Showing papers on "Conjunctive normal form published in 2018"

Boolean Decision Rules via Column Generation

Abstract: This paper considers the learning of Boolean rules in either disjunctive normal form (DNF, OR-of-ANDs, equivalent to decision rule sets) or conjunctive normal form (CNF, AND-of-ORs) as an interpretable model for classification. An integer program is formulated to optimally trade classification accuracy for rule simplicity. Column generation (CG) is used to efficiently search over an exponential number of candidate clauses (conjunctions or disjunctions) without the need for heuristic rule mining. This approach also bounds the gap between the selected rule set and the best possible rule set on the training data. To handle large datasets, we propose an approximate CG algorithm using randomization. Compared to three recently proposed alternatives, the CG algorithm dominates the accuracy-simplicity trade-off in 8 out of 16 datasets. When maximized for accuracy, CG is competitive with rule learners designed for this purpose, sometimes finding significantly simpler solutions that are no less accurate.

On the representation of Boolean and real functions as Hamiltonians for quantum computing

Abstract: Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli $Z$ operators (Ising spin operators) with the terms of the sum corresponding to the function's Fourier expansion. For many classes of functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses. We give composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks. We apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results. A primary goal of this paper is to provide a $\textit{design toolkit for quantum optimization}$ which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to demystify the various constructions appearing in the literature.

Algorithms for the Satisfiability Problem

Abstract: An instance of the satisfiability (SAT) problem is a Boolean formula that has three components [102, 191]: A set of n variables: x 1, x 2, x n . A set of literals. A literal is a variable (Q = x) or a negation of a variable \( \left( {Q = \bar x} \right)\). A set of m distinct clauses: C 1, C 2, ..., C m. Each clause consists of only literals combined by just logical or (V) connectives.

BOSPHORUS: Bridging ANF and CNF Solvers

Abstract: Algebraic Normal Form (ANF) and Conjunctive Normal Form (CNF) are commonly used to encode problems in Boolean algebra. ANFs are typically solved via Gr"obner basis algorithms, often using more memory than is feasible; while CNFs are solved using SAT solvers, which cannot exploit the algebra of polynomials naturally. We propose a paradigm that bridges between ANF and CNF solving techniques: the techniques are applied in an iterative manner to emph{learn facts} to augment the original problems. Experiments on over 1,100 benchmarks arising from four different applications domains demonstrate that learnt facts can significantly improve runtime and enable more benchmarks to be solved.

A Linear Time Algorithm for Computing #2SAT for Outerplanar 2-CNF Formulas

Abstract: Although the satisfiability problem for two Conjunctive Normal Form formulas (2SAT) is polynomial time solvable, it is well known that #2SAT, the counting version of 2SAT is #P-Complete. However, it has been shown that for certain classes of formulas, #2SAT can be computed in polynomial time. In this paper we show another class of formulas for which #2SAT can also be computed in lineal time, the so called outerplanar formulas, e.g. formulas whose signed primal graph is outerplanar. Our algorithm’s time complexity is given by \(O(n+m)\) where n is the number of variables and m the number of clauses of the formula.

Dualization Problem over the Product of Chains: Asymptotic Estimates for the Number of Solutions

Abstract: A key intractable problem in logical data analysis, namely, dualization over the product of partial orders, is considered. The important special case where each order is a chain is studied. If the cardinality of each chain is equal to two, then the considered problem is to construct a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form, which is equivalent to the enumeration of irreducible coverings of a Boolean matrix. The asymptotics of the typical number of irreducible coverings is known in the case where the number of rows in the Boolean matrix has a lower order of growth than the number of columns. In this paper, a similar result is obtained for dualization over the product of chains when the cardinality of each chain is higher than two. Deriving such asymptotic estimates is a technically complicated task, and they are required, in particular, for proving the existence of asymptotically optimal algorithms for the problem of monotone dualization and its generalizations.

Monotone Dualization Problem and Its Generalizations: Asymptotic Estimates of the Number of Solutions

Abstract: Issues related to the construction of efficient algorithms for intractable discrete problems are studied. Enumeration problems are considered. Their intractability has two aspects—exponential growth of the number of their solutions with increasing problem size and the complexity of finding (enumerating) these solutions. The basic enumeration problem is the dualization of a monotone conjunctive normal form or the equivalent problem of finding irreducible coverings of Boolean matrices. For the latter problem and its generalization for the case of integer matrices, asymptotics for the typical number of solutions are obtained. These estimates are required, in particular, to prove the existence of asymptotically optimal algorithms for monotone dualization and its generalizations.

