CNF Formulas Research Articles

Polynomial calculus for optimization

MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem.Weighted Polynomial Calculus is a natural generalization of the systems MaxSAT-Resolution and weighted Resolution. Unlike such systems, weighted Polynomial Calculus manipulates polynomials with coefficients in a finite field and either weights in N or Z. We show the soundness and completeness of weighted Polynomial Calculus via an algorithmic procedure.Weighted Polynomial Calculus, with weights in N and coefficients in F2, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in Z, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable.

Approximating Long Cycle Above Dirac’s Guarantee

Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit “natural” guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m/2+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m/2 +k$$\\end{document} clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n/2+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n/2 +k$$\\end{document} for some k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 1$$\\end{document} and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree δ(G)≤n/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta (G)\\le n/2$$\\end{document}, the length of a longest cycle L is at least 2δ(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\delta (G)$$\\end{document}. Thus the “essential” part in finding the longest cycle is in approximating the “offset” k=L-2δ(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k = L - 2 \\delta (G)$$\\end{document}. The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2δ(G)+Ω(f(k))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\delta (G)+\\Omega (f(k))$$\\end{document}.

Backdoor DNFs

We introduce backdoor DNFs, as a tool to measure the theoretical hardness of CNF formulas. Like backdoor sets and backdoor trees, backdoor DNFs are defined relative to a tractable class of CNF formulas. Each conjunctive term of a backdoor DNF defines a partial assignment that moves the input CNF formula into the base class. Backdoor DNFs are more expressive and potentially smaller than their predecessors backdoor sets and backdoor trees. We establish the fixed-parameter tractability of the backdoor DNF detection problem. Our results hold for the fundamental base classes Horn and 2CNF, and their combination. We complement our theoretical findings by an empirical study. Our experiments show that backdoor DNFs provide a significant improvement over their predecessors.

The phase transition in random regular exact (3s + s, k)-SAT problem

A CNF formula with each clause of length k and each variable occurring 4s times, where positive occurrences are 3s and negative occurrences are s, is a regular (3s + s, k)-CNF formula (F3s+s,k formula). The random regular exact (3s + s, k)-SAT problem is whether there exists a set of Boolean variable assignments such that exactly one literal is true for each clause in the F3s+s,k formula. By introducing a random instance generation model, the satisfiability phase transition of the solution is analyzed by using the first moment method, the second moment method, and the small subgraph conditioning method, which gives the phase transition point s* of the random regular exact (3s + s, k)-SAT problem for k≥3. When s < s*, F3s+s,k formula is satisfiable with high probability; when s > s*, F3s+s,k formula is unsatisfiable with high probability. Finally, through the experimental verification, the results show that the theoretical proofs are consistent with the experimental results.

A Compiler for Weak Decomposable Negation Normal Form

This paper integrates weak decomposable negation normal form (wDNNF) circuits, introduced by Akshay et al. in 2018, into the knowledge compilation map. This circuit type generalises decomposable negation normal form (DNNF) circuits in such a way that they allow a restricted form of sharing variables among the inputs of a conjunction node. We show that wDNNF circuits have the same properties as DNNF circuits regarding the queries and transformations presented in the knowledge compilation map, whilst being strictly more succinct than DNNF circuits (that is, they can represent Boolean functions compactly). We also present and evaluate a knowledge compiler, called Bella, for converting CNF formulae into wDNNF circuits. Our experiments demonstrate that wDNNF circuits are suitable for configuration instances.

A Novel, Efficient and Generalised Approach to Boolean Satisfiability

One question which has occupied mathematicians, engineers and scientists of this age is the problem of Boolean satisfiability (SAT) which asks whether a given CNF formula has at least a satisfying assignment. The goal of this work is to present a novel, efficient and general method of deciding SAT and finding all the satisfying assignments of CNF formulas. This method is powerful as it employs resolution identities in transforming clauses without a particular variable into clauses with that variable.

Exact satisfiability and phase transition analysis of the regular (k, d)-CNF formula

Exact satisfiability and phase transition analysis of the regular (k, d)-CNF formula

SAT backdoors: Depth beats size

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes.

Are hitting formulas hard for resolution?

