Propositional Proof Complexity Research Articles

On Proof Complexity of Resolution over Polynomial Calculus

The proof system Res (PC d,R ) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree d multivariate polynomials over a ring R with Boolean variables. Proving super-polynomial lower bounds for the size of Res ( PC 1, R )-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for Res ( PC 1,𝔽 ) when 𝔽 is a finite field, such as 𝔽 2 . In this article, we investigate Res ( PC d,R ) and tree-like Res ( PC d,R ) and prove size-width relations for them when R is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows: (1) We prove almost quadratic lower bounds for Res ( PC d ,𝔽)-refutations for every fixed d . The new lower bounds are for the following CNFs: (a) Mod q Tseitin formulas ( char (𝔽)≠ q ) and Flow formulas, (b) Random k -CNFs with linearly many clauses. (2) We also prove super-polynomial (more than n k for any fixed k ) and also exponential (2 nϵ for an ϵ > 0) lower bounds for tree-like Res ( PC d ,𝔽 )-refutations based on how big d is with respect to n for the following CNFs: (a) Mod q Tseitin formulas ( char (𝔽)≠ q ) and Flow formulas, (b) Random k -CNFs of suitable densities, (c) Pigeonhole principle and Counting mod q principle. The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case d =1. The lower bounds for the tree-like systems were known for the case d =1 (except for the Counting mod q principle, in which lower bounds for the case d > 1 were known too). Our lower bounds extend those results to the case where d > 1 and also give new proofs for the case d =1.

Guest Column

We survey lower-bound results in complexity theory that have been obtained via newfound interconnections between propositional proof complexity, boolean circuit complexity, and query/communication complexity. We advocate for the theory of total search problems (TFNP) as a unifying language for these connections and discuss how this perspective suggests a whole programme for further research.

Frege Systems for Quantified Boolean Logic

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C . Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC 0 [ p ] circuits. In contrast, any non-trivial lower bound for propositional AC 0 [ p ]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.

On transformations of constant depth propositional proofs

This paper studies the complexity of constant depth propositional proofs in the cedent and sequent calculus. We discuss the relationships between the size of tree-like proofs, the size of dag-like proofs, and the heights of proofs. The main result is to correct a proof construction in an earlier paper about transformations from proofs with polylogarithmic height and constantly many formulas per cedent.

MaxSAT Resolution With the Dual Rail Encoding

Conflict-driven clause learning (CDCL) is at the core of the success of modern SAT solvers. In terms of propositional proof complexity, CDCL has been shown as strong as general resolution. Improvements to SAT solvers can be realized either by improving existing algorithms, or by exploiting proof systems stronger than CDCL. Recent work proposed an approach for solving SAT by reduction to Horn MaxSAT. The proposed reduction coupled with MaxSAT resolution represents a new proof system, DRMaxSAT, which was shown to enable polynomial time refutations of pigeonhole formulas, in contrast with either CDCL or general resolution. This paper investigates the DRMaxSAT proof system, and shows that DRMaxSAT p-simulates general resolution, that AC0-Frege+PHP p-simulates DRMaxSAT, and that DRMaxSAT can not p-simulate AC0-Frege+PHP or the cutting planes proof system.

A Finite-Model-Theoretic View on Propositional Proof Complexity

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. By exploring these connections further, we establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and over finite fields.

Proof Complexity Modulo the Polynomial Hierarchy

We present and study a framework in which one can present alternation-based lower bounds on proof length in proof systems for quantified Boolean formulas. A key notion in this framework is that of proof system ensemble , which is (essentially) a sequence of proof systems where, for each, proof checking can be performed in the polynomial hierarchy. We introduce a proof system ensemble called relaxing QU-res that is based on the established proof system QU-resolution . Our main results include an exponential separation of the treelike and general versions of relaxing QU-res and an exponential lower bound for relaxing QU-res; these are analogs of classical results in propositional proof complexity.

