Formulas In Conjunctive Normal Form Research Articles

Classifying quantum data by dissipation

We investigate a general class of dissipative quantum circuit capable of computing arbitrary Conjunctive Normal Form (CNF) Boolean formulas. In particular, the clauses in a CNF formula define a local generator of Markovian quantum dynamics which acts on a network of qubits. Fixed points of this dynamical system encode the evaluation of the CNF formula. The structure of the corresponding quantum map partitions the Hilbert space into sectors, according to decoherence-free subspaces (DFSs) associated with the dissipative dynamics. These sectors then provide a natural and consistent way to classify quantum data (i.e. quantum states). Indeed, the attractive fixed points of the network allow one to learn the sector(s) for which some particular quantum state is associated. We show how this structure can be used to dissipatively prepare quantum states (e.g. entangled states), and outline how it may be used to generalize certain classical computational learning tasks.

Complementary Knowledge Compilation Using the Hyper Extension Rule

We introduce the concept of Complementary formula (COMF), which is a new and non-equivalent way for Knowledge compilation (KC). Based on the Hyper extension rule (HER) which is an expansion of Extension rule (ER), we design a compilation algorithm which can formula compile each Conjunctive normal form (CNF) formula to complementary Fully complementary connected diagram (c-FCCD), named as C2C (CNF formula to cFCCD). Theoretically, c-FCCD is a kind of complementary formulae of the input formulae and can support all queries and partial transformations in KC map. Experimentally, C2C is competitive with the EPCCL compilers KCER, C2E, UKCHER, DKCHER and IKCHER.

On the Parameterized Complexity of Finding Small Unsatisfiable Subsets of CNF Formulas and CSP Instances

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.

Automatic verification of single‐electron transistor arrays based on multiway decision graphs

The single-electron transistor (SET) is considered as a promising alternative to CMOS devices and integrated circuits due to its ultra-low-power consumption. The authors propose an automatic methodology by utilising SET-based multiway decision graph (MDG) for implementing SET architecture. MDG provides a powerful means of abstraction and can be used to manipulate a certain type of first-order logic formula called directed formula (DF). An automatic tool has been developed that is capable of transferring the SET array into MDG DF to generate all paths of all possible reachable states, drawing an abstract diamond-shaped network for the SET array, producing a reduced DF generated by the MDG tool and converting the reduced DF into a conjunctive normal form formula. Then, the authors use the satisfiability approach as a verification engine to verify the correctness of these conversions. The method is tested and verified using MCNC benchmarks. The results outperform approaches based on binary decision diagrams in terms of verification time, number of clauses and variables.

On semidefinite least squares and minimal unsatisfiability

This paper provides new results on the application of semidefinite optimization to satisfiability by studying the connection between semidefinite optimization and minimal unsatisfiability. We use a semidefinite least squares problem to assign weights to the clauses of a propositional formula in conjunctive normal form. We then show that these weights are a measure of the necessity of each clause in rendering the formula unsatisfiable, the weight of a necessary clause is strictly greater than the weight of any unnecessary clause. In particular, we show the following results: first, if a formula is minimal unsatisfiable, then all of its clauses have the same weight; second, if a clause does not belong to any minimal unsatisfiable subformula, then its weight is zero. An additional contribution of this paper is a demonstration of how the infeasibility of a semidefinite optimization problem can be tested using a semidefinite least squares problem by extending an earlier result for linear optimization. The connection between the semidefinite least squares problem and Farkas’ Lemma for semidefinite optimization is also discussed.

SYMMETRIES IN MODAL LOGICS

AbstractIn this paper we develop the theoretical foundations to exploit symmetries in modal logics. We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas using the framework provided by coinductive modal models introduced in [5]. Hence, the results apply to a wide class of modal logics including, for example, hybrid logics. We present two graph constructions that enable the reduction of symmetry detection in modal formulas to the graph automorphism detection problem, and we evaluate the graph constructions on modal benchmarks.