Hitting formulas, introduced by Iwama, are an unusual class of propositional CNF formulas. Not only is their satisfiability decidable in polynomial time, but even their models can be counted in closed form. This stands in stark contrast with other polynomial-time decidable classes, which usually have algorithms based on backtracking and resolution and for which model counting remains hard, like 2-SAT and Horn-SAT. However, those resolution-based algorithms usually easily imply an upper bound on resolution complexity, which is missing for hitting formulas. Are hitting formulas hard for resolution? In this paper we take the first steps towards answering this question. We show that the resolution complexity of hitting formulas is dominated by so-called irreducible hitting formulas, first studied by Kullmann and Zhao, that cannot be composed of smaller hitting formulas. However, by definition, large irreducible unsatisfiable hitting formulas are difficult to construct; it is not even known whether infinitely many exist. Building upon our theoretical results, we implement an efficient algorithm on top of the Nauty software package to enumerate all irreducible unsatisfiable hitting formulas with up to 14 clauses. We also determine the exact resolution complexity of the generated hitting formulas with up to 13 clauses by extending a known SAT encoding for our purposes. Our experimental results suggest that hitting formulas are indeed hard for resolution.

Learning Horn envelopes via queries from language models

We present an approach for systematically probing a trained neural network to extract a symbolic abstraction of it, represented as a Boolean formula. We formulate this task within Angluin's exact learning framework, where a learner attempts to extract information from an oracle (in our work, the neural network) by posing membership and equivalence queries. We adapt Angluin's algorithm for Horn formula to the case where the examples are labelled w.r.t. an arbitrary Boolean formula in CNF (rather than a Horn formula). In this setting, the goal is to learn the smallest representation of all the Horn clauses implied by a Boolean formula—called its Horn envelope—which in our case correspond to the rules obeyed by the network. Our algorithm terminates in exponential time in the worst case and in polynomial time if the target Boolean formula can be closely approximated by its envelope. We also show that extracting Horn envelopes in polynomial time is as hard as learning CNFs in polynomial time. To showcase the applicability of the approach, we perform experiments on BERT based language models and extract Horn envelopes that expose occupation-based gender biases.

Improved Algorithms for Maximum Satisfiability and Its Special Cases

The Maximum Satisfiability (MAXSAT) problem is an optimization version of the Satisfiability problem (SAT) in which one is given a CNF formula with n variables and needs to find the maximum number of simultaneously satisfiable clauses. Recent works achieved significant progress in proving new upper bounds on the worst-case computational complexity of MAXSAT. All these works reduce general MAXSAT to a special case of MAXSAT where each variable appears a small number of times. So, it is important to design fast algorithms for (n,k)-MAXSAT to construct an efficient exact algorithm for MAXSAT. (n,k)-MAXSAT is a special case of MAXSAT where each variable appears at most k times in the input formula. For the (n,3)-MAXSAT problem, we design a O*(1.1749^n) algorithm improving on the previous record running time of O*(1.191^n). For the (n,4)-MAXSAT problem, we construct a O*(1.3803^n) algorithm improving on the previous best running time of O*(1.4254^n). Using the results, we develop a O*(1.0911^L) algorithm for the MAXSAT where L is a length of the input formula which improves previous algorithm with O*(1.0927^L) running time.

Constraint Optimization over Semirings

Interpretations of logical formulas over semirings (other than the Boolean semiring) have applications in various areas of computer science including logic, AI, databases, and security. Such interpretations provide richer information beyond the truth or falsity of a statement. Examples of such semirings include Viterbi semiring, min-max or access control semiring, tropical semiring, and fuzzy semiring. The present work investigates the complexity of constraint optimization problems over semirings. The generic optimization problem we study is the following: Given a propositional formula phi over n variable and a semiring (K,+, . ,0,1), find the maximum value over all possible interpretations of phi over K. This can be seen as a generalization of the well-known satisfiability problem (a propositional formula is satisfiable if and only if the maximum value over all interpretations/assignments over the Boolean semiring is 1). A related problem is to find an interpretation that achieves the maximum value. In this work, we first focus on these optimization problems over the Viterbi semiring, which we call optConfVal and optConf. We first show that for general propositional formulas in negation normal form, optConfVal and optConf are in FP^NP. We then investigate optConf when the input formula phi is represented in the conjunctive normal form. For CNF formulae, we first derive an upper bound on the value of optConf as a function of the number of maximum satisfiable clauses. In particular, we show that if r is the maximum number of satisfiable clauses in a CNF formula with m clauses, then its optConf value is at most 1/4^(m-r). Building on this we establish that optConf for CNF formulae is hard for the complexity class FP^NP[log]. We also design polynomial-time approximation algorithms and establish an inapproximability for optConfVal. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings.

The 2CNF Boolean formula satisfiability problem and the linear space hypothesis

We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by natural size parameters using simultaneously polynomial time and sub-linear space. We are particularly focused on 2SAT3—a restricted variant of the 2CNF Boolean (propositional) formula satisfiability problem in which each variable of a given 2CNF formula appears at most 3 times in the form of literals—parameterized by the total number mvbl(ϕ) of variables of each given Boolean formula ϕ. We propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which asserts that (2SAT3,mvbl) cannot be solved in polynomial time using only “sub-linear” space (i.e., mvbl(x)εpolylog(|x|) space for a constant ε∈[0,1)) on all instances x. Immediate consequences of LSH include L≠NL, LOGDCFL≠LOGCFL, and SC≠NSC. For our investigation, we fully utilize a key notion of “short reductions”, under which the class PsubLIN of all parameterized polynomial-time sub-linear-space solvable problems is indeed closed.

