Resolution Proof Research Articles

Group Cancellation and Resolution

We establish a connection between the geometric methods developed in the combinatorial theory of small cancellation and the propositional resolution calculus. We define a precise correspondence between resolution proofs in logic and diagrams in small cancellation theory, and as a consequence, we derive that a resolution proof is a 2-dimensional process. The isoperimetric function defined on diagrams corresponds to the length of resolution proofs.

Many hard examples in exact phase transitions

This paper analyzes the resolution complexity of two random constraint satisfaction problem (CSP) models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CSPs and CNF formulas hard to solve, which can be useful in the experimental evaluation of CSP and SAT algorithms, but also propose models with both many hard instances and exact phase transitions. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.

Rank Bounds and Integrality Gaps for Cutting Planes Procedures

$ \def\GC{\mbox{GC}} \def\LS{\mbox{LS}} $ We present a new method for proving rank lower bounds for the cutting planes procedures of Gomory and Chvátal ($\GC$) and Lovász and Schrijver ($\LS$), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove near-optimal rank bounds for $\GC$ and $\LS$ proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of $\GC$ or $\LS$ procedures when applied to the standard MAXSAT linear relaxation does not reduce the integrality gap. Second, we give unsatisfiable examples that have constant rank $\GC$ and $\LS$ proofs but that require linear rank Resolution proofs. Third, we give examples where the $\GC$ rank is $O(\log n)$ but the $\LS$ rank is linear. Finally, we address the question of size versus rank; we show that, for both proof systems, rank does not accurately reflect proof size. Specifically, there are examples which have polynomial-size $\GC$/$\LS$ proofs but require linear rank.

The resolution complexity of random graphk-colorability

We consider the resolution proof complexity of propositional formulas which encode random instances of graph k-colorability. We obtain a tradeoff between the graph density and the resolution proof ...

Towards a clausal analysis of cut-elimination

In this paper we show that a large class of cut-elimination methods can be analysed by clause terms representing sets of characteristic clauses extractable from the original proof. Every reduction step of a cut-elimination procedure defines an operation on the corresponding clause term. Using this formal framework we prove that the methods of Gentzen and Tait and, more generally, every method based on a specific set of cut-reduction rules R , yield a resolution proof which is subsumed by a resolution proof of the characteristic clause set. As a consequence we obtain that CERES (a resolution-based method of cut-elimination) is never inferior to any method based on R . On the other hand we show that CERES is not optimal in general; instead there exist cut-reduction rules which efficiently simplify the set of characteristic clauses and thus produce much shorter proofs. Further improvements and pruning methods could thus be obtained by a structural (syntactic) analysis of the characteristic clause terms.

The resolution complexity of random graph k-colorability

We consider the resolution proof complexity of propositional formulas which encode random instances of graph k-colorability. We obtain a tradeoff between the graph density and the resolution proof complexity. For random graphs with linearly many edges we obtain linear-exponential lower bounds on the size of resolution refutations. For random graphs with n vertices and any ε > 0 , we obtain a lower-bound tradeoff between graph density and refutation size that implies subexponential lower bounds of the form 2 n δ for some δ > 0 for non- k-colorability proofs of graphs with n vertices and O ( n 3 / 2 - 1 / k - ε ) edges. We obtain sharper lower bounds for Davis–Putnam–DPLL proofs and for proofs in a system considered by McDiarmid. These proof complexity bounds imply that many natural algorithms for k-coloring or k-colorability have essentially the same exponential tradeoff lower bounds on their running times. We also show that very simple algorithms for k-colorability have upper bounds on their running times that are qualitatively similar to the lower bounds as a function of the graph density.

Resolution cannot polynomially simulate compressed-BFS

Many algorithms for Boolean satisfiability (SAT) work within the framework of resolution as a proof system, and thus on unsatisfiable instances they can be viewed as attempting to find proofs by resolution. However it has been known since the 1980s that every resolution proof of the pigeonhole principle (PHPnm), suitably encoded as a CNF instance, includes exponentially many steps [18]. Therefore SAT solvers based upon the DLL procedure [12] or the DP procedure [13] must take exponential time. Polynomial-sized proofs of the pigeonhole principle exist for different proof systems, but general-purpose SAT solvers often remain confined to resolution. This result is in correlation with empirical evidence. Previously, we introduced the Compressed-BFS algorithm to solve the SAT decision problem. In an earlier work [27], an implementation of a Compressed-BFS algorithm empirically solved $\overline{\mathrm{PHP}_{n}^{n+1}}$ instances in ?(n4) time. Here, we add to this claim, and show analytically that these instances are solvable in polynomial time by Compressed-BFS. Thus the class of tautologies efficiently provable by Compressed-BFS is different than that of any resolution-based procedure. We hope that the details of our complexity analysis shed some light on the proof system implied by Compressed-BFS. Our proof focuses on structural invariants within the compressed data structure that stores collections of sets of open clauses during the Compressed-BFS algorithm. We bound the size of this data structure, as well as the overall memory, by a polynomial. We then use this to show that the overall runtime is bounded by a polynomial.

