We compare the strength of the algebraic proof systems Sherali-Adams (SA) and Nullstellensatz (NS) with Frege-style proof systems. Unlike bounded-depth Frege, SA has polynomial-size proofs of the pigeonhole principle (PHP). A natural question is whether adding PHP to bounded-depth Frege is enough to simulate SA. We show that SA, with unary integer coefficients, lies strictly between tree-like depth-1 Frege + PHP and tree-like Resolution. We introduce a levelled version of PHP (L PHP) and we show that SA with integer coefficients lies strictly between tree-like depth-1 Frege + L PHP and Resolution. Analogous results are shown for NS using the bijective (i.e. onto and functional) pigeonhole principle and a leveled version of it.

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.

In recent years there has been an increasing interest in studying proof systems stronger than Resolution, with the aim of building more efficient SAT solvers based on them. In defining these proof systems, we try to find a balance between the power of the proof system (the size of the proofs required to refute a formula) and the difficulty of finding the proofs. In this paper we consider the proof systems circular Resolution, Sherali-Adams, Nullstellensatz and Weighted Resolution and we study their relative power from a theoretical perspective. We prove that circular Resolution, Sherali-Adams and Weighted Resolution are polynomially equivalent proof systems. We also prove that Nullstellensatz is polynomially equivalent to a restricted version of Weighted Resolution. The equivalences carry on also for versions of the systems where the coefficients/weights are expressed in unary. The practical interest in these systems comes from the fact that they admit efficient algorithms to find proofs in case these have small width/degree.

We characterize the strength of the algebraic proof systems Sherali-Adams () and Nullstellensatz () in terms of Frege-style proof systems. Unlike bounded-depth Frege, has polynomial-size proofs of the pigeonhole principle (). A natural question is whether adding to bounded-depth Frege is enough to simulate .

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is believed that most of these successful techniques exploit the underlying structure of industrial instances. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them. In this paper, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. Representing SAT instances as graphs, we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erdös-Rényi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of a CDCL SAT solver, and observe that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. Motivated by this observation, we finally present an application that exploits the community structure to detect relevant learned clauses, and we show that detecting these clauses results in an improvement on the performance of the SAT solver. Empirically, we observe that this improves the performance of several SAT solvers on industrial SAT formulas, especially on satisfiable instances.

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.

Abstract We extend results of Bonet, Buss and Pitassi on Bondy’s Theorem and of Nozaki, Arai and Arai on Bollobás’ Theorem by proving that Frankl’s Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parameter t , we prove that Frankl’s Theorem has polynomial size AC 0 -Frege proofs from instances of the pigeonhole principle.

This paper studies the relationship between resolution and conflict driven clause learning (CDCL) without restarts, and refutes some conjectured possible separations. We prove that the guarded, xor-ified pebbling tautology clauses, which Urquhart proved are hard for regular resolution, as well as the guarded graph tautology clauses of Alekhnovich, Johannsen, Pitassi, and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses. For the latter set of clauses, we extend this to prove that a CDCL search without restarts can refute these clauses in polynomial time, provided it makes the right choices for decision literals and clause learning. This holds even if the CDCL search is required to greedily process conflicts arising from unit propagation. This refutes the conjecture that the guarded graph tautology clauses or the guarded xor-ified pebbling tautology clauses can be used to separate CDCL without restarts from general resolution. Together with subsequent results by Buss and Kolodziejczyk, this means we lack any good conjectures about how to establish the exact logical strength of conflict-driven clause learning without restarts.

We show that two important problems that have applications in computational biology are ASP-complete, which implies that, given a solution to a problem, it is NP-complete to decide if another solution exists. We show first that a variation of BETWEENNESS, which is the underlying problem of questions related to radiation hybrid mapping, is ASP-complete. Subsequently, we use that result to show that QUARTET COMPATIBILITY, a fundamental problem in phylogenetics that asks whether a set of quartets can be represented by a parent tree, is also ASP-complete. The latter result shows that Steel’s QUARTET CHALLENGE, which asks whether a solution to QUARTET COMPATIBILITY is unique, is coNP-complete.

We present and implement a Weighted Partial MaxSAT solver based on successive calls to a SAT solver. We prove the correctness of our algorithm and compare our solver with other Weighted Partial MaxSAT solvers.

We show that subtree prune and regraft (uSPR) distance on unrooted trees is fixed parameter tractable with respect to the distance. We also make progress on a conjecture of Steel on the preservation of uSPR distance under chain reduction, improving on lower bounds of Hickey et al.

An exponential lower bound for the size of tree-like cutting planes refutations of a certain family of conjunctive normal form (CNF) formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and cutting planes. In both cases only superpolynomial separations were known [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467; J. Johannsen, Inform. Process. Lett., 67 (1998), pp. 37--41; P. Clote and A. Setzer, in Proof Complexity and Feasible Arithmetics, Amer. Math. Soc., Providence, RI, 1998, pp. 93--117]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [ Combinatorica, 19 (1999), pp. 403--435] are extended to monotone real circuits. An exponential separation is also proved between tree-like resolution and several refinements of resolution: negative resolution and regular resolution. Actually, this last separation also provides a separation between tree-like resolution and ordered resolution, and thus the corresponding superpolynomial separation of [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [A. Goerdt, Ann. Math. Artificial Intelligence, 6 (1992), pp. 169--184].