Clause Elimination for SAT and QSAT

The famous archetypical NP-complete problem of Boolean satisfiability (SAT) and its PSPACE-complete generalization of quantified Boolean satisfiability (QSAT) have become central declarative programming paradigms through which real-world instances of various computationally hard problems can be efficiently solved. This success has been achieved through several breakthroughs in practical implementations of decision procedures for SAT and QSAT, that is, in SAT and QSAT solvers. Here, simplification techniques for conjunctive normal form (CNF) for SAT and for prenex conjunctive normal form (PCNF) for QSAT---the standard input formats of SAT and QSAT solvers---have recently proven very effective in increasing solver efficiency when applied before (i.e., in preprocessing) or during (i.e., in inprocessing) satisfiability search. In this article, we develop and analyze clause elimination procedures for pre- and inprocessing. Clause elimination procedures form a family of (P)CNF formula simplification techniques which remove clauses that have specific (in practice polynomial-time) redundancy properties while maintaining the satisfiability status of the formulas. Extending known procedures such as tautology, subsumption, and blocked clause elimination, we introduce novel elimination procedures based on asymmetric variants of these techniques, and also develop a novel family of so-called covered clause elimination procedures, as well as natural liftings of the CNF-level procedures to PCNF. We analyze the considered clause elimination procedures from various perspectives. Furthermore, for the variants not preserving logical equivalence under clause elimination, we show how to reconstruct solutions to original CNFs from satisfying assignments to simplified CNFs, which is important for practical applications for the procedures. Complementing the more theoretical analysis, we present results on an empirical evaluation on the practical importance of the clause elimination procedures in terms of the effect on solver runtimes on standard real-world application benchmarks. It turns out that the importance of applying the clause elimination procedures developed in this work is empirically emphasized in the context of state-of-the-art QSAT solving.

Model Counting of Monotone Conjunctive Normal Form Formulas with Spectra

Model counting is the #P problem of counting the number of satisfying solutions of a given propositional formula. Here we focus on a restricted variant of this problem, where the input formula is monotone (i.e., there are no negations). A monotone conjunctive normal form (CNF) formula is sufficient for modeling various graph problems, e.g., the vertex covers of a graph. Even for this restricted case, there is no known efficient approximation scheme. We show that the classical Spectra technique that is widely used in network reliability can be adapted for counting monotone CNF formulas. We prove that the proposed algorithm is logarithmically efficient for random monotone 2-CNF instances. Although we do not prove the efficiency of Spectra for k-CNF where k > 2, our experiments show that it is effective in practice for such formulas.

Extracting verified decision procedures: DPLL and Resolution

This article is concerned with the application of the program extraction technique to a new class of problems: the synthesis of decision procedures for the classical satisfiability problem that are correct by construction. To this end, we formalize a completeness proof for the DPLL proof system and extract a SAT solver from it. When applied to a propositional formula in conjunctive normal form the program produces either a satisfying assignment or a DPLL derivation showing its unsatisfiability. We use non-computational quantifiers to remove redundant computational content from the extracted program and translate it into Haskell to improve performance. We also prove the equivalence between the resolution proof system and the DPLL proof system with a bound on the size of the resulting resolution proof. This demonstrates that it is possible to capture quantitative information about the extracted program on the proof level. The formalization is carried out in the interactive proof assistant Minlog.

Space Complexity in Polynomial Calculus

During the last 10 to 15 years, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important concern in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of papers on space in resolution, which is now reasonably well-understood from this point of view. For other proof systems of interest, however, such as polynomial calculus or cutting planes, progress has been more limited. Essentially nothing has been known about space complexity in cutting planes, and for polynomial calculus the only lower bound has been for conjunctive normal form (CNF) formulas of unbounded width in [Alekhnovich et al., SIAM J. Comput., 31 (2002), pp. 1184--1211], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any $k$-CNF formula in constant space. In this paper, we prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied by Alekhnovich et al.: (1) We prove an $\omega(n)$ space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas $PHP_{m}^{n}$ with $m$ pigeons and $n$ holes, and show that this is tight. (2) For PCR, we prove an $\omega(n)$ space lower bound for a bitwise encoding of the functional pigeonhole principle. These formulas have width O(log n), and hence this is an exponential improvement over Alekhnovich et al. measured in the width of the formulas. (3) We then present another encoding of the pigeonhole principle that has constant width, and prove an $\omega(n)$ space lower bound in PCR for these formulas as well. (4) Finally, we prove that any $k$-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity.