Translation of resolution proofs into short first-order proofs without choice axioms

We present a way of transforming a resolution-style proof containing Skolemization into a natural deduction proof without Skolemization. The size of the proof increases only moderately (polynomially). This makes it possible to translate the output of a resolution theorem prover into a purely first-order proof that is moderate in size.

Homomorphism Proof Systems for Unsatisfiable Formulas

A homomorphism ? of CNF formulas from H to F is a function mapping the set of literals in H to the set of literals in F and it preserves complements and clauses. If the formula H is homomorphic to the formula F, then the unsatisfiability of H implies the unsatisfiability of F. A CNF formula F is minimally unsatisfiable if F is unsatisfiable and the resulting formula deleting any one clause from F is satisfiable. MU(1) is a class of minimally unsatisfiable formulas with the deficiency of the number of clauses and variables to be one. A triple (H,?,F) is called a homomorphism proof from H of F if ? is a homomorphism from H to F. In this paper, a method from the basic matrix of MU(1) formula is used to prove that a tree resolution proof for an unsatisfiable formula F can be transformed into a homomorphism proof from a MU(1) formula for F. Whence, the homomorphism proof system from formulas in MU(1) is complete, and this proof system and the tree resolution proof system can be transformed mutually in polynomial time on the size of proof.

Towards Understanding and Harnessing the Potential of Clause Learning

Efficient implementations of DPLL with the addition of clause learning are the fastest complete Boolean satisfiability solvers and can handle many significant real-world problems, such as verification, planning and design. Despite its importance, little is known of the ultimate strengths and limitations of the technique. This paper presents the first precise characterization of clause learning as a proof system (CL), and begins the task of understanding its power by relating it to the well-studied resolution proof system. In particular, we show that with a new learning scheme, CL can provide exponentially shorter proofs than many proper refinements of general resolution (RES) satisfying a natural property. These include regular and Davis-Putnam resolution, which are already known to be much stronger than ordinary DPLL. We also show that a slight variant of CL with unlimited restarts is as powerful as RES itself. Translating these analytical results to practice, however, presents a challenge because of the nondeterministic nature of clause learning algorithms. We propose a novel way of exploiting the underlying problem structure, in the form of a high level problem description such as a graph or PDDL specification, to guide clause learning algorithms toward faster solutions. We show that this leads to exponential speed-ups on grid and randomized pebbling problems, as well as substantial improvements on certain ordering formulas.

Regular Resolution Lower Bounds For The Weak Pigeonhole Principle

We prove that any regular resolution proof for the weak pigeon hole principle, with n holes and any number of pigeons, is of length $$\Omega {\left( {2^{{n^{\varepsilon } }} } \right)}$$, (for some global constant e > 0).

A PROOF PROCEDURE FOR TEMPORAL LOGIC PROGRAMMING

In this paper, we propose a new resolution proof procedure for the branching-time logic programming language Cactus. The particular strength of the new proof procedure, called CSLD-resolution, is that it can handle, in a more general way, open-ended queries, i.e. goal clauses that include atoms which do not refer to specific moments in time, without the need of enumerating all their canonical instances. We also prove soundness, completeness and independence of the computation rule for CSLD-resolution. The new proof procedure overcomes the limitations of a family of proof procedures for temporal logic programming languages, which were based on the notions of canonical program and goal clauses. Moreover, it applies directly to Chronolog programs and it can be easily extended to apply to multi-dimensional logic programs as well as to Chronolog(MC) programs.

Resolution lower bounds for perfect matching principles

For an arbitrary hypergraph H , let PM( H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM( H) must have size exp Ω δ( H) λ( H)r( H)( log n( H))(r( H)+ log n( H)) , where n( H) is the number of vertices, δ( H) is the minimal degree of a vertex, r( H) is the maximal size of an edge, and λ( H) is the maximal number of edges incident to two different vertices. For ordinary graphs G our general bound considerably simplifies to exp Ω δ(G) ( log n(G)) 2 (implying an exp(Ω(δ(G) 1/3)) lower bound that depends on the minimal degree only). As a direct corollary, every resolution proof of the functional onto version of the pigeonhole principle onto− FPHP n m must have size exp Ω n ( log m) 2 (which becomes exp Ω(n 1/3) when the number of pigeons m is unbounded). This in turn immediately implies an exp(Ω(t/n 3)) lower bound on the size of resolution proofs of the principle asserting that the circuit size of the Boolean function f n in n variables is greater than t. In particular, Resolution does not possess efficient proofs of NP ⊈ P/poly . These results relativize, in a natural way, to a more general principle M(U| H) asserting that H contains a matching covering all vertices in U⊆V( H) .

