Resolution Refutation Research Articles

Hardness of Random Reordered Encodings of Parity for Resolution and CDCL

Parity reasoning is challenging for Conflict-Driven Clause Learning (CDCL) SAT solvers. This has been observed even for simple formulas encoding two contradictory parity constraints with different variable orders (Chew and Heule 2020). We provide an analytical explanation for their hardness by showing that they require exponential resolution refutations with high probability when the variable order is chosen at random. We obtain this result by proving that these formulas, which are known to be Tseitin formulas, have Tseitin graphs of linear treewidth with high probability. Since such Tseitin formulas require exponential resolution refutations, our result follows. We generalize this argument to a new class of formulas that capture a basic form of parity reasoning involving a sum of two random parity constraints with random orders. Even when the variable order for the sum is chosen favorably, these formulas remain hard for resolution. In contrast, we prove that they have short DRAT refutations. We show experimentally that the running time of CDCL SAT solvers on both classes of formulas grows exponentially with their treewidth.

On computing small variable disjunction branch-and-bound trees

This article investigates smallest branch-and-bound trees and their computation. We first revisit the notion of hiding sets to deduce lower bounds on the size of branch-and-bound trees for certain binary programs, using both variable disjunctions and general disjunctions. We then provide exponential lower bounds for variable disjunctions by a disjoint composition of smaller binary programs. Moreover, we investigate the complexity of finding small branch-and-bound trees using variable disjunctions: We show that it is not possible to approximate the size of a smallest branch-and-bound tree within a factor of 215n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\smash {2^{\\frac{1}{5}n}}$$\\end{document} in time O(2δn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(2^{\\delta n})$$\\end{document} with δ<15\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta <\ frac{1}{5}$$\\end{document}, unless the strong exponential time hypothesis fails. Similarly, for any ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon > 0$$\\end{document}, no polynomial time 2(12-ε)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\smash {2^{(\\frac{1}{2} - \\varepsilon )n}}$$\\end{document}-approximation is possible, unless P=NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {P} = \ ext {NP} $$\\end{document}. We also show that computing the size of a smallest branch-and-bound tree exactly is #P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\#P} $$\\end{document}-hard. Similar results hold for estimating the size of the tree produced by branching rules like most-infeasible branching. Finally, we discuss that finding small branch-and-bound trees generalizes finding short treelike resolution refutations, and thus non-automatizability results transfer from this setting.

Number of Variables for Graph Differentiation and the Resolution of Graph Isomorphism Formulas

We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution. Using this connection, we obtain upper and lower bounds for refuting graph isomorphism formulas in (normal) resolution. In particular, we show that if k is the minimum number of variables needed to distinguish two graphs with n vertices each, then there is an n O ( k ) resolution refutation size upper bound for the corresponding isomorphism formula, as well as lower bounds of 2 k -1 and k for the treelike resolution size and resolution clause space for this formula. We also show a (normal) resolution size lower bound of exp (Ω ( k 2 / n )) for the case of colored graphs with constant color class sizes. Applying these results, we prove the first exponential lower bound for graph isomorphism formulas in the proof system SRC-1, a system that extends resolution with a global symmetry rule, thereby answering an open question posed by Schweitzer and Seebach.

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.

Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

Tseitin-formulas are systems of parity constraints whose structure is described by a graph. These formulas have been studied extensively in proof complexity as hard instances in many proof systems. In this paper, we prove that a class of unsatisfiable Tseitin-formulas of bounded degree has regular resolution refutations of polynomial length if and only if the treewidth of all underlying graphs G for that class is in O(log |V (G)|). It follows that unsatisfiable Tseitin-formulas with polynomial length of regular resolution refutations are completely determined by the treewidth of the underlying graphs when these graphs have bounded degree. To prove this, we show that any regular resolution refutation of an unsatisfiable Tseitin-formula with graph G of bounded degree has length 2Ω(tw(G))/|V (G)|, thus essentially matching the known 2O(tw(G))poly(|V (G)|) upper bound. Our proof first connects the length of regular resolution refutations of unsatisfiable Tseitin-formulas to the size of representations of satisfiable Tseitin-formulas in decomposable negation normal form (DNNF). Then we prove that for every graph G of bounded degree, every DNNF-representation of every satisfiable Tseitin-formula with graph G must have size 2Ω(tw(G)) which yields our lower bound for regular resolution.

Proofs and Certificates for Max-SAT

Current Max-SAT solvers are able to efficiently compute the optimal value of an input instance but they do not provide any certificate of its validity. In this paper, we present a tool, called MS-Builder, which generates certificates for the Max-SAT problem in the particular form of a sequence of equivalence-preserving transformations. To generate a certificate, MS-Builder iteratively calls a SAT oracle to get a SAT resolution refutation which is handled and adapted into a sound refutation for Max-SAT. In particular, we prove that the size of the computed Max-SAT refutation is linear with respect to the size of the initial refutation if it is semi-read-once, tree-like regular, tree-like or semi-tree-like. Additionally, we propose an extendable tool, called MS-Checker, able to verify the validity of any Max-SAT certificate using Max-SAT inference rules. Both tools are evaluated on the unweighted and weighted benchmark instances of the 2020 Max-SAT Evaluation.