A SAT-Based Approach for SDN Rule Table Distribution

Abstract: In Software-Defined Networking (SDN) it is important to efficiently partition the rule table into sub-tables and distribute them to the multiple switches over the network. In this paper we proposed an optimal rule table distribution strategy by applying satisfiability (SAT)-based approach. N-coloring problem for partitioning is formulated as conjunctive normal form (CNF), and by repeatedly running SAT solver we can obtain maximum number of partitions.

Boolean Decision Rules via Column Generation

Abstract: This paper considers the learning of Boolean rules in either disjunctive normal form (DNF, OR-of-ANDs, equivalent to decision rule sets) or conjunctive normal form (CNF, AND-of-ORs) as an interpretable model for classification. An integer program is formulated to optimally trade classification accuracy for rule simplicity. Column generation (CG) is used to efficiently search over an exponential number of candidate clauses (conjunctions or disjunctions) without the need for heuristic rule mining. This approach also bounds the gap between the selected rule set and the best possible rule set on the training data. To handle large datasets, we propose an approximate CG algorithm using randomization. Compared to three recently proposed alternatives, the CG algorithm dominates the accuracy-simplicity trade-off in 7 out of 15 datasets. When maximized for accuracy, CG is competitive with rule learners designed for this purpose, sometimes finding significantly simpler solutions that are no less accurate.

Modeling Attacks in Computer Networks Using Boolean Constraint Propagation

Abstract: We propose a new approach to modeling the processes of the development of attacks in computer networks. This approach is based on the idea that it is possible to associate with a considered computer network a Discrete Dynamical System (DDS) of automaton type. Under the standard assumptions, employed in computer security, such DDS has a single stationary point and cannot have cycles of length greater than one. With each DDS of such kind one can naturally link an effectively computed discrete function. As a result of propositional encoding of an algorithm, defining this function a Boolean formula in a Conjunctive Normal Form, is constructed. By applying to it the state-of-the-art SAT solvers it is possible to make conclusions, regarding different properties of a considered DDS. In the present paper we use the formulas of such kind to effectively construct the attack graphs for computer networks. We show that for this purpose it is sufficient to employ the simple algorithm known as the Unit Propagation rule. We compare the effectiveness of the proposed method for constructing the attack graphs with the well-known MulVAlsoftware system and show that our method outperforms MulVAlby a number of criteria.

Tractable Learning and Inference for Large-Scale Probabilistic Boolean Networks

Abstract: Probabilistic Boolean Networks (PBNs) have been previously proposed so as to gain insights into complex dy- namical systems. However, identification of large networks and of the underlying discrete Markov Chain which describes their temporal evolution, still remains a challenge. In this paper, we introduce an equivalent representation for the PBN, the Stochastic Conjunctive Normal Form (SCNF), which paves the way to a scalable learning algorithm and helps predict long- run dynamic behavior of large-scale systems. Moreover, SCNF allows its efficient sampling so as to statistically infer multi- step transition probabilities which can provide knowledge on the activity levels of individual nodes in the long run.

Diagnosing a Strong-Fault Model by Conflict and Consistency.

Abstract: The diagnosis method for a weak-fault model with only normal behaviors of each component has evolved over decades. However, many systems now demand a strong-fault models, the fault modes of which have specific behaviors as well. It is difficult to diagnose a strong-fault model due to its non-monotonicity. Currently, diagnosis methods usually employ conflicts to isolate possible fault and the process can be expedited when some observed output is consistent with the model’s prediction where the consistency indicates probably normal components. This paper solves the problem of efficiently diagnosing a strong-fault model by proposing a novel Logic-based Truth Maintenance System (LTMS) with two search approaches based on conflict and consistency. At the beginning, the original a strong-fault model is encoded by Boolean variables and converted into Conjunctive Normal Form (CNF). Then the proposed LTMS is employed to reason over CNF and find multiple minimal conflicts and maximal consistencies when there exists fault. The search approaches offer the best candidate efficiency based on the reasoning result until the diagnosis results are obtained. The completeness, coverage, correctness and complexity of the proposals are analyzed theoretically to show their strength and weakness. Finally, the proposed approaches are demonstrated by applying them to a real-world domain—the heat control unit of a spacecraft—where the proposed methods are significantly better than best first and conflict directly with A* search methods.

