Formulas In Conjunctive Normal Form Research Articles

Engineering an Exact Pseudo-Boolean Model Counter

Model counting, a fundamental task in computer science, involves determining the number of satisfying assignments to a Boolean formula, typically represented in conjunctive normal form (CNF). While model counting for CNF formulas has received extensive attention with a broad range of applications, the study of model counting for Pseudo-Boolean (PB) formulas has been relatively overlooked. Pseudo-Boolean formulas, being more succinct than propositional Boolean formulas, offer greater flexibility in representing real-world problems. Consequently, there is a crucial need to investigate efficient techniques for model counting for PB formulas. In this work, we propose the first exact Pseudo-Boolean model counter, PBCount , that relies on knowledge compilation approach via algebraic decision diagrams. Our extensive empirical evaluation shows that PBCount can compute counts for 1513 instances while the current state-of-the-art approach could only handle 1013 instances. Our work opens up several avenues for future work in the context of model counting for PB formulas, such as the development of preprocessing techniques and exploration of approaches other than knowledge compilation.

IASCAR: Incremental Answer Set Counting by Anytime Refinement

Abstract Answer set programming (ASP) is a popular declarative programming paradigm with various applications. Programs can easily have many answer sets that cannot be enumerated in practice, but counting still allows quantifying solution spaces. If one counts under assumptions on literals, one obtains a tool to comprehend parts of the solution space, so-called answer set navigation. However, navigating through parts of the solution space requires counting many times, which is expensive in theory. Knowledge compilation compiles instances into representations on which counting works in polynomial time. However, these techniques exist only for conjunctive normal form (CNF) formulas, and compiling ASP programs into CNF formulas can introduce an exponential overhead. This paper introduces a technique to iteratively count answer sets under assumptions on knowledge compilations of CNFs that encode supported models. Our anytime technique uses the inclusion–exclusion principle to improve bounds by over- and undercounting systematically. In a preliminary empirical analysis, we demonstrate promising results. After compiling the input (offline phase), our approach quickly (re)counts.

On the Parallel Parameterized Complexity of MaxSAT Variants

In the maximum satisfiability problem (max-sat) we are given a propositional formula in conjunctive normal form and have to find an assignment that satisfies as many clauses as possible. We study the parallel parameterized complexity of various versions of max-sat and provide the first constant-time algorithms parameterized either by the solution size or by the allowed excess relative to some guarantee. For the dual parameterized version where the parameter is the number of clauses we are allowed to leave unsatisfied, we present the first parallel algorithm for max-2sat (known as almost-2sat). The difficulty in solving almost-2sat in parallel comes from the fact that the iterative compression method, originally developed to prove that the problem is fixed-parameter tractable at all, is inherently sequential. We observe that a graph flow whose value is a parameter can be computed in parallel and develop a parallel algorithm for the vertex cover problem parameterized above the size of a given matching. Finally, we study the parallel complexity of max-sat parameterized by the vertex cover number, the treedepth, the feedback vertex set number, and the treewidth of the input’s incidence graph. While max-sat is fixedparameter tractable for all of these parameters, we show that they allow different degrees of possible parallelization. For all four we develop dedicated parallel algorithms that are constructive, meaning that they output an optimal assignment – in contrast to results that can be obtained by parallel meta-theorems, which often only solve the decision version.

Proving the infeasibility of Horn formulas through read-once resolution

In this paper, we study Horn formulas from the perspective of read-once resolution refutations (RORs). A Horn formula is a Boolean formula in conjunctive normal form (CNF), in which each clause contains at most one positive literal. Horn formulas are used in a number of domains, including program verification, logic programming, and econometrics. In particular, deduction in ProLog is based on unification. Unification is based on resolution and instantiation. Resolution is a system used to prove the infeasibility of Boolean formulas. It is important to note that resolution is both sound and complete. However, resolution is inefficient in the following sense: There exist CNF formulas with resolution refutations whose lengths are bounded below by an exponential function of the input size. At the same time, these formulas admit shorter (polynomially bounded) proofs of infeasibility in other proof systems, such as Frege Systems. Despite this inefficiency, resolution is simple and easy to implement and hence used in a wide variety of theorem provers. In this paper, we study two variants of resolution. These are read-once resolution (ROR) and read-once unit resolution (UROR). Both ROR and UROR are sound. However, they are incomplete since there exist infeasible Boolean formulas which do not have either an ROR or a UROR. In this paper, we look at the problems of determining if a Horn formula has an ROR or a UROR. We also examine the problem of finding the optimal length ROR of a Horn formula from both the computational complexity and the approximation perspectives. Finally, we analyze the copy complexity of Horn formulas with respect to URORs.