On Minimum Representations of Matched Formulas

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. Each matched CNF is trivially satisfiable (each clause can be satisfied by its representative variable). Another property which is easy to see, is that the class of matched CNFs is not closed under partial assignment of truth values to variables. This latter property leads to a fact (proved here) that given two matched CNFs it is co-NP complete to decide whether they are logically equivalent. The construction in this proof leads to another result: a much shorter and simpler proof of the fact that the Boolean minimization problem for matched CNFs is a complete problem for the second level of the polynomial hierarchy. The main result of this paper deals with the structure of clause minimum CNFs. We prove here that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

Computational hardness of enumerating groundstates of the antiferromagnetic Ising model in triangulations

Satisfying spin-assignments of triangulations of a surface are states of minimum energy of the antiferromagnetic Ising model on triangulations which correspond (via geometric duality) to perfect matchings in cubic bridgeless graphs. In this work we show that it is NP-complete to decide whether or not a triangulation of a surface admits a satisfying spin-assignment, and that it is #P-complete to determine the number of such assignments. Our results imply that the determination of even the entropy of the Ising model on triangulations at the thermodynamical limit is already #P-hard.The aforementioned claims are derived via elaborate (and atypical) reductions that map a Boolean formula in conjunctive normal form into triangulations of orientable closed surfaces. Moreover, the novel reduction technique enables us to prove that even very constrained versions of #MaxCut are already #P-hard.

Algorithms for computing minimal equivalent subformulas

Knowledge representation and reasoning using propositional logic is an important component of AI systems. A propositional formula in Conjunctive Normal Form (CNF) may contain redundant clauses — clauses whose removal from the formula does not affect the set of its models. Identification of redundant clauses is important because redundancy often leads to unnecessary computation, wasted storage, and may obscure the structure of the problem. A formula obtained by the removal of all redundant clauses from a given CNF formula F is called a Minimal Equivalent Subformula (MES) of F. This paper proposes a number of efficient algorithms and optimization techniques for the computation of MESes. Previous work on MES computation proposes a simple algorithm based on iterative application of the definition of a redundant clause, similar to the well-known deletion-based approach for the computation of Minimal Unsatisfiable Subformulas (MUSes). This paper observes that, in fact, most of the existing algorithms for the computation of MUSes can be adapted to the computation of MESes. However, some of the optimization techniques that are crucial for the performance of the state-of-the-art MUS extractors cannot be applied in the context of MES computation, and thus the resulting algorithms are often not efficient in practice. To address the problem of efficient computation of MESes, the paper develops a new class of algorithms that are based on the iterative analysis of subsets of clauses, and a lightweight pruning technique based on the computation of backbones. The experimental results, obtained on representative problem instances, confirm the effectiveness of the proposed methods. The experimental results also reveal that many CNF instances obtained from the practical applications of SAT exhibit a large degree of redundancy.

3-SAT Faster and Simpler---Unique-SAT Bounds for PPSZ Hold in General

The PPSZ algorithm by Paturi, Pudlák, Saks, and Zane [J. ACM, 52 (2005), pp. 337--364] is the fastest known algorithm for Unique $k$-SAT, where the input formula does not have more than one satisfying assignment. For $k\geq 5$ the same bounds hold for general $k$-SAT. We show that this is also the case for k=3,4, using a slightly modified PPSZ algorithm. We do the analysis by defining a cost for satisfiable conjunctive normal form formulas, which we prove to decrease in each PPSZ step by a certain amount. This improves our previous best bounds with Moser and Scheder [Proceedings of STACS, 2011, pp. 237--248] for 3-SAT to $O(1.308^n)$ and for 4-SAT to $O(1.469^n)$. Furthermore, our analysis is much simpler than the existing analysis of PPSZ for general $k$-SAT.

DNA strand displacement system running logic programs

The paper presents a DNA-based computing model which is enzyme-free and autonomous, not requiring a human intervention during the computation. The model is able to perform iterated resolution steps with logical formulae in conjunctive normal form. The implementation is based on the technique of DNA strand displacement, with each clause encoded in a separate DNA molecule. Propositions are encoded assigning a strand to each proposition p, and its complementary strand to the proposition ¬p; clauses are encoded comprising different propositions in the same strand. The model allows to run logic programs composed of Horn clauses by cascading resolution steps. The potential of the model is demonstrated also by its theoretical capability of solving SAT. The resulting SAT algorithm has a linear time complexity in the number of resolution steps, whereas its spatial complexity is exponential in the number of variables of the formula.