Sub-exponential complexity of regular linear CNF formulas.

Abstract: The study of regular linear conjunctive normal form (LCNF) formulas is of interest because exact satisfiability (XSAT) is known to be NP-complete for this class of formulas. In a recent paper it was shown that the subclass of regular exact LCNF formulas (XLCNF) is of sub-exponential complexity, i.e. XSAT can be determined in sub-exponential time. Here I show that this class is just a subset of a larger class of LCNF formulas which display this very kind of complexity. To this end I introduce the property of disjointedness of LCNF formulas, measured, for a single clause C, by the number of clauses which have no variable in common with C. If for a given LCNF formula F all clauses have the same disjointedness d we call F d-disjointed and denote the class of such formulas by dLCNF. XLCNF formulas correspond to the special cased=0. One main result of the paper is that the class of all monotone l-regular LCNF formulas which are d-disjointed, with d smaller than some upper bound D, is of sub-exponential complexity. This result can be generalized to show that all monotone, l-regular LCNF formulas F which have a bounded mean disjointedness, are of sub-exponential XSAT-complexity, as well.

Automated Theorem Proving with Extensions of First-Order Logic

Abstract: Automated theorem provers are computer programs that check whether a logical conjecture follows from a set of logical statements. The conjecture and the statements are expressed in the language of some formal logic, such as first-order logic. Theorem provers for first-order logic have been used for automation in proof assistants, verification of programs, static analysis of networks, and other purposes. However, the efficient usage of these provers remains challenging. One of the challenges is the complexity of translating domain problems to first-order logic. Not only can such translation be cumbersome due to semantic differences between the domain and the logic, but it might inadvertently result in problems that provers cannot easily handle. The work presented in the thesis addresses this challenge by developing an extension of first-order logic named FOOL. FOOL contains syntactical features of programming languages and more expressive logics, is friendly for translation of problems from various domains, and can be efficiently supported by existing theorem provers. We describe the syntax and semantics of FOOL and present a simple translation from FOOL to plain first-order logic. We describe an efficient clausal normal form transformation algorithm for FOOL and based on it implement a support for FOOL in the Vampire theorem prover. We illustrate the efficient use of FOOL for program verification by describing a concise encoding of next state relations of imperative programs in FOOL. We show a usage of features of FOOL in problems of static analysis of networks. We demonstrate the efficiency of automated theorem proving in FOOL with an extensive set of experiments. In these experiments we compare the performance of Vampire on a large collection of problems from various sources translated to FOOL and ordinary first-order logic. Finally, we fix the syntax for FOOL in TPTP, the standard language of first-order theorem provers.

The complete set of minimal simple graphs that support unsatisfiable 2-CNFs

Abstract: A propositional logic sentence in conjunctive normal form that has clauses of length two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We first show that every such sentence that has been reduced, that is, which is unchanged under application of certain tautologies, is equisatisfiable to a 2-CNF whose associated multigraph is, in fact, a simple graph. Our main result is a complete characterization of graphs that can support unsatisfiable 2-CNF sentences. We show that a simple graph can support an unsatisfiable reduced 2-CNF sentence if and only if it contains any one of four specific small graphs as a topological minor. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if all subdivisions of those four graphs are forbidden as subgraphs of of the original graph. We conclude with a discussion of why the Robertson-Seymour graph minor theorem does not apply in our approach.

Reconstruction of Boolean Formulas in Conjunctive Normal Form

Abstract: A long-standing open problem in graph theory is to prove or disprove the graph reconstruction conjecture proposed by Kelly and Ulam in the 1940s. This conjecture roughly states that every graph on at least three vertices is uniquely determined by its vertex-deleted subgraphs. We adapt the idea of reconstruction for Boolean formulas in conjunctive normal form (CNFs) and formulate the reconstruction conjecture for CNFs: every CNF with at least four clauses is uniquely determined by its clause-deleted subformulas. Our main results can be summarized as follows. First, we prove that our conjecture is equivalent to a well-studied variation of the graph reconstruction conjecture, namely, the edge-reconstruction conjecture for hypergraphs. Second, we prove that the number of satisfying assignments of a CNF is reconstructible, i.e., this number can be computed from the clause-deleted subformulas. Third, we show that every CNF with m clauses over n variables is reconstructible if \(2^{m-1} > 2^n \cdot n!\).