On Probabilistic Generalization of Backdoors in Boolean Satisfiability

The paper proposes a probabilistic generalization of the well-known Strong Backdoor Set (SBS) concept applied to the Boolean Satisfiability Problem (SAT). We call a set of Boolean variables B a ρ-backdoor, if for a fraction of at least ρ of possible assignments of variables from B, assigning their values to variables in a Boolean formula in Conjunctive Normal Form (CNF) results in polynomially solvable formulas. Clearly, a ρ-backdoor with ρ=1 is an SBS. For a given set B it is possible to efficiently construct an (ε, δ)-approximation of parameter ρ using the Monte Carlo method. Thus, we define an (ε, δ)-SBS as such a set B for which the conclusion "parameter ρ deviates from 1 by no more than ε" is true with probability no smaller than 1 - δ. We consider the problems of finding the minimum SBS and the minimum (ε, δ)-SBS. To solve the former problem, one can use the algorithm described by R. Williams, C. Gomes and B. Selman in 2003. In the paper we propose a new probabilistic algorithm to solve the latter problem, and show that the asymptotic estimation of the worst-case complexity of the proposed algorithm is significantly smaller than that of the algorithm by Williams et al. For practical applications, we suggest a metaheuristic optimization algorithm based on the penalty function method to seek the minimal (ε, δ)-SBS. Results of computational experiments show that the use of (ε, δ)-SBSes found by the proposed algorithm allows speeding up solving of test problems related to equivalence checking and hard crafted and combinatorial benchmarks compared to state-of-the-art SAT solvers.

Technical Perspective

Model counting is the problem of approximately counting the number |Sol(Φ)| of satisfying assignments to a given model Φ, which could, for example, be a formula in conjunctive normal form (CNF) or a formula in disjunctive normal form (DNF).

Fundamental limitations on efficiently forecasting certain epidemic measures in network models

The ongoing COVID-19 pandemic underscores the importance of developing reliable forecasts that would allow decision makers to devise appropriate response strategies. Despite much recent research on the topic, epidemic forecasting remains poorly understood. Researchers have attributed the difficulty of forecasting contagion dynamics to a multitude of factors, including complex behavioral responses, uncertainty in data, the stochastic nature of the underlying process, and the high sensitivity of the disease parameters to changes in the environment. We offer a rigorous explanation of the difficulty of short-term forecasting on networked populations using ideas from computational complexity. Specifically, we show that several forecasting problems (e.g., the probability that at least a given number of people will get infected at a given time and the probability that the number of infections will reach a peak at a given time) are computationally intractable. For instance, efficient solvability of such problems would imply that the number of satisfying assignments of an arbitrary Boolean formula in conjunctive normal form can be computed efficiently, violating a widely believed hypothesis in computational complexity. This intractability result holds even under the ideal situation, where all the disease parameters are known and are assumed to be insensitive to changes in the environment. From a computational complexity viewpoint, our results, which show that contagion dynamics become unpredictable for both macroscopic and individual properties, bring out some fundamental difficulties of predicting disease parameters. On the positive side, we develop efficient algorithms or approximation algorithms for restricted versions of forecasting problems.

On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing

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 Boolean 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, i.e., as hard as computing its number of satisfying assignments. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses each acting on a fixed number of bits as is common in constraint satisfaction problems. We show composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks, which are particularly suitable for direct implementation as classical software. We further 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 to quantum algorithms for optimization. A goal of this work is to provide a 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 provide a unified framework for the various constructions appearing in the literature.

QuantifyML: How Good is my Machine Learning Model?

The efficacy of machine learning models is typically determined by computing their accuracy on test data sets. However, this may often be misleading, since the test data may not be representative of the problem that is being studied. With QuantifyML we aim to precisely quantify the extent to which machine learning models have learned and generalized from the given data. Given a trained model, QuantifyML translates it into a C program and feeds it to the CBMC model checker to produce a formula in Conjunctive Normal Form (CNF). The formula is analyzed with off-the-shelf model counters to obtain precise counts with respect to different model behavior. QuantifyML enables i) evaluating learnability by comparing the counts for the outputs to ground truth, expressed as logical predicates, ii) comparing the performance of models built with different machine learning algorithms (decision-trees vs. neural networks), and iii) quantifying the safety and robustness of models.