Verification of Reconfigurable Binary Decision Diagram-Based Single-Electron Transistor Arrays

Recently, single-electron transistors (SETs) have been attracting substantial attention and are considered candidate devices for future integrated circuits due to their ultralow power consumption. To realize SETs, a binary decision diagram-based SET array is proposed as a suitable candidate for implementing Boolean circuits. Then, some works started developing computer-aided design techniques for this new architecture. However, most of them focused on the development of mapping techniques. How to verify the mapping results is still an open problem. Thus, in this paper, we address this problem and develop a satisfiability (SAT)-based verification method. We propose a transformation approach to model the functionality of a mapped SET array as a conjunctive normal form formula. Then, the problem that whether the SET array is functionally equivalent to its specification circuit can be solved with a SAT solver. The experimental results show that the proposed method can successfully verify correct and incorrect SET array implementations with reasonable verification time.

An Improved Satisfiability Algorithm for Nested Canalyzing Functions and its Application to Determining a Singleton Attractor of a Boolean Network

We study the problem of finding a 0-1 assignment to Boolean variables satisfying a given set of nested canalyzing functions, a class of Boolean functions that is known to be of interest in biology. For this problem, an extension of the satisfiability problem for a conjunctive normal form formula, an O(min(2(k), 2((k+m)/2))poly(m)) time algorithm has been known, where m and k are the number of nested canalyzing functions and variables, respectively. Here we present an improved O(min(2(k), 1.325(k+m), 2(m))poly(m)) time algorithm for this problem. We also study the problem of finding a singleton attractor of a Boolean network consisting of n nested canalyzing functions. Although an O(1.799(n)) time algorithm was proposed in a previous study, it was implicitly assumed that the network does not contain any positive self-loops. By utilizing the improved satisfiability algorithm for nested canalyzing functions, while allowing for the presence of positive self-loops, we show that the general case can be solved in O(1.871(n)) time.

Complexity issues related to propagation completeness

Knowledge compilation is a process of adding more information to a knowledge base in order to make it easier to deduce facts from the compiled base than from the original one. One type of knowledge compilation occurs when the knowledge in question is represented by a Boolean formula in conjunctive normal form (CNF). The goal of knowledge compilation in this case is to add clauses to the input CNF until a logically equivalent propagation complete CNF is obtained. A CNF is called propagation complete if after any partial substitution of truth values all logically entailed literals can be inferred from the resulting CNF formula by unit propagation. The key to this type of knowledge compilation is the ability to generate so-called empowering clauses. A clause is empowering for a CNF if it is an implicate and for some partial substitution of truth values it enlarges the set of entailed literals inferable by unit propagation.In this paper we study several complexity issues related to empowering implicates, propagation completeness, and its relation to resolution proofs. We show several results: (a) given a CNF and a clause it is co-NP complete to decide whether the clause is an empowering implicate of the CNF, (b) given a CNF it is NP-complete to decide whether there exists an empowering implicate for it and thus it is co-NP complete to decide whether a CNF is propagation complete, and (c) there exist CNFs to which an exponential number of clauses must be added to make them propagation complete.

Symmetries in Modal Logics

We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas. Our framework uses the coinductive models and, hence, the results apply to a wide class of modal logics including, for example, hybrid logics. Our main result shows that the symmetries of a modal formula preserve entailment.

Formal Methods for Hopfield-Like Networks

Building a meaningful model of biological regulatory network is usually done by specifying the components (e.g. the genes) and their interactions, by guessing the values of parameters, by comparing the predicted behaviors to the observed ones, and by modifying in a trial-error process both architecture and parameters in order to reach an optimal fitness. We propose here a different approach to construct and analyze biological models avoiding the trial-error part, where structure and dynamics are represented as formal constraints. We apply the method to Hopfield-like networks, a formalism often used in both neural and regulatory networks modeling. The aim is to characterize automatically the set of all models consistent with all the available knowledge (about structure and behavior). The available knowledge is formalized into formal constraints. The latter are compiled into Boolean formula in conjunctive normal form and then submitted to a Boolean satisfiability solver. This approach allows to formulate a wide range of queries, expressed in a high level language, and possibly integrating formalized intuitions. In order to explore its potential, we use it to find cycles for 3-nodes networks and to determine the flower morphogenesis regulatory network of Arabidopsis thaliana. Applications of this technique are numerous and concern the building of models from data as well as the design of biological networks possessing specified behaviors.