On Strong Normalization in Proof-Graphs for Propositional Logic

Traditional proof theory of Propositional Logic deals with proofs whose size can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The use of proof-graphs, instead of trees or lists, for representing proofs is getting popular among proof-theoreticians. Proof-graphs serve as a way to study complexity of propositional proofs and to provide more efficient theorem provers, concerning size of propositional proofs.Fpl-graphs were initially developed for minimal implicational logic representing proofs through references rather than copy. Thus, formulas and sub-deductions preserved in the graph structure, can be shared deleting unnecessary sub-deductions resulting in the reduced proof. In this work, we consider full minimal propositional logic and show how to reduce (eliminating maximal formulas) these representations such that strong normalization theorem can be proved by simply counting the number of maximal formulas in the original derivation. In proof-graphs, the main reason for obtaining the strong normalization property using such simple complexity measure is a direct consequence of the fact that each formula occurs only once in the proof-graph and the case of the hidden maximum formula that usually occurs in the tree-form derivation is already represented in the fpl-graph.

Guest Column

This essay is a highly personal and biased account of some main concepts and several research directions in modern propositional proof complexity. Special attention will be paid to connections with other disciplines.

A New Kind of Tradeoffs in Propositional Proof Complexity

We exhibit an unusually strong tradeoff in propositional proof complexity that significantly deviates from the established pattern of almost all results of this kind. Namely, restrictions on one resource (width, in our case) imply an increase in another resource (tree-like size) that is exponential not only with respect to the complexity of the original problem, but also to the whole class of all problems of the same bit size. More specifically, we show that for any parameter k = k ( n ), there are unsatisfiable k -CNFs that possess refutations of width O ( k ), but such that any tree-like refutation of width n 1 − ϵ / k must necessarily have doubly exponential size exp ( n Ω( k ) ). This means that there exist contradictions that allow narrow refutations, but in order to keep the size of such a refutation even within a single exponent, it must necessarily use a high degree of parallelism. Our construction and proof methods combine, in a non-trivial way, two previously known techniques: the hardness escalation method based on substitution formulas and expansion. This combination results in a hardness compression approach that strives to preserve hardness of a contradiction while significantly decreasing the number of its variables.

Short Proofs for the Determinant Identities

We study arithmetic proof systems ${\mathbb P}_c({\mathbb F})$ and $ {\mathbb P}_f({\mathbb F})$ operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field ${\mathbb F}$. We establish a series of structural theorems about these proof systems, the main one stating that ${\mathbb P}_c({\mathbb F})$ proofs can be balanced: if a polynomial identity of syntactic degree $ d $ and depth $k$ has a ${\mathbb P}_c({\mathbb F})$ proof of size $s$, then it also has a ${\mathbb P}_c({\mathbb F})$ proof of size $ {\rm poly}(s,d) $ in which every circuit has depth $ O(k+\log^2 d + \log d\cdot \log s) $. As a corollary, we obtain a quasi-polynomial simulation of ${\mathbb P}_c({\mathbb F})$ by ${\mathbb P}_f({\mathbb F})$. Using these results we obtain the following: consider the identities $\det(XY) = \det(X)\cdot\det(Y) \mbox{ and } \det(Z)= z_{11}\cdots z_{nn},$ where $X,Y$, and $ Z$ are $n\times n$ square matrices and $Z$ is a triangular matrix with $z_{11},\dots, z_{nn}$ on the diagonal (and $ \det $ is the determinant polynomial). Then we can construct a polynomial-size arithmetic circuit $\det$ such that the above identities have ${\mathbb P}_c({\mathbb F})$ proofs of polynomial size using circuits of $ O(\log^2 n)$ depth. Moreover, there exists an arithmetic formula $ \det $ of size $n^{O(\log n)}$ such that the above identities have ${\mathbb P}_f({\mathbb F})$ proofs of size $n^{O(\log n)}$. This yields a solution to a basic open problem in propositional proof complexity, namely, whether there are polynomial-size $\mathbf{NC}^2$-Frege proofs for the determinant identities and the hard matrix identities, as considered, e.g., in Soltys and Cook [Ann. Pure Appl. Logic, 130 (2004), pp. 277--323] (cf. Beame and Pitassi [Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 65 (1998), pp. 66--89]). We show that matrix identities like $ AB=I \rightarrow BA=I $ (for matrices over the two element field) as well as basic properties of the determinant have polynomial-size $\mathbf{NC}^2$-Frege proofs and quasi-polynomial-size Frege proofs.

Partially definable forcing and bounded arithmetic

We describe a method of forcing against weak theories of arithmetic and its applications in propositional proof complexity.