Copy complexity of Horn formulas with respect to unit read-once resolution

In this paper, we discuss the copy complexity of Horn formulas with respect to unit resolution. A Horn formula is a boolean formula in conjunctive normal form (CNF) with at most one positive literal per clause. Horn formulas find applications in a number of domains, such as program verification (abstract interpretation) and logic programming (answer set programming). Quantified Horn clauses are used extensively in temporal verification of universal properties. Resolution is one of the oldest proof systems (refutation systems) for the boolean satisfiability problem (SAT), when the input is presented in conjunctive normal form (CNF). It is both sound and complete, although inefficient, when compared to other stronger proof systems for boolean formulas. Despite its inefficiency, the simple nature of resolution makes it an integral part of several theorem provers. Unit resolution is a restricted form of resolution in which each resolution step needs to use a clause with only one literal (unit literal clause). While not complete for general CNF formulas, unit resolution is complete for Horn formulas. Read-once resolution is a form of resolution in which each clause (input or derived) may be used in at most one resolution step. As with unit resolution, read-once resolution is incomplete in general and complete for Horn clauses. This paper focuses on a combination of unit resolution and read-once resolution called unit read-once resolution. Unit read-once resolution is incomplete for Horn clauses. In this paper, we study the copy complexity problem in Horn formulas with respect to unit read-once resolution. Briefly, the copy complexity of a formula with respect to unit read-once resolution, is the smallest number k, such that replicating each clause k times guarantees the existence of a unit read-once resolution refutation (UROR). This paper focuses on two problems related to the copy complexity of Horn formulas with respect to unit read-once resolution. We first relate the copy complexity of Horn formulas with respect to unit read-once resolution to the copy complexity of the corresponding Horn constraint system with respect to the addition rule. We also examine a form of copy complexity in which we permit replication of derived clauses, in addition to the input clauses. Finally, we provide a polynomial time algorithm for the problem of checking if a 2-CNF formula has a UROR.

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form.

The satisfiability (SAT) problem is a core problem in computer science. Existing studies have shown that most industrial SAT instances can be effectively solved by modern SAT solvers while random SAT instances cannot. It is believed that the structural characteristics of different SAT formula classes are the reasons behind this difference. In this paper, we study the structural properties of propositional formulas in conjunctive normal form (CNF) by the principle of structural entropy of formulas. First, we used structural entropy to measure the complex structure of a formula and found that the difficulty solving the formula is related to the structural entropy of the formula. The smaller the compressing information of a formula, the more difficult it is to solve the formula. Secondly, we proposed a -approximation strategy to approximate the structural entropy of large formulas. The experimental results showed that the proposed strategy can effectively approximate the structural entropy of the original formula and that the approximation ratio is more than 92%. Finally, we analyzed the structural properties of a formula in the solution process and found that a local search solver tends to select variables in different communities to perform the next round of searches during a search and that the structural entropy of a variable affects the probability of the variable being flipped. By using these conclusions, we also proposed an initial candidate solution generation strategy for a local search for SAT, and the experimental results showed that this strategy effectively improves the performance of the solvers CCAsat and Sparrow2011 when incorporated into these two solvers.

An Anonymous Credential System with Constant-Size Attribute Proofs for CNF Formulas with Negations

To enhance the user’s privacy in electronic ID, anonymous credential systems have been researched. In the anonymous credential system, a trusted issuing organization first issues a certificate certifying the user’s attributes to a user. Then, in addition to the possession of the certificate, the user can anonymously prove only the necessary attributes. Previously, an anonymous credential system was proposed, where CNF (Conjunctive Normal Form) formulas on attributes can be proved. The advantage is that the attribute proof in the authentication has the constant size for the number of attributes that the user owns and the size of the proved formula. Thus, various expressive logical relations on attributes can be efficiently verified. However, the previous system has a limitation: the proved CNF formulas cannot include any negation. Therefore, in this paper, we propose an anonymous credential system with constant-size attribute proofs such that the user can prove CNF formulas with negations. For the proposed system, we extend the previous accumulator for the limited CNF formulas to verify CNF formulas with negations.