Approximate counting and NP search problems

We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory [Formula: see text] of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory [Formula: see text], this shows that [Formula: see text] does not prove every [Formula: see text] sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496–525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the “fixing lemma” from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185–239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a [Formula: see text] oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic.

Correction to: Near-Optimal Lower Bounds on Regular Resolution Refutations of Tseitin Formulas for All Constant-Degree Graphs

Correction to: Near-Optimal Lower Bounds on Regular Resolution Refutations of Tseitin Formulas for All Constant-Degree Graphs

Lower Bounds on OBDD Proofs with Several Orders

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε &gt; 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε &gt; 0 is a constant. The lower bound is based on multiparty communication complexity.

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.

Parameterized and exact algorithms for finding a read-once resolution refutation in 2CNF formulas

Parameterized and exact algorithms for finding a read-once resolution refutation in 2CNF formulas

Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space

We show a new connection between the clause space measure in tree-like resolution and the reversible pebble game on graphs. Using this connection, we provide several formula classes for which there is a logarithmic factor separation between the clause space complexity measure in tree-like and general resolution. We also provide upper bounds for tree-like resolution clause space in terms of general resolution clause and variable space. In particular, we show that for any formula F, its tree-like resolution clause space is upper bounded by space(pi)(log({rm time}(pi)), where pi is any general resolution refutation of F. This holds considering as space(pi) the clause space of the refutation as well as considering its variable space. For the concrete case of Tseitin formulas, we are able to improve this bound to the optimal bound space(pi)log n, where n is the number of vertices of the corresponding graph

Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted $${\rm Res}({\rm lin}_R)$$ , this refutation system operates with disjunctions of linear equations with Boolean variables over a ring R, to refute unsatisfiable sets of such disjunctions. Beginning in the work of Raz & Tzameret (2008), through the work of Itsykson & Sokolov (2020) which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf. Garlik & Kołodziejczyk 2018; Itsykson & Sokolov 2020; Krajícek 2017; Krajícek & Oliveira 2018) made it evident that establishing lower bounds against general $${\rm Res}({\rm lin}_R)$$ refutations is a challenging and interesting task since the system captures a ``minimal'' extension of resolution with counting gates for which no super-polynomial lower bounds are known to date. We provide the first super-polynomial size lower bounds against general (dag-like) resolution over linear equations refutations in the large characteristic regime. In particular, we prove that the subset-sum principle $$1+\sum\nolimits_{i=1}^{n}2^i x_i = 0$$ requires refutations of exponential size over $$\mathbb{Q}$$ . We use a novel lower bound technique: We show that under certain conditions every refutation of a subset-sum instance $$f=0$$ must pass through a fat clause consisting of the equation $$f=\alpha$$ for every $$\alpha$$ in the image of f under Boolean assignments, or can be efficiently reduced to a proof containing such a clause. We then modify this approach to prove exponential lower bounds against tree-like refutations of any subset-sum instance that depends on n variables, hence also separating tree-like from dag-like refutations over the rationals. We then turn to the finite fields regime, showing that the work of Itsykson & Sokolov (2020), where tree-like lower bounds over $$\mathbb{F}_2$$ were obtained, can be carried over and extended to every finite field. We establish new lower bounds and separations as follows: (i) For every pair of distinct primes $$p,q$$ , there exist CNF formulas with short tree-like refutations in $${\rm Res}({\rm lin}{\mathbb{F}_p})$$ that require exponential-size tree-like $${\rm Res}({\rm lin}{\mathbb{F}_q})$$ refutations; (ii) random k-CNF formulas require exponential-size tree-like $${\rm Res}({\rm lin}{\mathbb{F}_p})$$ refutations, for every prime p and constant k; and (iii) exponential-size lower bounds for tree-like $${\rm Res}({\rm lin}{\mathbb{F}})$$ refutations of the pigeonhole principle, for every field $$\mathbb{F}$$ .

Automating Resolution is NP-Hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatable unless P = NP. Indeed, we show that it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatable in subexponential time or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively.

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.

