SAT Solving Using XOR-OR-AND Normal Forms

When studying the propositional satisfiability problem (SAT), that is, the problem of deciding whether a propositional formula is satisfiable, it is typically assumed that the formula is given in the conjunctive normal form (CNF). Also most software tools for deciding satisfiability of a formula (SAT solvers) assume that their input is in CNF. An important reason for this is that it is simpler to develop efficient data structures and algorithms for CNF than for arbitrary formulas. On the other hand, using CNF makes efficient modeling of an application cumbersome. Therefore one often employs a more general formula representation in modeling and then transforms the formula into CNF for SAT solvers. Transforming a propositional formula in CNF either increases the formula size exponentially or requires the use of auxiliary variables, which can have an negative effect on the performance of a SAT solver in the worst-case. Moreover, by translating to CNF one often loses information about the structure of the original problem. In this chapter we survey methods for solving propositional satisfiability problems when the input formula is not given in CNF but as a general formula or even more compactly as a Boolean circuit. We show how the techniques applied in CNF level Davis-Putnam-Loveland-Logemann algorithm generalize to Boolean circuits and how the problem structure available in the circuit form can be exploited. Then we consider a closely related area of automatic test pattern generation (ATPG) for digital circuits and review classical ATPG algorithms, formulation of ATPG as a SAT problem, and advanced techniques for SAT-based ATPG.

When studying the propositional satisfiability problem (SAT), that is, the problem of deciding whether a propositional formula is satisfiable, it is typically assumed that the formula is given in the conjunctive normal form (CNF). Also most software tools for deciding satisfiability of a formula (SAT solvers) assume that their input is in CNF. An important reason for this is that it is simpler to develop efficient data structures and algorithms for CNF than for arbitrary formulas. On the other hand, using CNF makes efficient modeling of an application cumbersome. Therefore one often employs a more general formula representation in modeling and then transforms the formula into CNF for SAT solvers. Transforming a propositional formula in CNF either increases the formula size exponentially or requires the use of auxiliary variables, which can have an negative effect on the performance of a SAT solver in the worst-case. Moreover, by translating to CNF one often loses information about the structure of the original problem. In this chapter we survey methods for solving propositional satisfiability problems when the input formula is not given in CNF but as a general formula or even more compactly as a Boolean circuit. We show how the techniques applied in CNF level Davis-Putnam-Loveland-Logemann algorithm generalize to Boolean circuits and how the problem structure available in the circuit form can be exploited. Then we consider a closely related area of automatic test pattern generation (ATPG) for digital circuits and review classical ATPG algorithms, formulation of ATPG as a SAT problem, and advanced techniques for SAT-based ATPG.

Algebraic Normal Form (ANF) and Conjunctive Normal Form (CNF) are commonly used to encode problems in Boolean algebra. ANFs are typically solved via Grobner ¨ 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 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.

Resolution is the rule of inference at the basis of most procedures for automated reasoning. In these procedures, the input formula is first translated into an equisatisfiable formula in conjunctive normal form (CNF) and then represented as a set of clauses. Deduction starts by inferring new clauses by resolution, and goes on until the empty clause is generated or satisfiability of the set of clauses is proven, e.g., because no new clauses can be generated. In this paper, we restrict our attention to the problem of evaluating Quantified Boolean Formulas (QBFs). In this setting, the above outlined deduction process is known to be sound and complete if given a formula in CNF and if a form of resolution, called ``Q-resolution'', is used. We introduce Q-resolution on terms, to be used for formulas in disjunctive normal form. We show that the computation performed by most of the available procedures for QBFs --based on the Davis-Logemann-Loveland procedure (DLL) for propositional satisfiability-- corresponds to a tree in which Q-resolution on terms and clauses alternate. This poses the theoretical bases for the introduction of learning, corresponding to recording Q-resolution formulas associated with the nodes of the tree. We discuss the problems related to the introduction of learning in DLL based procedures, and present solutions extending state-of-the-art proposals coming from the literature on propositional satisfiability. Finally, we show that our DLL based solver extended with learning, performs significantly better on benchmarks used in the 2003 QBF solvers comparative evaluation.