NAE-resolution: A new resolution refutation technique to prove not-all-equal unsatisfiability

AbstractIn this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

Complexity of Unordered CNF Games

The classic TQBF problem is to determine who has a winning strategy in a game played on a given conjunctive normal form formula (CNF), where the two players alternate turns picking truth values for the variables in a given order, and the winner is determined by whether the CNF gets satisfied. We study variants of this game in which the variables may be played in any order, and each turn consists of picking a remaining variable and a truth value for it. For the version where the set of variables is partitioned into two halves and each player may only pick variables from his or her half, we prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for unbounded-width CNFs (Schaefer, STOC 1976). For the general unordered version (where each variable can be picked by either player), we also prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for 6-CNFs (Ahlroth and Orponen, MFCS 2012) and PSPACE-complete for positive 11-CNFs (Schaefer, STOC 1976).

ADDMC: Weighted Model Counting with Algebraic Decision Diagrams

We present an algorithm to compute exact literal-weighted model counts of Boolean formulas in Conjunctive Normal Form. Our algorithm employs dynamic programming and uses Algebraic Decision Diagrams as the main data structure. We implement this technique in ADDMC, a new model counter. We empirically evaluate various heuristics that can be used with ADDMC. We then compare ADDMC to four state-of-the-art weighted model counters (Cachet, c2d, d4, and miniC2D) on 1914 standard model counting benchmarks and show that ADDMC significantly improves the virtual best solver.

Exact Learning

A major problem in computational learning theory is whether the class of formulas in conjunctive normal form (CNF) is efficiently learnable. Although it is known that this class cannot be polynomially learned using either membership or equivalence queries alone, it is open whether the CNF class can be polynomially learned using both types of queries. One of the most important results concerning a restriction of the CNF class is that propositional Horn formulas are polynomial time learnable in Angluin’s exact learning model with membership and equivalence queries. In this work, we push this boundary and show that the class of multivalued dependency formulas (MVDF), which non-trivially extends propositional Horn, is polynomially learnable from interpretations. We then provide a notion of reduction between learning problems in Angluin’s model, showing that a transformation of the algorithm suffices to efficiently learn multivalued database dependencies from data relations. We also show via reductions that our main result extends well known previous results and allows us to find alternative solutions for them.

A resolution calculus for MinSAT

Abstract The logical calculus for SAT are not valid for MaxSAT and MinSAT because they preserve satisfiability but not the number of unsatisfied clauses. To overcome this drawback, a MaxSAT resolution rule preserving the number of unsatisfied clauses was defined in the literature. This rule is complete for MaxSAT when it is applied following a certain strategy. In this paper we first prove that the MaxSAT resolution rule also provides a complete calculus for MinSAT if it is applied following the strategy proposed here. We then describe an exact variable elimination algorithm for MinSAT based on that rule. Finally, we show how the results for Boolean MinSAT can be extended to solve the MinSAT problem of the multiple-valued clausal forms known as signed conjunctive normal form formulas.

Efficient Verified (UN)SAT Certificate Checking

SAT solvers decide the satisfiability of Boolean formulas in conjunctive normal form. They are commonly used for software and hardware verification. Modern SAT solvers are highly complex and optimized programs. As a single bug in the solver may invalidate the verification of many systems, SAT solvers output certificates for their answer, which are then checked independently. However, even certificate checking requires highly optimized non-trivial programs. This paper presents the first SAT solver certificate checker that is formally verified down to the integer sequence representing the formula. Our tool supports the full DRAT standard, and is even faster than the unverified state-of-the-art tool drat-trim, on a realistic set of benchmarks drawn from the 2016 and 2017 SAT competitions. An optional multi-threaded mode further reduces the runtime, in particular for big certificates.

A SAT Approach to Branchwidth

Branch decomposition is a prominent method for structurally decomposing a graph, a hypergraph, or a propositional formula in conjunctive normal form. The width of a branch decomposition provides a measure of how well the object is decomposed. For many applications, it is crucial to computing a branch decomposition whose width is as small as possible. We propose an approach based on Boolean Satisfiability (SAT) to finding branch decompositions of small width. The core of our approach is an efficient SAT encoding that determines with a single SAT-call whether a given hypergraph admits a branch decomposition of a certain width. For our encoding, we propose a natural partition-based characterization of branch decompositions. The encoding size imposes a limit on the size of the given hypergraph. To break through this barrier and to scale the SAT approach to larger instances, we develop a new heuristic approach where the SAT encoding is used to locally improve a given candidate decomposition until a fixed-point is reached. This new SAT-based local improvement method scales now to instances with several thousands of vertices and edges.

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.