An Isomorphism-Invariant Distance Function on Propositional Formulas in CNF

We define a distance function on propositional formulas in CNF as a measure of non-isomorphism of formulas: the larger the distance between two formulas is, the further they are from being isomorphic. This distance induces a metric on isomorphism classes of formulas. We show how this distance can be used for SAT solving, namely for per-instance algorithm selection where there is a “portfolio” of SAT solvers and there is a “meta-solver” that chooses a solver from the portfolio for a given input formula.

One-Way Functions and (Im)perfect Obfuscation

A program obfuscator takes a program and outputs a “scrambled” version of it, where the goal is that the obfuscated program will not reveal much about its structure beyond what is apparent from executing it. There are several ways of formalizing this goal. Specifically, in indistinguishability obfuscation, first defined by Barak et al. (CRYPTO 2001), the requirement is that the results of obfuscating any two functionally equivalent programs (circuits) will be computationally indistinguishable. Recently, a fascinating candidate construction for indistinguishability obfuscation was proposed by Garg et al. (FOCS 2013). This has led to a flurry of discovery of intriguing constructions of primitives and protocols whose existence was not previously known (for instance, fully deniable encryption by Sahai and Waters, STOC 2014). Most of them explicitly rely on additional hardness assumptions, such as one-way functions. Our goal is to get rid of this extra assumption. We cannot argue that indistinguishability obfuscation of all polynomial-time circuits implies the existence of one-way functions, since if P = NP, then program obfuscation (under the indistinguishability notion) is possible. Instead, the ultimate goal is to argue that if P 6= NP and program obfuscation is possible, then one-way functions exist. Our main result is that if NP 6⊆ io-BPP and there is an efficient (even imperfect) indistinguishability obfuscator, then there are one-way functions. In addition, we show that the existence of an indistinguishability obfuscator implies (unconditionally) the existence of SZKarguments for NP. This, in turn, provides an alternative version of our main result, based on the assumption of hard-on-the average NP problems. To get some of our results we need obfuscators for simple programs such as 3CNF formulas. ∗This paper incorporates the manuscript of Moran and Rosen [MR13]. †Weizmann Institute of Science. Email: {ilan.komargodski,moni.naor,eylon.yogev}@weizmann.ac.il. Supported in part by a grant from the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation, BSF, IMOS and the Citi Foundation. Moni Naor is the incumbent of the Judith Kleeman Professorial Chair. ‡IDC Herzliya. Email: talm@idc.ac.il. Supported by ISF grant no. 1790/13 and by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 293843 §Cornell University. Email: rafael@cs.cornell.edu. Supported in part by a Alfred P. Sloan Fellowship, Microsoft New Faculty Fellowship, NSF Award CNS1217821, NSF CAREER Award CCF-0746990, NSF Award CCF-1214844, AFOSR YIP Award FA9550-10-1-0093, and DARPA and AFRL under contract FA8750-11-20211. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Defense Advanced Research Projects Agency or the US Government. ¶IDC Herzliya. Email: alon.rosen@idc.ac.il. Supported by ISF grant no. 1255/12 and by the ERC under the EU’s Seventh Framework Programme (FP/2007-2013) ERC Grant Agreement n. 307952.

Combining bounded solving and controllable randomization for approximate model counting

ABSTRACT Propositional model counting is the problem of computing the satisfying assignments count of a given CNF formula. Due to the weak solving ability of the existing exact model counters in large-scale problems, approximate model counting has been proposed as a practical alternative to exact model counting. Most of the best current approaches are based on XOR constraints for approximate model counting, also known as the hashing-based method. In this paper, a new approximate model counter using short XOR constraints is proposed, which integrates bounded solving and controllable randomisation. By constantly adding short XOR constants, the solution space has been reduced to a smaller one. However, a smaller scale of solution space may result in worse result accuracy. Therefore, by limiting the scale of solution space that applies XOR constraints, bounded solving effectively increases the accuracy of a model counter. Controllable randomisation makes use of backbone variables and the constraint level among variables, such that the short XOR constraints can reach the same reduction effect on solution space as the long ones. It improves the quality of XOR constraints as well as the SAT solving efficiency. Experimentally, we demonstrate that in almost every benchmark used in well-known ApproxMC2 and STAC_CNF, our approach outperforms the existing approximate model counters in both accuracy and efficiency.