Given a propositional formula $F$ in Conjunctive Normal Form (CNF), a SAT solver decides whether it is satisfiable or not. It is often required to find a solution to a satisfiable CNF formula F, which optimizes a given Pseudo-Boolean objective function Ψ, that is, to extend SAT to optimization. MaxSAT is a widely used extension of SAT to optimization. A MaxSAT solver can be applied to optimize a Pseudo-Boolean objective function Ψ, given a CNF formula F, whenever Ψ is a linear function. MaxSAT has a diverse plethora of applications, including applications in computer-aided design, artificial intelligence, planning, scheduling and bioinformatics. A variety of approaches to MaxSAT have been developed over the last two decades. In this tutorial, we focus on anytime MaxSAT algorithms, where an anytime algorithm is expected to find better and better solutions, the longer it keeps running. The anytime property is crucial in industrial applications, since it allows the user to: 1) get an approximate solution even for very difficult instances, and 2) trade quality for performance by regulating the timeout. Anytime MaxSAT solvers have been evaluated at yearly MaxSAT Evaluations since 2011 in the so-called incomplete tracks. We trace the evolvement of anytime MaxSAT algorithms over the last decade and lay out the algorithms, applied by the winners of MaxSAT Evaluation 2020. Furthermore, we touch upon anytime algorithms for optimization problems beyond MaxSAT, such as bit-vector optimization and the problem of optimizing an arbitrary not-necessarily-linear function, given a CNF formula. Finally, we discuss challenges and future work.

We show that a conjunctive normal form (CNF) formula F is unsatisfiable iff there is a set of points of the Boolean space that is stable with respect to F. So testing the satisfiability of a CNF formula reduces to looking for a stable set of points (SSP). We give some properties of SSPs and describe a simple algorithm for constructing an SSP for a CNF formula. Building an SSP can be viewed as a “natural” way of search space traversal. This naturalness of search space examination allows one to make use of the regularity of CNF formulas to be checked for satisfiability. We illustrate this point by showing that if a CNF F formula is symmetric with respect to a group of permutations, it is very easy to make use of this symmetry when constructing an SSP. As an example, we show that the unsatisfiability of pigeon-hole CNF formulas can be proven by examining only a set of points whose size is quadratic in the number of holes.

In this paper, we discuss a simple, Monte Carlo algorithm for the problem of checking whether a Quantified Boolean Formula (QBF) in Conjunctive Normal Form (CNF), with at most two literals per clause has a model. The term k-CNF is used to describe boolean formulas in CNF, with at most k literals per clause and the problem of checking whether a given k-CNF formula is satisfiable is called the k-SAT problem. A QBF is a boolean formula, accompanied by a quantifier string which imposes a linear ordering on the variables of that formula. The problem of finding a model for a QBF formula in CNF, with at at most k literals per clause is called the QkSAT problem. The QkSAT problem is PSPACE-complete, for k≥3. However, the Q2SAT problem can be decided in polynomial time; the graph-based procedure, discussed in [1], is the first such algorithm for this problem. This procedure requires the construction of a global implication graph, corresponding to the input formula and searching for certain paths in this graph. Hence the complete set of clauses must be part of the input. We propose an incremental, randomized approach for the Q2SAT problem that is essentially local in nature, in that the complete clausal set need not be provided at any time, in the presence of a verifier. We show that the randomized algorithm can be analyzed as a one-dimensional random walk, with one reflecting barrier and one absorbing barrier. On a Q2SAT instance with m clauses on n variables, our coin-flipping algorithm runs in time O(n2 · V(m, n)), where V(m, n) is the time required to verify that a given model satisfies the formula. Additionally, if the instance is satisfiable, the probability that our algorithm fails to find a model is less than one half. The design and analysis of a randomized algorithm for a problem, is important from both the theoretical and the practical perspectives. Randomized approaches tend to be simple and elegant, thereby making the process of checking correctness, effortless as well. The randomized approach discussed in this paper lays the groundwork for analyzing a number of problems related to 2CNF formulas and directed graphs. We remark that our work in this paper is the first randomized algorithm for a class of QBFs.

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 ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas (Formula presented.) with m pigeons and n holes, and show that this is tight. (2) For PCR, we prove an ω(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 ω(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.

We show that a conjunctive normal form (CNF) formula F is unsatisfiable if and only if there is a set of points of the Boolean space that is stable with respect to F. So testing the satisfiability of a CNF formula reduces to looking for a stable set of points (SSP). We give some properties of SSPs and describe a simple algorithm for constructing an SSP for a CNF formula. Building an SSP can be viewed as a "natural" way of search space traversal. This naturalness of search space examination allows one to make use of the regularity of CNF formulas to be checked for satisfiability. We illustrate this point by showing that if a CNF F formula is symmetric with respect to a group of permutations, it is very easy to make use of this symmetry when constructing an SSP. As an example, we show that the unsatisfiability of pigeon-hole CNF formulas can be proven by examining only a set of points whose size is quadratic in the number of holes. Finally, we introduce the notion of an SSP with excluded directions and sketch a procedure of satisfiability testing based on the construction of such SSPs.