Resolution lower bounds for the weak pigeonhole principle

We prove that any Resolution proof for the weak pigeonhole principle, with n holes and any number of pigeons, is of length Ω(2 n ϵ ), (for some global constant ϵ > 0). One corollary is that a certain propositional formulation of the statement NP ⊄ P / poly does not have short Resolution proofs.

On the automatizability of resolution and related propositional proof systems

A propositional proof system is automatizable if there is an algorithm that, given a tautology, produces a proof in time polynomial in the size of its smallest proof. This notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound. This new notion is called weak automatizability. Among other characterizations, we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time. After studying the robustness of the definition, we prove the equivalence between: (i) Resolution is weakly automatizable, (ii) Res( k ) is weakly automatizable, and (iii) Res( k ) has feasible interpolation, when k >1. In order to prove this result, we show that Res(2) has polynomial-size proofs of the reflection principle of Resolution, which is a version of consistency. We also show that Res( k ), for every k >1, proves its consistency in polynomial size, while Resolution does not. In fact, we show that Resolution proofs of its own consistency require almost exponential size. This gives a better lower bound for the monotone interpolation of Res(2) and a separation from Resolution as a byproduct. Our techniques also give us a way to obtain a large class of examples that have small Resolution refutations but require relatively large width. This answers a question of Alekhnovich and Razborov related to whether Resolution is automatizable in quasipolynomial-time.

Homomorphisms of conjunctive normal forms

We study homomorphisms of propositional formulas in CNF generalizing symmetries considered by Krishnamurthy. If ϕ : H→F is a homomorphism, then unsatisfiability of H implies unsatisfiability of F. Homomorphisms from F to a subset F′ of F (endomorphisms) are of special interest, since in such cases F and F′ are satisfiability-equivalent. We show that the smallest subsets F′ of a formula F for which an endomorphism F→ F′ exists are mutually isomorphic. Furthermore, we study connections between homomorphisms and autark assignments. We introduce the concept of “proof by homomorphism” which is based on the observation that there exist sets Γ of unsatisfiable formulas such that (i) formulas in Γ can be recognized in polynomial time, and (ii) for every unsatisfiable formula F there exist some H∈ Γ and a homomorphism ϕ : H→F . We identify several sets Γ of unsatisfiable formulas satisfying (i) and (ii) for which proofs by homomorphism w.r.t. Γ and tree resolution proofs can be simulated by each other in polynomial time.

Space complexity of random formulae in resolution

AbstractWe study the space complexity of refuting unsatisfiable random k‐CNFs in the Resolution proof system. We prove that for Δ ≥ 1 and any ϵ > 0, with high probability a random k‐CNF over n variables and Δn clauses requires resolution clause space of Ω(n/Δ1+ϵ). For constant Δ, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3‐CNFs with clause density Δ ≫ n. This bound is nearly tight. Specifically, we show that with high probability, a random 3‐CNF with Δn clauses requires treelike refutation size of exp(Ω(n/Δ1+ϵ)), for any ϵ > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2‐player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G‐PHP, a generalization of the Pigeonhole principle based on bipartite graphs G. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 92–109, 2003

A new framework for declarative programming

We propose a new framework for the syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over τ-categories: finite product categories with canonical structure. Constraint information is directly built-in to the notion of signature via categorical syntax. Many-sorted equational are a special case of the formalism which combines features of uniform logic programming languages (moduels and hypothetical implication) with those of constraint logic programming. Using the cannoical structure supplied by τ-categories, we define a diagrammatic generalization of formulas, goals, programs and resolution proofs up to equality (rather than just up to isomorphism). We extend the Kowalski-van Emden fixed point interpretation, a cornerstone of declarative semantics, to an operational, non-ground, categorical semantics based on indexing over sorts and programs. We also introduce a topos-theoretic declarative semantics and show soundness and completeness of resolution proofs and of a sequent calculus over the categorical signature. We conclude with a discussion of semantic perspectives on uniform logic programming.

Resolution lower bounds for the weak functional pigeonhole principle

We show that every resolution proof of the functional version FPHP n m of the pigeonhole principle (in which one pigeon may not split between several holes) must have size exp( Ω(n/( log m) 2)) . This implies an exp( Ω(n 1/3)) bound when the number of pigeons m is arbitrary.

Resolution Proofs of Matching Principles

This paper discusses current techniques for proving lower bounds on the size of resolution proofs from sets of clauses expressing assertions about matchings in bipartite graphs (a class including the well-known pigeon-hole clauses). The techniques are illustrated by demonstrating an improved lower bound for some examples discussed by Kleine Buning. The final section discusses some problems where current techniques appear inadequate, and new ideas seem to be required.