Random resolution refutations

We study the random resolution refutation system defined in Buss et al. (J Symb Logic 79(2):496–525, 2014). This attempts to capture the notion of a resolution refutation that may make mistakes but is correct most of the time. By proving the equivalence of several different definitions, we show that this concept is robust. On the other hand, if $${{\bf P} \neq {\bf NP}}$$ , then random resolution cannot be polynomially simulated by any proof system in which correctness of proofs is checkable in polynomial time. We prove several upper and lower bounds on the width and size of random resolution refutations of explicit and random unsatisfiable CNF formulas. Our main result is a separation between polylogarithmic width random resolution and quasipolynomial size resolution, which solves the problem stated in Buss et al. (2014). We also prove exponential size lower bounds on random resolution refutations of the pigeonhole principle CNFs, and of a family of CNFs which have polynomial size refutations in constant-depth Frege.

Proof Complexity Meets Algebra

We analyze how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional, algebraic, and semialgebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence, and addition of constants to a core preserve the proof complexity of the CSP. As a result, for those proof systems, the classes of constraint languages for which small unsatisfiability certificates exist can be characterized algebraically. We illustrate our results by a gap theorem saying that a constraint language either has resolution refutations of constant width or does not have bounded-depth Frege refutations of subexponential size. The former holds exactly for the widely studied class of constraint languages of bounded width. This class is also known to coincide with the class of languages with refutations of sublinear degree in Sums of Squares and Polynomial Calculus over the real field, for which we provide alternative proofs. We then ask for the existence of a natural proof system with good behavior with respect to reductions and simultaneously small-size refutations beyond bounded width. We give an example of such a proof system by showing that bounded-degree Lovász-Schrijver satisfies both requirements. Finally, building on the known lower bounds, we demonstrate the applicability of the method of reducibilities and construct new explicit hard instances of the graph three-coloring problem for all studied proof systems.

Short refutations for an equivalence‐chain principle for constant‐depth formulas

AbstractWe consider tautologies expressing equivalence‐chain properties in the spirit of Thapen and Krajíček, which are candidates for exponentially separating depth k and depth Frege proof systems. We formulate a special case where the initial member of the equivalence chain is fully specified and the equivalence‐chain implications are actually equivalences. This special case is shown to lead to polynomial size resolution refutations. Thus it cannot be used for separating depth k and depth propositional systems. We state some Håstad switching lemma conditions that restrict the possible propositional proofs in more general situations.

On approximating optimal weight “no”-certificates in weighted difference constraint systems

This paper is concerned with the design and analysis of approximation algorithms for the problem of determining the least weight refutation in a weighted difference constraint system. Recall that a difference constraint is a linear constraint of the form $$x_{i}-x_{j} \le b_{ij}$$ and a conjunction of such constraints is called a difference constraint system (DCS). In a weighted DCS (WDCS), a positive weight is associated with each constraint. Every infeasible constraint system has a refutation, which attests to its infeasibility. In the case of a DCS, this refutation is a subset of the input constraints, which when added together produces a contradiction of the form $$0 \le -b$$ , $$b> 0$$ . It follows that every refutation acts as a “no”-certificate. The length of a refutation is the number of constraints used in the derivation of a contradiction. Associated with a DCS $$\mathbf{D: A\cdot x \le b}$$ is its constraint network $$\mathbf{G= \langle V,E, b \rangle }$$ . It is well-known that $$\mathbf{D}$$ is infeasible if and only if $$\mathbf{G}$$ contains a simple, negative cost cycle. Previous research has established that every negative cost cycle of length k in $$\mathbf{G}$$ corresponds exactly to a refutation of $$\mathbf{D}$$ using k constraints. It follows that the shortest refutation of $$\mathbf{D}$$ (i.e., the refutation which uses the fewest number of constraints) corresponds to the length of the shortest negative cycle in $$\mathbf{G}$$ . The constraint network of a WDCS is represented by a constraint network $$\mathbf{G = \langle V, E, b, l \rangle }$$ , where $$\mathbf{l}:\mathbf{E \rightarrow \mathbb {N}}$$ represents a function which associates a positive, integral length with each edge in $$\mathbf{G}$$ . In the case of a WDCS, the weight of a refutation is defined as the sum of the lengths of the edges corresponding to the refutation. The problem of finding the minimum weight refutation in a WDCS is called the weighted optimal length resolution refutation (WOLRR) problem and is known to be NP-hard. In this paper, we describe a pseudo-polynomial time algorithm for the WOLRR problem and convert it into a fully polynomial time approximation scheme (FPTAS).

Cliques enumeration and tree-like resolution proofs

We show the close connection between the enumeration of cliques in a k-clique free graph G, the running time of DPLL-style algorithms for k-clique problem, and the length of tree-like resolution refutations for formula Clique(G,k), which claims that G has a k-clique. The length of any such tree-like refutation is within a “fixed parameter tractable” factor from the number of cliques in the graph. We then proceed to drastically simplify the proofs of the lower bounds for the length of tree-like resolution refutations of Clique(G,k) shown in Beyersdorff et at. 2013, Lauria et al. 2017, which now reduce to a simple estimate of said quantity.