The Transition Phenomenon of (1,0)-d-Regular (k, s)-SAT

For a d-regular (k,s)-CNF formula, a problem is to determine whether it has a (1,0)-super solution. If so, it is called (1,0)-d-regular (k,s)-SAT. A (1,0)-super solution is an assignment that satisfies at least two literals of each clause. When the value of any one of the variables is flipped, the (1,0)-super solution is still a solution. Super solutions have gained significant attention for their robustness. Here, a d-regular (k,s)-CNF formula is a special CNF formula with clauses of size exactly k, in which each variable appears exactly s-times, and the absolute frequency difference between positive and negative occurrences of each variable is at most a nonnegative integer d. Obviously, the structure of a d-regular (k,s)-CNF formula is much more regular than other formulas. In this paper, we certify that, for k≥5, there is a critical function φ(k,d) such that, if s≤φ(k,d), all d-regular (k,s)-CNF formulas have a (1,0)-super solution; otherwise (1,0)-d-regular (k,s)-SAT is NP-complete. By the Lopsided Local Lemma, we get an existence condition of (1,0)-super solutions and propose an algorithm to find the lower bound of φ(k,d).

Lower Bounds on Intermediate Results in Bottom-Up Knowledge Compilation

Bottom-up knowledge compilation is a paradigm for generating representations of functions by iteratively conjoining constraints using a so-called apply function. When the input is not efficiently compilable into a language - generally a class of circuits - because optimal compiled representations are provably large, the problem is not the compilation algorithm as much as the choice of a language too restrictive for the input. In contrast, in this paper, we look at CNF formulas for which very small circuits exists and look at the efficiency of their bottom-up compilation in one of the most general languages, namely that of structured decomposable negation normal forms (str-DNNF). We prove that, while the inputs have constant size representations as str-DNNF, any bottom-up compilation in the general setting where conjunction and structure modification are allowed takes exponential time and space, since large intermediate results have to be produced. This unconditionally proves that the inefficiency of bottom-up compilation resides in the bottom-up paradigm itself.

Hardness of uncertain segment cover, contiguous SAT and visibility with uncertain obstacles

In this paper, we prove that the uncertain segment cover problem is NP-hard. The uncertain segment cover problem, which appears in solving the question of visibility of a segment in the plane from a given point in the presence of uncertain obstacles, asks if a given interval can be covered by a collection of uncertain sub-intervals having two possible positions. This is done by showing that this problem is equivalent to a special version of the SAT problem that we will call contiguous SAT. In a contiguous SAT problem a CNF formula with a given ordering on its clauses is given such that any literal appears contiguously. We prove that contiguous SAT problem is NP-hard and as a consequence it follows that the uncertain segment cover problem is NP-hard. The approximation solution of this problem and its hardness are also studied.

CNF Satisfiability in a Subspace and Related Problems

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over \({\mathbb {F}}_2\) each of which is a product of affine forms. We focus on the case of k-CNF formulas (the \({k}-\textsc {Sub}-\textsc {Sat}\) problem). Clearly, \({k}-\textsc {Sub}-\textsc {Sat}\) is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that \({2}-\textsc {Sub}-\textsc {Sat}\) is NP-hard, and \({\small \mathrm {W}}[1]\)-hard when parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-\({2}-\textsc {Sub}-\textsc {Sat}\) is NP-hard to approximate better than the trivial 3/4 ratio even on satisfiable instances. On the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with running time \(O^*(1.5)^r\) for \({2}-\textsc {Sub}-\textsc {Sat}\), where r is the subspace dimension, as well as an \(O^*(1.4312)^n\) time algorithm where n is the number of variables. Turning to \({k}-\textsc {Sub}-\textsc {Sat}\) for \(k \geqslant 3\), while known algorithms for solving a system of degree k polynomial equations already imply a solution with running time \(\approx 2^{r(1-1/2k)}\), we explore a more combinatorial approach. Based on an analysis of critical variables (a key notion underlying the randomized k-SAT algorithm of Paturi, Pudlak, and Zane), we give an algorithm with running time \(\approx {n\atopwithdelims (){\leqslant t}} 2^{n-n/k}\) where n is the number of variables and t is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for \({k}-\textsc {Sub}-\textsc {Sat}\) with running time \(\approx 2^{n-n/2k}\). This algorithm is in fact oblivious to the structure of the subspace, and extends when the subspace-membership constraint is replaced by any constraint for which partial satisfying assignments can be efficiently completed to a full satisfying assignment. Finally, for systems of O(n) polynomial equations in n variables over \({\mathbb {F}}_2\), we give a fast exponential algorithm when each polynomial has bounded degree irreducible factors (but can otherwise have large degree) using a degree reduction trick.