Probabilistic Logic for Intelligent Systems

Abstract: Given a knowledge base in Conjunctive Normal Form for use by an intelligent agent, with probabilities assigned to the conjuncts, the probability of any new query sentence can be determined by solving the Probabilistic Satisfiability Problem (PSAT). This involves finding a consistent probability distribution over the atoms (if they are independent) or complete conjunction set of the atoms. We show how this problem can be expressed and solved as a set of nonlinear equations derived from the knowledge base sentences and standard probability of logical sentences. Evidence is given that numerical gradient descent algorithms can be used more effectively then other current methods to find PSAT solutions.

A Linear Complementarity Theorem to solve any Satisfiability Problem in conjunctive normal form in polynomial time.

Abstract: Any satisfiability problem in conjunctive normal form can be solved in polynomial time by reducing it to a 3-sat formulation and transforming this to a Linear Complementarity problem (LCP) which is then solved as a linear program (LP) Any instance in this problem class, reduced to a LCP may be solved, provided certain necessary and sufficient conditions hold The proof that these conditions will be satisfied for all problems in this class is the contribution of this paper and this derivation requires a nonlinear Instrumentalist methodology rather than a Realistic one and confirms the advantages of a Variational Inequalities implementation

Complexity of the CNF-satisfiability problem.

Abstract: This paper is devoted to the complexity of the Boolean satisfiability problem. We consider a version of this problem, where the Boolean formula is specified in the conjunctive normal form. We prove an unexpected result that the CNF-satisfiability problem can be solved by a deterministic Turing machine in polynomial time.

Knowing and/or Believing a Think: Deriving Knowledge Using RDF CFL

Abstract: From the web discussion on a difference between knowing and believing, we have chosen in this paper the statements fulfilling enough our seeing the topic, corresponding to our knowledge level of cognitive science. The aim of paper shows one of the capabilities of our Resource Description Framework Clausal Form Logic (RDF CFL) graph language using as an example a well- known Castaněda’s puzzle. RDF CFL is an appropriate tool that contains a package of inference methods working especially in closed-worlds that have been developed in the clausal form of first order predicate logics.

Efficient Encodings of Conditional Cardinality Constraints.

Abstract: In the encoding of many real-world problems to propositional satisfiability, the cardinality constraint is a recurrent constraint that needs to be managed effectively. Several efficient encodings have been proposed while missing that such a constraint can be involved in a more general propositional formulation. To avoid combinatorial explosion, Tseitin principle usually used to translate such general propositional formula to Conjunctive Normal Form (CNF), introduces fresh propositional variables to represent sub-formulas and/or complex contraints. Thanks to Plaisted and Greenbaum improvement, the polarity of the sub-formula $\Phi$ is taken into account leading to conditional constraints of the form $y\rightarrow \Phi$, or $\Phi\rightarrow y$, where $y$ is a fresh propositional variable. In the case where $\Phi$ represents a cardinality constraint, such translation leads to conditional cardinality constraints subject of the present paper. We first show that when all the clauses encoding the cardinality constraint are augmented with an additional new variable, most of the well-known encodings cease to maintain the generalized arc consistency property. Then, we consider some of these encodings and show how they can be extended to recover such important property. An experimental validation is conducted on a SAT-based pattern mining application, where such conditional cardinality constraints is a cornerstone, showing the relevance of our proposed approach.

Moving Between Conjunctive and Disjunctive Normal Forms

Abstract: As discussed in Chap. 1 (on inequality systems with logical connectives), disjunctive sets have many equivalent forms, of which the two extremes are the conjunctive normal form (CNF) and the disjunctive normal form (DNF). Although these two normal forms are at the opposite ends of the variety of equivalent forms, they share a property not common to all forms: each of them is an intersection of unions of polyhedra.