Boolean satisfiability (SAT) and its extensions have become a core technology in many application domains, such as planning and formal verification, and continue finding various new application domains today. The SAT-based approach divides into three steps: encoding, preprocessing, and search. It is often argued that by encoding arbitrary Boolean formulas in conjunctive normal form (CNF), structural properties of the original problem are not reflected in the CNF. This should result in the fact that CNF-level preprocessing and SAT solver techniques have an inherent disadvantage compared to related techniques applicable on the level of more structural SAT instance representations such as Boolean circuits. Motivated by this, various simplification techniques and intricate CNF encodings for circuit-level SAT instance representations have been proposed. On the other hand, based on the highly efficient CNF-level clause learning SAT solvers, there is also strong support for the claim that CNF is sufficient as an input format for SAT solvers. In this work we study the effect of CNF-level simplification techniques, focusing on SatElite-style variable elimination (VE) and what we call blocked clause elimination (BCE). We show that BCE is surprisingly effective both in theory and in practice on CNF formulas resulting from a standard CNF encoding for circuits: without explicit knowledge of the underlying circuit structure, it achieves the same level of simplification as a combination of circuit-level simplifications and previously suggested polarity-based CNF encodings. We also show that VE can achieve many of the same effects as BCE, but not all. On the other hand, it turns out that VE and BCE are indeed partially orthogonal techniques. We also study the practical effects of combining BCE and VE for reducing the size of formulas and on the running times of state-of-the-art SAT solvers. Furthermore, we address the problem of how to construct original witnesses to satisfiable CNF formulas when applying a combination of BCE and VE.

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.

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.

Satisfiability of Boolean formulas (SAT) is one of the most prominent NP-complete problems. We consider k-SAT, the decision problem that asks whether formulas in conjunctive normal form (CNF) with clauses of size at most k are satisfiable, and the more general problem called (d, k)ClSP (clause satisfaction problem) where the variables are d-valued instead of Boolean. For k-SAT and (d, k)-ClSP many algorithms have been presented whose running time is “moderately exponential” in the number of variables of the input formula. One of the fastest randomized algorithm for k-SAT is the PPSZ algorithm by Paturi, Pudlak, Saks, and Zane (FOCS 1998). We re-analyze the PPSZ algorithm and show that the bounds shown in the case where the input formula has at most one satisfying assignment (Unique k-SAT) hold in general, which was previously only known for k ≥ 5. We also show how to generalize PPSZ to (d, k)-ClSP, improving on the previous algorithms for most considered values of (d, k). Furthermore, we present a new algorithm based on PPSZ with exponentially better bounds for 3-SAT. For general k we show that in order to improve on PPSZ for k-SAT, it is enough to improve on PPSZ for Unique k-SAT.

Modern conflict-driven clause learning (CDCL) SAT solvers are very good in solving conjunctive normal form (CNF) formulas. However, some application problems involve lots of parity (xor) constraints which are not necessarily efficiently handled if translated into CNF. This paper studies solving CNF formulas augmented with xor-clauses in the DPLL(XOR) framework where a CDCL SAT solver is coupled with a separate xor-reasoning module. New techniques for analyzing xor-reasoning derivations are developed, allowing one to obtain smaller CNF clausal explanations for xor-implied literals and also to derive and learn new xor-clauses. It is proven that these new techniques allow very short unsatisfiability proofs for some formulas whose CNF translations do not have polynomial size resolution proofs, even when a very simple xor-reasoning module capable only of unit propagation is applied. The efficiency of the proposed techniques is evaluated on a set of challenging logical cryptanalysis instances.

The minimization of propositional formulae is a classical problem in logic, whose first algorithms date back at least to the 1950s with the works of Quine and Karnaugh. Most previous work in the area has focused on obtaining minimal, or quasi-minimal, formulae in conjunctive normal form (CNF) or disjunctive normal form (DNF), with applications in hardware design. In this paper, we are interested in the problem of obtaining an equivalent formula in any format, also allowing connectives that are not present in the original formula. We are primarily motivated in applying minimization algorithms to generate natural language translations of the original formula, where using shorter equivalents as input may result in better translations. Buchfuhrer and Umans have proved that the (decisional version of the) problem is Σp2-complete. We analyze three possible (practical) approaches to solving the problem. First, using brute force, generating all possible formulae in increasing size and checking if they are equivalent to the original formula by testing all possible variable assignments. Second, generating the Tseitin coding of all the formulae and checking equivalence with the original using a SAT solver. Third, encoding the problem as a Quantified Boolean Formula (QBF), and using a QBF solver. Our results show that the QBF approach largely outperforms the other two.