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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. [ABSTRACT FROM AUTHOR]

Satisfiability is considered the canonical NP‐complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large‐scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real‐world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random k$$ k $$‐SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random k$$ k $$‐SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT‐solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time. [ABSTRACT FROM AUTHOR]

This paper investigates a problem that lies at the intersection of three research areas, namely automated negotiation, vehicle routing, and multi-objective optimization. Specifically, it investigates the scenario that multiple competing logistics companies aim to cooperate by delivering truck loads for one another, in order to improve efficiency and reduce the distance they drive. In order to do so, these companies need to find ways to exchange their truck loads such that each of them individually benefits. We present a new heuristic algorithm that, given one set of orders for each company, tries to find the set of all truck load exchanges that are Pareto-optimal and individually rational. Unlike existing approaches, it does this without relying on any kind of trusted central server, so the companies do not need to disclose their private cost models to anyone. The idea is that the companies can then use automated negotiation techniques to negotiate which of these truck load exchanges will truly be carried out. Furthermore, this paper presents a new, multi-objective, variant of And/Or search that forms part of our approach, and it presents experiments based on real-world data, as well as on the commonly used Li & Lim data set. These experiments show that our algorithm is able to find hundreds of solutions within a matter of minutes. Finally, this paper presents an experiment with several state-of-the-art negotiation algorithms to show that the combination of our search algorithm with automated negotiation is viable. [ABSTRACT FROM AUTHOR]

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called scale-free random SAT instances. This is based on the use of a non-uniform probability distribution P (i) ∼ i − β to select variable i, where β is a parameter of the model. This results in formulas where the number of occurrences k of variables follows a power-law distribution P (k) ∼ k − δ , where δ = 1 + 1 / β . This property has been observed in most real-world SAT instances. For β = 0 , our model extends classical random SAT instances. We prove the existence of a SAT–UNSAT phase transition phenomenon for scale-free random 2-SAT instances with β < 1 / 2 when the clause/variable ratio is m / n = 1 − 2 β (1 − β) 2 . We also prove that scale-free random k-SAT instances are unsatisfiable with a high probability when the number of clauses exceeds ω (n (1 − β) k) . The proof of this result suggests that, when β > 1 − 1 / k , the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes. [ABSTRACT FROM AUTHOR]

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. We also provide a nominal unification algorithm for higher-order expressions with recursive let and atom-variables, where we show that it also runs in nondeterministic polynomial time. In addition we prove that there is a guessing strategy for nominal unification with letrec and atom-variable that is a trade-off between exponential growth and non-determinism. Nominal matching with variables representing partial letrec-environments is also shown to be in NP. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We present a rule-based Huet's style anti-unification algorithm for simply typed lambda-terms, which computes a least general higher-order pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo $$\alpha $$ -equivalence and variable renaming. With a minor modification, the algorithm works for untyped lambda-terms as well. The time complexity of both algorithms is linear. [ABSTRACT FROM AUTHOR]

The Maximum Satisfiability (MaxSAT) problem is an optimization variant of the Satisfiability (SAT) problem. Several combinatorial optimization problems can be translated into a MaxSAT formula. Among exact MaxSAT algorithms, SAT-based MaxSAT algorithms are the best performing approaches for real-world problems. We have extended the WPM2 algorithm by adding several improvements. In particular, we show that by solving some subproblems of the original MaxSAT instance we can dramatically increase the efficiency of WPM2. This led WPM2 to achieve the best overall results at the international MaxSAT Evaluation 2013 (MSE13) on industrial instances. Then, we present additional techniques and heuristics to further exploit the information retrieved from the resolution of the subproblems. We exhaustively analyze the impact of each improvement what contributes to our understanding of why they work. This architecture allows to convert exact algorithms into efficient incomplete algorithms. The resulting solver had the best results on industrial instances at the incomplete track of the latest international MSE. [ABSTRACT FROM AUTHOR]

We study anti-unification for unranked terms and hedges that may contain term and hedge variables. The anti-unification problem of two hedges ${\tilde{s}}_1$ and ${\tilde{s}}_2$ is concerned with finding their generalization, a hedge ${\tilde{q}}$ such that both ${\tilde{s}}_1$ and ${\tilde{s}}_2$ are instances of ${\tilde{q}}$ under some substitutions. Hedge variables help to fill in gaps in generalizations, while term variables abstract single (sub)terms with different top function symbols. First, we design a complete and minimal algorithm to compute least general generalizations. Then, we improve the efficiency of the algorithm by restricting possible alternatives permitted in the generalizations. The restrictions are imposed with the help of a rigidity function, which is a parameter in the improved algorithm and selects certain common subsequences from the hedges to be generalized. The obtained rigid anti-unification algorithm is further made more precise by permitting combination of hedge and term variables in generalizations. Finally, we indicate a possible application of the algorithm in software engineering. [ABSTRACT FROM AUTHOR]

Nominal logic is an extension of first-order logic with equality, name-binding, renaming via name-swapping and freshness of names. Contrarily to lambda-terms, in nominal terms, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of the lambda-calculus. Despite these differences, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be quadratically reduced to a particular fragment of higher-order unification problems: higher-order pattern unification. We also prove that the translation preserves most generality of unifiers. [ABSTRACT FROM AUTHOR]

Bounded Second-Order Unification is a decidable variant of undecidable Second-Order Unification. Stratified Context Unification is a decidable restriction of Context Unification, whose decidability is a long-standing open problem. This paper is a join of two separate previous, preliminary papers on NP-completeness of Bounded Second-Order Unification and Stratified Context Unification. It clarifies some omissions in these papers, joins the algorithmic parts that construct a minimal solution, and gives a clear account of a method of using singleton tree grammars for compression that may have potential usage for other algorithmic questions in related areas. [ABSTRACT FROM PUBLISHER]

Monadic second-order unification is second-order unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NP-complete, where we use the technique of compressing solutions using singleton context-free grammars. We prove that monadic second-order matching is also NP-complete. [ABSTRACT FROM AUTHOR]

Monadic second-order unification is second-order unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NP-complete, where we use the technique of compressing solutions using singleton context-free grammars. We prove that monadic second-order matching is also NP-complete. [ABSTRACT FROM AUTHOR]

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