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 -Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC 0 -Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-sized TC 0 -Frege. As a corollary, we obtain that TC 0 -Frege, as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integers is hard). We also show under the same hardness assumption that the k-provability problem for Frege systems is hard.

In this paper; we show how to extend the argument due to Bonet, Pitassi and Raz to show that bounded-depth Frege proofs do not have feasible interpolation, assuming that factoring of Blum integers or computing the Diffie-Hellman function is sufficiently hard. It follows as a corollary that bounded-depth Frege is not automatizable; in other words, there is no deterministic polynomial-time algorithm that will output a short proof if one exists. A notable feature of our argument is its simplicity.

The paper is concerned with the complexity of proofs and of searching for proofs in two propositional proof systems: Resolution and Polynomial Calculus (PC). For the former system we show that the recently proposed algorithm of E. Ben-Sasson and A. Wigderson (1999) for searching for proofs cannot give better than weakly exponential performance. This is a consequence of showing optimality of their general relationship, referred to as size-width trade-off. We moreover obtain the optimality of the size width trade-off for the widely used restrictions of resolution: regular, Davis-Putnam, negative, positive and linear. As for the second system, we show that the direct translation to polynomials of a CNF formula having short resolution proofs, cannot be refuted in PC with degree less than /spl Omega/ (log n). A consequence of this is that the simulation of resolution by PC of M. Clegg, J. Edmonds and R. Impagliazzo (1996) cannot be improved to better than quasipolynomial in the case where we start with small resolution proofs. We conjecture that the simulation of M. Clegg et al. is optimal.

Most phylogenetics literature and construction methods based uponcharacters presume monomorphism (one state per character per species), yet polymorphism (multiple states per character per species) is well documented in both biology and historical linguistics. In this paper we consider the problem of inferring evolutionary trees for polymorphic characters. We show efficient algorithms for the construction of perfect phylogenies from polymorphic data. These methods have been used to help construct the evolutionary tree proposed by Warnow, Ringe, and Taylor for the Indo-European family of languages and presented by invitation at the National Academy of Sciences in November 1995.

We prove an exponential lower bound for tree-like cutting planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between tree-like and dag-like proofs for both cutting planes and resolution; in both cases only superpolynomial separations were known before. In order to prove this, we extend the lower bounds on the depth of monotone circuits of R. Raz and P. McKenzie (1997) to monotone real circuits. In the case of resolution, we further improve this result by giving an exponential separation of tree-like resolution front (dag-like) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a Davis-Puatam resolution proof. Finally, we prove an exponential separation between Davis-Putnam resolution and unrestricted resolution proofs; only a superpolynomial separations was previously known.

Evolutionary tree reconstruction is a challenging problem with important applications in Biology and Liiguistics.In Biology, one of the most promising approaches to tree reconstruction is to 6nd the "maximum parsimony" tree, while in Liignistics, the use of the "m&mum compatibility" method has been very useful.However, these problems are NP-hard, and current approaches to solving these problems amount to heuristic searches through the space of possible tree topologies (a search which can, on large trees, take months to complete).In this paper, we present a new technique, Uptimmnl l+ee Refinement, for reconstructing very large trees.Our technique is motivated by recent experimental studies which have shown that certain polynomial time methods reliably return contractions of the true tree.We study the use of this technique in solving maximum parsimony and maximum compatibility and present both hardness results and polynomial time algorithms.

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC/sup 0/-Frege systems. More specifically, we show that unless factoring is feasible, neither Frege nor TC/sup 0/-Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-size TC/sup 0/-Frege. In particular, we show how to carry out the proof for the Chinese Remainder Theorem, which may be of independent interest. As a corollary, we obtain that TC/sup 0/-Frege as well as any proof system that polynomially simulates it, is not automatizable (under a hardness assumption).

Abstract We consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

Most phylogenetics literature and construction methods

We consider small-weight Cutting Planes (CP* ) proofs;Our proofs also work for some generalizations of the C'P* proof system.In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

The serial transitive closure problem is the problem, given a directed graph G and a list of edges, called closure edges, which are in the transitive closure of the graph, to generate all the closure edges from edges in G. A nearly linear upper bound is given on the number of steps in optimal solutions to the serial transitive closure problem for the case of graphs that are trees. “Nearly linear” means $O(n \cdot \alpha (n))$, where $\alpha $ is the inverse Ackermann function. This upper bound is optimal to within a constant factor.

Abstract We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems , and nested deduction Frege systems , which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof. A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O (α( n )) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O ( n . α( n )). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O ( n log n ) lines and of height O (log n ). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O ( n log n ).

Proof systems for propositional logic called simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule, are introduced. Upper bounds are given on the lengths of proofs in these systems compared to lengths in Frege proof systems. As an application, a near-linear simulation of the propositional Gentzen sequent calculus by Frege proofs is presented. It is shown that a general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege systems where by 'nearly linear' is meant that the ratio of proof lengths is O( alpha (n)), where alpha is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, and hence a Frege proof system can simulate the propositional sequent calculus with proof lengths bounded by O(n alpha (n)). As a technical tool, the serial transitive closure problem is introduced. Given a directed graph and a list of closure edges in the transitive closure of the graph, the problem is to derive all the closure edges. A nearly linear bound is given on the number of steps in such a derivation when the graph is treelike.>