Search

Quantum Annealers are basically quantum computers that with high probability can optimize certain quadratic functions on Boolean variables in constant time. These functions are basically the Hamiltonian of Ising models that reach the ground energy state, with a high probability, after an annealing process. They have been proposed as a way to solve SAT. These Hamiltonians can be seen as Max2XOR problems, i.e. as the problem of finding an assignment that maximizes the number of XOR clauses of at most 2 variables that are satisfied. In this paper, we present several gadgets to reduce SAT to Max2XOR. We show how they can be used to translate SAT instances to initial configurations of a quantum annealer., Comment: arXiv admin note: text overlap with arXiv:2204.01774

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 MaxSAT-Resolution and weighted Resolution that manipulates polynomials with coefficients in a finite field and either weights in ? or ?. We show the soundness and completeness of these systems via an algorithmic procedure. Weighted Polynomial Calculus, with weights in ? and coefficients in ??, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in ?, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable.

The minimization of propositional formulae is a classical problem in logic, whose first algorithms date back at least to the 1950s with the works of Quine and Karnaugh. Most previous work in the area has focused on obtaining minimal, or quasi-minimal, formulae in conjunctive normal form (CNF) or disjunctive normal form (DNF), with applications in hardware design. In this paper, we are interested in the problem of obtaining an equivalent formula in any format, also allowing connectives that are not present in the original formula. We are primarily motivated in applying minimization algorithms to generate natural language translations of the original formula, where using shorter equivalents as input may result in better translations. Recently, Buchfuhrer and Umans have proved that the (decisional version of the) problem is $\Sigma_2^p$-complete. We analyze three possible (practical) approaches to solving the problem. First, using brute force, generating all possible formulae in increasing size and checking if they are equivalent to the original formula by testing all possible variable assignments. Second, generating the Tseitin coding of all the formulae and checking equivalence with the original using a SAT solver. Third, encoding the problem as a Quantified Boolean Formula (QBF), and using a QBF solver. Our results show that the QBF approach largely outperforms the other two.

[EN]In the last decades, we have witnessed a remarkable success of algorithms solving the Boolean Satisfiability problem (SAT) on instances encoding application or real-world problems arising from a very diverse number of domains, such as hardware and software verification, planning or cryptography. These algorithms are the so known Conflict-Driven Clause Learning (CDCL) SAT solvers. Interestingly enough, the reasons for the success of these solvers on this diverse range of problems are not completely understood yet. A common issue when facing this open challenge is the heterogeneity of this set of benchmarks. Another problem is the limited number of existing instances. In this context, random models of SAT formulas capturing features shared by the majority of these application benchmarks become crucial, for both theoretical and practical purposes. On the one hand, it is undoubtedly necessary to have random models where theoretical properties, like hardness, can be studied. Therefore, realistic random SAT models may contribute to explain the success of these solvers on these industrial problems. On the other hand, the limited number of benchmarks and their hardness in practice makes the evaluation of new solving techniques a costly task. Therefore, these realistic random SAT generators can provide an unlimited number of pseudo-industrial random SAT instances with some desired properties. In this work, we present a random SAT instances generator based on the notion of locality. This notion is complementary to the popularity of variables, which is present in the scale-free structure, observable in actual application problems and achievable by previous generators. Our random SAT model combines both locality and popularity, and we show that they are two decisive dimensions of attractiveness among the variables of a formula, and how CDCL SAT solvers take advantage of them. Locality is closely related to the community structure, another important feature of application SA

In the literature we find reductions from 3SAT to Max2SAT. These reductions are based on the usage of a gadget, i.e., a combinatorial structure that allows translating constraints of one problem to constraints of another. Unfortunately, the generation of these gadgets lacks an intuitive or efficient method. In this paper, we provide an efficient and constructive method for Reducing SAT to Max2SAT and show empirical results of how MaxSAT solvers are more efficient than SAT solvers solving the translation of hard formulas for Resolution.

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.

Representing some problems with XOR clauses (parity constraints) can allow to apply more efficient reasoning techniques. In this paper, we present a gadget for translating SAT clauses into Max2XOR constraints, i.e., XOR clauses of at most 2 variables equal to zero or to one. Additionally, we present new resolution rules for the Max2XOR problem which asks for which is the maximum number of constraints that can be satisfied from a set of 2XOR equations.

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.

Accés lliure al text del llibre a la web de l'editor, Knowledge, reasoning and learning (KRL) play a central role in artificial intelligence (AI) and are instrumental in solving many AI complex problems. In such problems we may have massive amounts of data and imprecise models, and the goal is to create AI systems that scale well in such scenarios, which often requires combining KRL techniques. In this chapter, we first present KRL from an historical perspective and then identify future research directions in the KRL domain in which CSIC is or could be very competitive. In particular, we present challenges integrating KRL from several perspectives, such as: General planning : finding AI agents that combine learning and planning techniques to be able to learn solutions to a problem and generalize them to others. Problem solving : create algorithms with learning capabilities for solving complex optimization problems with huge amounts of data. Learning adaptable AI agents from a reduced amount of data, keeping the most information possible from data, while being computationally and sample efficient for adapting to changing situations. Enhancing logics for conditional, causal reasoning and their integration with machine learning (ML). Integrating uncertainty, similarities, knowledge and learning by using stochastic methods that better represent the variability in real-world scenarios. These challenges are considered key topics in the current AI research and, with CSIC already having a good knowledge on them, pushing research in those directions could place CSIC as a reference institution in the world on these fields. We detail a plan and resources needed to boost research on the integration of KRL techniques., Postprint (published version)

[EN]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-SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random k-SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT-solvers. On the other hand, modeling locality with an underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time. A key ingredient for the result on geometric random k-SAT can be found in the complexity of higher-order Voronoi diagrams. As an additional technical contribution, we show a linear upper bound on the number of non-empty Voronoi regions, that holds for points with random positions in a very general setting. In particular, it covers arbitrary p-norms, higher dimensions, and weights affecting the area of influence of each point multiplicatively. This is in stark contrast to quadratic lower bounds for the worst case.

We investigate a problem that lies at the intersection of three research areas, namely Automated Negotiation, Vehicle Routing, and Multi-Objective Optimization. Specifically, we investigate 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 to deliver for each company, tries to find the set of all order-exchanges that are Pareto-optimal and individually rational. Furthermore, we present experiments based on real-world test data from two major logistics companies, which show that our algorithm is able to find hundreds of solutions in a matter of minutes.

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., Comment: 37 pages, 9 figures, This paper is an extended version of the conference publication: Manfred Schmidt-Schau{\ss} and Temur Kutsia and Jordi Levy and Mateu Villaret and Yunus Kutz, Nominal Unification of Higher Order Expressions with Recursive Let, LOPSTR-16, Lecture Notes in Computer Science 10184, Springer, p 328 -344, 2016. arXiv admin note: text overlap with arXiv:1608.03771

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-SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random k-SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT-solvers. On the other hand, modeling locality with an underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time. A key ingredient for the result on geometric random k-SAT can be found in the complexity of higher-order Voronoi diagrams. As an additional technical contribution, we show a linear upper bound on the number of non-empty Voronoi regions, that holds for points with random positions in a very general setting. In particular, it covers arbitrary p-norms, higher dimensions, and weights affecting the area of influence of each point multiplicatively. This is in stark contrast to quadratic lower bounds for the worst case., Comment: 53 pages, 2 figures

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. Among those proof systems we can mention Circular Resolution, MaxSAT Resolution with Extensions and MaxSAT Resolution with the Dual-Rail encoding. In this paper we study the relative power of those proof systems from a theoretical perspective. We prove that Circular Resolution and MaxSAT Resolution with extension are polynomially equivalent proof systems. This result is generalized to arbitrary sets of inference rules with proof constructions based on circular graphs or based on weighted clauses. We also prove that when we restrict the Split rule (that both systems use) to bounded size clauses, these two restricted systems are also equivalent. Finally, we show the relationship between these two restricted systems and Dual-Rail MaxSAT Resolution.

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. As extension a nominal unification algorithm for higher-order expressions with recursive let and atom-variables is constructed, where we show that it also runs in non-deterministic polynomial time.

Phase-transition in random SAT formulas is one of the properties best studied by theoretical SAT researchers. There exists a constant rk depending on k such that, if we choose randomly a k-SAT formula over n variables and m clauses, it will be satisfiable with high probability, if m/n < r, and unsatisfiable, otherwise. However, this criterion is useless in practice, because real-world or industrial instances have some properties not shown in random formulas. In the last years, several models of realistic random formulas have been proposed. Here we discuss about the phase transition in these models, and about the size of unsatisfiability proofs. We observe that in these models, like in real-world formulas, there is not a sharp phase transition, the transition occurs for smaller values of r, and the proofs on unsatisfiable formulas are smaller than in the classical random model. We also discuss about the strategies used by modern SAT solvers to exploit these properties.

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.

We study anti-unification for possibly cyclic, unranked term-graphs and develop an algorithm, which computes a minimal complete set of generalizations for them. For bisimilar graphs the algorithm computes the join in the lattice generated by a functional bisimulation. These results generalize anti-unification for ranked and unranked terms to the corresponding term-graphs, and solve also anti-unification problems for rational terms and dags. Our results open a way to widen anti-unification based code clone detection techniques from a tree representation to a graph representation of the code.

We study anti-unification for possibly cyclic, unranked term-graphs and develop an algorithm, which computes a minimal complete set of generalizations for them. For bisimilar graphs the algorithm computes the join in the lattice generated by a functional bisimulation. These results generalize anti-unification for ranked and unranked terms to the corresponding term-graphs, and solve also anti-unification problems for rational terms and dags. Our results open a way to widen anti-unification based code clone detection techniques from a tree representation to a graph representation of the code.

We study anti-unification for possibly cyclic, unranked term-graphs and develop an algorithm, which computes a minimal complete set of generalizations for them. For bisimilar graphs the algorithm computes the join in the lattice generated by a functional bisimulation. These results generalize anti-unification for ranked and unranked terms to the corresponding term-graphs, and solve also anti-unification problems for rational terms and dags. Our results open a way to widen anti-unification based code clone detection techniques from a tree representation to a graph representation of the code.

The success of portfolio approaches in SAT solving relies on the observation that different SAT solvers may dramatically change their performance depending on the class of SAT instances they are trying to solve. In these approaches, a set of features of the problem is used to build a prediction model, which classifies instances into classes, and computes the fastest algorithm to solve each of them. Therefore, the set of features used to build these classifiers plays a crucial role. Traditionally, portfolio SAT solvers include features about the structure of the problem and its hardness. Recently, there have been some attempts to better characterize the structure of industrial SAT instances. In this paper, we use some structure features of industrial SAT instances to build some classifiers of industrial SAT families of instances. Namely, they are the scale-free structure, the community structure and the self similar structure. First, we measure the effectiveness of these classifiers by comparing them to other sets of SAT features commonly used in portfolio SAT solving approaches. Then, we evaluate the performance of this set of structure features when used in a real portfolio SAT solver. Finally, we analyze the relevance of these features on the analyzed classifiers., QC 20170821

Despite the success of CDCL SAT solvers solving industrial problems, there are still many open questions to explain such success. In this context, the generation of random SAT instances having computational properties more similar to real-world problems becomes crucial. Such generators are possibly the best tool to analyze families of instances and solvers behaviors on them. In this paper, we present a random SAT instances generator based on the notion of locality. We show that this is a decisive dimension of attractiveness among the variables of a formula, and how CDCL SAT solvers take advantage of it. To the best of our knowledge, this is the first random SAT model that generates both scale-free structure and community structure at once.

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 \emph{scale-free random SAT instances}. It is based on the use of a non-uniform probability distribution $P(i)\sim i^{-\beta}$ to select variable $i$, where $\beta$ is a parameter of the model. This results into formulas where the number of occurrences $k$ of variables follows a power-law distribution $P(k)\sim k^{-\delta}$ where $\delta = 1 + 1/\beta$. This property has been observed in most real-world SAT instances. For $\beta=0$, 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 $\beta<1/2$ when the clause/variable ratio is $m/n=\frac{1-2\beta}{(1-\beta)^2}$. We also prove that scale-free random k-SAT instances are unsatisfiable with high probability when the number of clauses exceeds $\omega(n^{(1-\beta)k})$. %This implies that the SAT/UNSAT phase transition phenomena vanishes when $\beta>1-1/k$, and formulas are unsatisfiable due to a small core of clauses. The proof of this result suggests that, when $\beta>1-1/k$, 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.

Nowadays, many real-world problems are encoded into SAT instances and efficiently solved by modern SAT solvers. These solvers, usually known as Conflict-Driven Clause Learning (CDCL) SAT solvers, include a variety of sophisticated techniques, such as clause learning, lazy data structures, conflict-based adaptive branching heuristics, or random restarts, among others. However, the reasons of their efficiency in solving real-world, or industrial, SAT instances are still unknown. The common wisdom in the SAT community is that these technique exploit some hidden structure of real-world problems. In this thesis, we characterize some important features of the underlying structure of industrial SAT instances. Namely, they are the community structure and the self-similar structure. We observe that most industrial SAT formulas, viewed as graphs, have these two properties. This means that (i) in a graph with a clear community structure, i.e. having high modularity, we can find a partition of its nodes into communities such that most edges connect nodes of the same community; and (ii) in a graph with a self-similar pattern, i.e. being fractal, its shape is kept after re-scalings, i.e., grouping sets of nodes into a single node. We also analyze how these structures are affected by the effects of CDCL techniques during the search. Using the previous structural studies, we propose three applications. First, we face the problem of generating pseudo-industrial random SAT instances using the notion of modularity. Our model generates instances similar to (classical) random SAT formulas when the modularity is low, but when this value is high, our model is also adequate to model realistic pseudo-industrial problems. Second, we propose a method based on the community structure of the instance to detect relevant learnt clauses. Our technique augments the original instance with this set of relevant clauses, and this results into an overall improvement of the efficiency of several...

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine their complexity. © Springer International Publishing AG 2017.

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. © 2015, Springer Science+Business Media New York.

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems., Comment: Pre-proceedings paper presented at the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh, Scotland UK, 6-8 September 2016 (arXiv:1608.02534)

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there are few works trying to exactly characterize the main features of this structure. The research community on complex networks has developed techniques of analysis and algorithms to study real-world graphs that can be used by the SAT community. 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, inspired by the results on complex networks, 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. In our analysis, we represent SAT instances as graphs, and we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erd\"os-R\'enyi 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. In particular, we use the community structure to detect that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. This is, learned clauses tend to contain variables of distinct communities.

Nowadays, modern SAT solvers are able to efficiently solve many industrial, or real-world, SAT instances. However, the process of development and testing of new SAT solving techniques is conditioned to the finite and reduced number of known industrial benchmarks. Therefore, new models of random SAT instances generation that capture realistically the features of real-world problems can be beneficial to the SAT community. In many works, the structure of industrial instances has been analyzed representing them as graphs and studying some of their properties, like modularity. In this work, we use the notion of modularity to define a new model of generation of random SAT instances with community structure, called Community Attachment. For high values of modularity (i.e., clear community structure), we realistically model pseudo-industrial random SAT formulas. This model also generates SAT instances very similar to classical random formulas using a low value of modularity. We also prove that the phase transition point, if exists, is independent on the modularity. We evaluate the adequacy of this model to real industrial SAT problems in terms of SAT solvers performance, and show that modern solvers do actually exploit this community structure. Finally, we use this generator to observe the connections between the modularity of the instance and some components of the solver, such as the variable branching heuristics or the clause learning mechanism.

We study nominal anti-unification, which is concerned with computing least general generalizations for given terms-in-context. In general, the problem does not have a least general solution, but if the set of atoms permitted in generalizations is finite, then there exists a least general generalization which is unique modulo variable renaming and alpha-equivalence. We present an algorithm that computes it. The algorithm relies on a subalgorithm that constructively decides equivariance between two terms-in-context. We prove soundness and completeness properties of both algorithms and analyze their complexity. Nominal anti-unification can be applied to problems where generalization of first-order terms is needed (inductive learning, clone detection, etc.), but bindings are involved.

We study nominal anti-unification, which is concerned with computing least general generalizations for given terms-in-context. In general, the problem does not have a least general solution, but if the set of atoms permitted in generalizations is finite, then there exists a least general generalization which is unique modulo variable renaming and alpha-equivalence. We present an algorithm that computes it. The algorithm relies on a subalgorithm that constructively decides equivariance between two terms-in-context. We prove soundness and completeness properties of both algorithms and analyze their complexity. Nominal anti-unification can be applied to problems where generalization of first-order terms is needed (inductive learning, clone detection, etc.), but bindings are involved.

Nowadays, Conflict-Driven Clause Learning (CDCL) techniques are one of the key components of modern SAT solvers specialized in industrial instances. Last years, one of the focuses has been put on strategies to select which learnt clauses are removed during the search. Originally, one need for removing clauses was motivated by the finiteness of memory. Recently, it has been shown that more aggressive clause deletion policies may improve solvers performance, even when memory is sufficient. Also, the utility of learnt clauses has been related to the modular structure of industrial SAT instances. In this paper, we show that augmenting SAT instances with learnt clauses does not always make them easier for the SAT solver. In fact, it makes worse the solver performance in many cases. However, we identify a set of highly useful learnt clauses, and we show that augmenting SAT instances with this set of clauses contributes to improve the solver performance in many cases, especially in satisfiable formulas. These clauses are related to the community structure of the formula, and they can be computed in a fast preprocessing step. This would suggest that the community structure may play an important role in clause deletion policies.

The success of portfolio approaches in SAT solving relies on the observation that different SAT solving techniques perform better on different SAT instances. The Algorithm Selection Problem faces the problem of choosing, using a prediction model, the best algorithm from a predefined set, to solve a particular instance of a problem. Using Machine Learning techniques, this prediction is performed by analyzing some features of the instance and using an empirical hardness model, previously built, to select the expected fastest algorithm to solve such instance. Recently, there have been some attempts to characterize the structure of industrial SAT instances. In this paper, we use some structural features of industrial SAT instances to build some classifiers of industrial SAT families of instances. Namely, they are the scale-free structure, the community structure and the self-similar structure. We measure the effectiveness of these classifiers by comparing them to other sets of SAT instances features commonly used in portfolio SAT solving approaches. Finally, we evaluate the performance of this set of structural features when used in a real portfolio SAT solver.

Nowadays, many industrial SAT instances can be solved efficiently by modern SAT solvers. However, the number of real-world instances is finite. Therefore, the process of development and test of SAT solving techniques can benefit of new models of random formulas that capture more realistically the features of real-world problems. In many works, the structure of industrial instances has been analyzed representing them as graphs and studying some of their properties, like modularity. In this paper, we use modularity, or community structure, to define a new model of pseudo-industrial random SAT instances, called Community Attachment. We prove that the phase transition point, if exists, is independent on the modularity. We evaluate the adequacy of this model to real industrial problems in terms of SAT solvers performance, and show that modern solvers do actually exploit this community structure.

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there is not a precise definition of the notion of structure. Recently, there have been some attempts to analyze this structure in terms of complex networks, with the long-term aim of explaining the success of SAT solving techniques, and possibly improving them. We study the fractal dimension of SAT instances with the aim of complementing the model that describes the structure of industrial instances. We show that many industrial families of formulas are self-similar, with a small fractal dimension. We also show how this dimension is affected by the addition of learnt clauses during the execution of SAT solvers.

Second-Order Unification is a problem that naturally arises when applying automated deduction techniques with variables denoting predicates. The problem is undecidable, but a considerable effort has been made in order to find decidable fragments, and understand the deep reasons of its complexity. Two variants of the problem, Bounded Second-Order Unification and Linear Second-Order Unification ¿where the use of bound variables in the instantiations is restricted¿, have been extensively studied in the last two decades. In this paper we summarize some decidability/undecidability/complexity results, trying to focus on those that could be more interesting for a wider audience, and involving less technical details.

We study anti-unification for unranked terms and hedges that may contain term and hedge variables. The anti-unification problem of two hedges -1 and -2 is concerned with finding their generalization, a hedge q such that both -1 and -2 are instances of 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. © 2013 The Author(s).

We present a rule-based Huet's style anti-unification algorithm for simply-typed lambda-terms in eta-long beta-normal form, 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. The algorithm computes it in cubic time within linear space. It has been implemented and the code is freely available.

We present a rule-based Huet's style anti-unification algorithm for simply-typed lambda-terms in eta-long beta-normal form, 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. The algorithm computes it in cubic time within linear space. It has been implemented and the code is freely available.

Signed clausal forms offer a suitable logical framework for automated reasoning in multiple-valued logics. It turns out that the satisfiability problem of any finitely-valued propositional logic, as well as of certain infinitely-valued logics, can be easily reduced, in polynomial time, to the satisfiability problem of signed clausal forms. On the other hand, signed clausal forms are a powerful knowledge representation language for constraint programming, and have shown to be a practical and competitive approach to solving combinatorial decision problems. Motivated by the existing theoretical and practical results for the satisfiability problem of signed clausal forms, as well as by the recent logical and algorithmic results on the Boolean maximum satisfiability problem, in this paper we investigate the maximum satisfiability problem of propositional signed clausal forms from the logical and practical points of view. From the logical perspective, our aim is to define complete inference systems taking as a starting point the resolution-style calculi defined for the Boolean CNF case. The result is the definition of two sound and complete resolution-style rules, called signed binary resolution and signed parallel resolution for maximum satisfiability. From the practical perspective, our main motivation is to use the language of signed clausal forms along with the newly defined inference systems as a generic approach to solve combinatorial optimization problems, and not just for solving decision problems as so far. The result is the establishment of a link between signed logic and constraint programming that provides a concise and elegant logical framework for weighted constraint programming. © 2012 Elsevier Inc. All rights reserved.

Many industrial optimization problems can be translated to MaxSAT. Although the general problem is NP hard, like SAT, many practical problems may be solved using modern MaxSAT solvers. In this paper we present several algorithms specially designed to deal with industrial or real problems. All of them are based on the idea of solving MaxSAT through successive calls to a SAT solver. We show that this SAT-based technique is efficient in solving industrial problems. In fact, all state-of-the-art MaxSAT solvers that perform well in industrial instances are based on this technique. In particular, our solvers won the 2009 partial MaxSAT and the 2011 weighted partial MaxSAT industrial categories of the MaxSAT evaluation. We prove the correctness of all our algorithms. We also present a complete experimental study comparing the performance of our algorithms with latest MaxSAT solvers. © 2013 Elsevier B.V.

Weighted Partial MaxSAT (WPMS) is an optimization variant of the Satisfiability (SAT) problem. Several combinatorial optimization problems can be translated into WPMS. In this paper we extend the state-of-the-art WPM2 algorithm by adding several improvements, and implement it on top of an SMT solver. In particular, we show that by focusing search on solving to optimality subformulas of the original WPMS instance we increase the efficiency of WPM2. From the experimental evaluation we conducted on the PMS and WPMS instances at the 2012 MaxSAT Evaluation, we can conclude that the new approach is both the best performing for industrial instances, and for the union of industrial and crafted instances.

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. © 2012 ACM 1529-3785/2012/04-ART10 $10.00.

We study anti-unification for unranked terms and hedges that may contain term and hedge variables. The anti-unification problem of two hedges ~s_1 and ~s_2 is concerned with finding their generalization, a hedge ~q such that both ~s_1 and ~s_2 are instances of ~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 that is a parameter in the improved algorithm and selects certain common subsequences from the hedges to be generalized. Finally, we indicate a possible application of the algorithm in software engineering.

We study anti-unification for unranked terms and hedges that may contain term and hedge variables. The anti-unification problem of two hedges ~s_1 and ~s_2 is concerned with finding their generalization, a hedge ~q such that both ~s_1 and ~s_2 are instances of ~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 that is a parameter in the improved algorithm and selects certain common subsequences from the hedges to be generalized. Finally, we indicate a possible application of the algorithm in software engineering.

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. © The Author 2010. Published by Oxford University Press. All rights reserved.

Nominal Unification is an extension of first-order unification where terms can contain binders and unification is performed modulo alpha-equivalence. Here we prove that the existence of nominal unifiers can be decided in quadratic time. First, we linearly-reduce nominal unification problems to a sequence of freshness and equalities between atoms, modulo a permutation, using ideas as Paterson and Wegman for first-order unification. Second, we prove that solvability of these reduced problems may be checked in quadratic time. Finally, we point out how using ideas of Brown and Tarjan for unbalanced merging, we could solve these reduced problems more efficiently.