reducing the expressivity of the information that the logical structure holds. This study proposed an additional optimization layer of Discrete Hopfield Neural Network called the logic phase that controls the distribution of negative literals in the logical structure. Hence, a new variant of non-systematic satisfiability named Weighted Random 2 Satisfiability was formulated. Thus, a proposed searching technique called the binary Artificial Bee Colony algorithm will ensure the correct distribution of the negative literals. It is worth mentioning that the binary Artificial Bee Colony has flexible and less free parameters where the modifications tackled on the objective function. Specifically, this study utilizes a binary Artificial Bee Colony algorithm by modifying the updating rule equation by using not and (NAND) logic gate operator. The performance of the binary Artificial Bee Colony will be compared with other variants of binary Artificial Bee Colony algorithms of different logic gate operators and conventional binary algorithms such as the Particle Swarm Optimization, Exhaustive Search, and Genetic Algorithm. The experimental results and comparison show that the proposed algorithm is compatible in finding the correct logical structure according to the initiate ratio of negative literal.

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.

A novel SAT solver for the Van der Waerden numbers

This paper introduces a new efficient satisfiability problem (SAT) solver, negative-literal Van der Waerden numbers SAT solver (NegVanSAT). It is a modification of thewell-known SAT solver MINISAT where the constructor of the literals has been adjusted to start with the negated literals first. It reduces the calculations needed to solve a problem. The NegVanSAT is specifically designed for solving the satisfiability problem of finding Van der Waerden numbers, which are known to be very difficult to compute. Comparisons between the MINISAT and the proposed NegVanSAT show that the latter outperforms the MINISAT in finding many of them.

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.

Causal Graph Justifications of Logic Programs

AbstractIn this work we propose a multi-valued extension of logic programs under the stable models semantics where each true atom in a model is associated with a set of justifications. These justifications are expressed in terms of causal graphs formed by rule labels and edges that represent their application ordering. For positive programs, we show that the causal justifications obtained for a given atom have a direct correspondence to (relevant) syntactic proofs of that atom using the program rules involved in the graphs. The most interesting contribution is that this causal information is obtained in a purely semantic way, by algebraic operations (product, sum and application) on a lattice of causal values whose ordering relation expresses when a justification is stronger than another. Finally, for programs with negation, we define the concept of causal stable model by introducing an analogous transformation to Gelfond and Lifschitz's program reduct. As a result, default negation behaves as “absence of proof” and no justification is derived from negative literals, something that turns out convenient for elaboration tolerance, as we explain with a running example.

Maximum number of fixed points in AND–OR–NOT networks

We are interested in the number of fixed points in AND–OR–NOT networks, i.e. Boolean networks in which the update function of each component is either a conjunction or a disjunction of positive or negative literals. As the main result, we prove that the maximum number of fixed points in a loop-less connected AND–OR–NOT network with n components is at most the maximum number of maximal independent sets in a loop-less connected graph with n vertices, a quantity already known.

Solving min ones 2-sat as fast as vertex cover

The problem of finding a satisfying assignment that minimizes the number of variables that are set to 1 is NP-complete even for a satisfiable 2-SAT formula. We call this problem min ones 2-sat. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k.In this paper, we present a polynomial-time reduction from min ones 2-sat to vertex cover without increasing the parameter and ensuring that the number of vertices in the reduced instance is equal to the number of variables of the input formula. Consequently, we conclude that this problem also has a simple 2-approximation algorithm and a 2k−clogk-variable kernel subsuming (or, in the case of kernels, improving) the results known earlier. Further, the problem admits algorithms for the parameterized and optimization versions whose runtimes will always match the runtimes of the best-known algorithms for the corresponding versions of vertex cover.Finally we show that the optimum value of the LP relaxation of the min ones 2-sat and that of the corresponding vertex cover are the same. This implies that the (recent) results of vertex cover version parameterized above the optimum value of the LP relaxation of vertex cover carry over to the min ones 2-sat version parameterized above the optimum of the LP relaxation of min ones 2-sat.

Translation Algorithm for Negative Literals in Conformant Planning

The encoded negative literals in a conformant planning task will result in increasing state spaces. Getting a compact representation of state spaces is one of the most important issues in conformant planning. In this paper, a translation algorithm for negative literals is proposed to reduce the state spaces in a conformant planning task. The relationship between encoded literals is analyzed in detail. Based on the one-of relaxation technique in domain language, the algorithm is used to express the uncertain initial states and action effects in conformant planning. It converts formula one-of into a set of mutually exclusive literals with the relationship of mutual. The experiment study shows the efficiency of the proposed algorithm in pruning the state space in conformant planning tasks.

SIMILARITY-BASED RELATIONS IN DATALOG PROGRAMS

We consider similarity-based relational databases that allow to retrieve approximate data, find data within a given range of distance or similarity, and support imprecise queries. We focus on the recently introduced relational algebra with similarities on [Formula: see text]-relations, which are annotated with multi-dimensional similarity values with each dimension referring to a single attribute. The codomains of the annotated relations are De Morgan frames, and the annotations express the relevance of the tuples as answers to a similarity-based query. In this paper, we study Datalog programs on [Formula: see text]-relations, with and without negation. We describe the least-fixpoint algorithm for safe and rectified Datalog programs on [Formula: see text]-relations with finite support but without negative literals in the body. We further describe the perfect-minimal-fixpoint algorithm of a Datalog program on [Formula: see text]-relations with finite support and negative literals in the body when rules are safe, rectified and stratified. We introduce the idea of controlling the calculation of the annotations such that the tuples that enter an IDB relation last will be announced less desirable than those that enter first. For this we define a damping function that augments/diminishes the individual annotations that contribute to the final annotations of tuples. With a damping function, for instance, long chains of inferences may be made significantly less desirable or even totally undesirable.

Constraint Satisfaction Problems in Clausal Form I: Autarkies and Deficiency

We consider the problem of generalising boolean formulas in conjunctive normal form by allowing non-boolean variables, with the goal of maintaining combinatorial properties. Requiring that a literal involves only a single variable, the most general form of literals are the wellknown “signed literals”, corresponding to unary constraints in CSP. However we argue that only the restricted form of “negative monosigned literals” and the resulting generalised clause-sets, corresponding to “sets of no-goods” in the AI literature, maintain the essential properties of boolean conjunctive normal forms. In this first part of a mini-series of two articles, we build up a solid foundation for (generalised) clause-sets, including the notion of autarky systems, the interplay between autarkies and resolution, and basic notions of (DP-)reductions. As a basic combinatorial parameter of generalised clause-sets we introduce the (generalised) notion of deficiency, which in the boolean case is the difference between the number of clauses and the number of variables. Autarky theory plays a fundamental role here, and we concentrate especially on matching autarkies (based on matching theory). A natural task is to determine the structure of (matching) lean clause-sets, which do not admit non-trivial (matching) autarkies. A central result is the computation of the lean kernel (the largest lean subset) of a (generalised) clause-set in polynomial time for bounded maximal deficiency.

Tautologies over implication with negative literals

We consider logical expressions built on the single binary connector of implication and a finite number of literals (Boolean variables and their negations). We prove that asymptotically, when the number of variables becomes large, all tautologies have the following simple structure: either a premise equal to the goal, or two premises which are opposite literals (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)