Fork-forests in bi-colored complete bipartite graphs

Motivated by the problem in (Tveretina et al., 2009) [8], which studies problems from propositional proof complexity, 2-edge colorings of complete bipartite graphs are investigated. It is shown that if the edges of G=Kn,n are colored with black and white such that the number of black edges differs from the number of white edges by at most 1, then there are at least n(1−1/2) vertex-disjoint forks with centers in the same partite set of G. Here, a fork is a graph formed by two adjacent edges of different colors. The bound is sharp. Moreover, an algorithm running in time O(n2lognnα(n2,n)logn) and giving a largest such fork forest is found.

Towards [formula omitted]–[formula omitted] via proof complexity and search

This is a survey of work on proof complexity and proof search from a logico-algorithmic viewpoint, as motivated by the P versus NP problem. We discuss propositional proof complexity, Cook’s program, proof automatizability, proof search, algorithms for satisfiability, and the state of the art of our (in)ability to separate P and NP.

Proof systems that take advice

One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [J. Symbolic Logic, 1979], where they defined propositional proof systems as poly-time computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Krajı´ček [J. Symbolic Logic, 2007] have recently started the investigation of proof systems which are computed by poly-time functions using advice.In this paper we concentrate on three fundamental questions regarding this new model. First, we investigate whether a given language L admits a polynomially bounded proof system with advice. Depending on the complexity of the underlying language L and the amount and type of the advice used by the proof system, we obtain different characterizations for this problem. In particular, we show that this question is tightly linked with the question whether L has small nondeterministic instance complexity.The second question concerns the existence of optimal proof systems with advice. For propositional proof systems, Cook and Krajı´ček gave a surprising positive answer which we extend to all languages.These results show that providing proof systems with advice yields a more powerful model, but this model is also less directly applicable in practice. Our third question therefore asks whether the usage of advice in propositional proof systems can be simplified or even eliminated. While in principle, the advice can be very complex, we show that propositional proof systems with logarithmic advice are also computable in poly-time with access to a sparse NP-oracle. Employing a recent technique of Buhrman and Hitchcock [CCC, 2008] we also manage to transfer the advice from the proof to the proven formula, which leads to a more practical computational model.

A note on propositional proof complexity of some Ramsey-type statements

A Ramsey statement denoted \({n \longrightarrow (k)^2_2}\) says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(n k) and with terms of size \({\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}\) . Let r k be the minimal n for which the statement holds. We prove that RAM(r k , k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák’s work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4k, k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R.

Complexity of propositional proofs under a promise

We study—within the framework of propositional proof complexity—the problem of certifying unsatisfiability of CNF formulas under the promise that any satisfiable formula has many satisfying assignments, where many stands for an explicitly specified function Λ in the number of variables n . To this end, we develop propositional proof systems under different measures of promises (i.e., different Λ) as extensions of resolution. This is done by augmenting resolution with axioms that, roughly, can eliminate sets of truth assignments defined by Boolean circuits. We then investigate the complexity of such systems, obtaining an exponential separation in the average case between resolution under different size promises: (1) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is ϵ…2 n , for any constant 0<ϵ<1. (2) There are no subexponential size resolution refutations for random 3CNF formulas, when the promise is 2 Δ n , for any constant 0<δ<1 (and the number of clauses is O ( n 3/2-ϵ ), for 0<ϵ<1/2). “ Goods Satisfactory or Money Refunded ” —The Eaton Promise

On meta complexity of propositional formulas and propositional proofs

A new approach to defining complexity of propositional formulas and proofs is suggested. Instead of measuring the size of these syntactical structures in the propositional language, the article suggests to define the complexity by the size of external descriptions of such constructions. The main result is a lower bound on proof complexity with respect to this new definition of complexity.

The Complexity of Propositional Proofs

AbstractPropositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

Tuples of Disjoint $\mathsf{NP}$ -Sets

Disjoint \(\mathsf{NP}\) -pairs are a well studied complexity-theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint \(\mathsf{NP}\) -pairs to disjoint k-tuples of \(\mathsf{NP}\) -sets for k≥2. We define subclasses of the class of all disjoint k-tuples of \(\mathsf{NP}\) -sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint \(\mathsf{NP}\) -pairs exist if and only if complete disjoint k-tuples of \(\mathsf{NP}\) -sets exist for all k≥2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems.