Satisfiability is considered the canonical NP-complete problem and is used as a starting point fo... more 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 an 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. Our bound is linear in the total weight. This is in stark contrast to quadratic lower bounds for the worst case.
Journal of artificial intelligence research/The journal of artificial intelligence research, Feb 11, 2024
In recent years there has been an increasing interest in studying proof systems stronger than Res... more 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.
Satisfiability is considered the canonical NP-complete problem and is used as a starting point fo... more 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.
Many studies focus on the generation of hard SAT instances. The hardness is usually measured by t... more Many studies focus on the generation of hard SAT instances. The hardness is usually measured by the time it takes SAT solvers to solve the instances. In this preliminary study, we focus on the generation of instances that have computational properties that are more similar to real-world instances. In particular, instances with the same degree of difficulty, measured in terms of the tree-like resolution space complexity. It is known that industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. One of the reasons may be their relatively small space complexity, compared with randomly generated instances. We provide two generation methods of k-SAT instances, called geometrical and the geo-regular, as generalizations of the uniform and regular k-CNF generators. Both are based on the use of a geometric probability distribution to select variables. We study the phase transition phenomena and the hardness of the generated instances as a function of the number of variables and the base of the geometric distribution. We prove that, with these two parameters we can adjust the difficulty of the problems in the phase transition point. We conjecture that this will allow us to generate random instances more similar to industrial instances, of interest for testing purposes.
The success of portfolio approaches in SAT solving relies on the observation that different SAT s... more 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 selfsimilar 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.
Nominal Logic is an extension of first-order logic with equality, name-binding, name-swapping, an... more Nominal Logic is an extension of first-order logic with equality, name-binding, name-swapping, and freshness of names. Contrarily to higher-order logic, bound variables are treated as atoms, and only free variables are proper unknowns in nominal unification. This allows "variable capture", breaking a fundamental principle of lambda-calculus. Despite this difference, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: higher-order patterns unification. This reduction proves that nominal unification can be decided in quadratic deterministic time. This research has been partially founded by the CICYT research project TIN2007-68005-C04-01/02/03.
Zenodo (CERN European Organization for Nuclear Research), Jul 8, 2023
Some applications of artificial intelligence make it desirable that logical formulae be converted... more Some applications of artificial intelligence make it desirable that logical formulae be converted computationally to comprehensible natural language sentences. As there are many logical equivalents to a given formula, finding the most suitable equivalent to be used as input for such a "logic-to-text" generation system is a difficult challenge. In this paper, we focus on the role of brevity: Are the shortest formulae the most suitable? We focus on propositional logic (PL), framing formula minimization (i.e., the problem of finding the shortest equivalent of a given formula) as a Quantified Boolean Formulae (QBFs) satisfiability problem. We experiment with several generators and selection strategies to prune the resulting candidates. We conduct exhaustive automatic and human evaluations of the comprehensibility and fluency of the generated texts. The results suggest that while, in many cases, minimization has a positive impact on the quality of the sentences generated, formula minimization may ultimately not be the best strategy.
UPCommons institutional repository (Universitat Politècnica de Catalunya), Jun 10, 2016
Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is ... more 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.
Nowadays, Conflict-Driven Clause Learning (CDCL) techniques are one of the key components of mode... more 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.
Society for Industrial and Applied Mathematics eBooks, 2021
Satisfiability is considered the canonical NP-complete problem and is used as a starting point fo... more 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 an 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. Our bound is linear in the total weight. This is in stark contrast to quadratic lower bounds for the worst case.
Representing some problems with XOR clauses (parity constraints) can allow to apply more efficien... more 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 recursi... more 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 garbagefree 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.
The minimization of propositional formulae is a classical problem in logic, whose first algorithm... more 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 Σ p 2-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.
Phase-transition in random SAT formulas is one of the properties best studied by theoretical SAT ... more 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.
Frontiers in artificial intelligence and applications, Oct 17, 2022
Obviously, we do not prove P = NP in this article. In fact, the title only refers to the first pa... more Obviously, we do not prove P = NP in this article. In fact, the title only refers to the first part, where the proof that we present contains an error that, to make reading more attractive, is only revealed in the second part. In the second part, we describe how the reduction of SAT to Max2XOR and the proof system presented in the first part-although they do not solve one of the Millennium Prize Problems-may trigger new complementary ways of solving the SAT problem.
We focus on the random generation of SAT instances that have properties similar to real-world ins... more 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=1−2β(1−β)2. We also p...
Satisfiability is considered the canonical NP-complete problem and is used as a starting point fo... more 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 an 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. Our bound is linear in the total weight. This is in stark contrast to quadratic lower bounds for the worst case.
Journal of artificial intelligence research/The journal of artificial intelligence research, Feb 11, 2024
In recent years there has been an increasing interest in studying proof systems stronger than Res... more 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.
Satisfiability is considered the canonical NP-complete problem and is used as a starting point fo... more 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.
Many studies focus on the generation of hard SAT instances. The hardness is usually measured by t... more Many studies focus on the generation of hard SAT instances. The hardness is usually measured by the time it takes SAT solvers to solve the instances. In this preliminary study, we focus on the generation of instances that have computational properties that are more similar to real-world instances. In particular, instances with the same degree of difficulty, measured in terms of the tree-like resolution space complexity. It is known that industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. One of the reasons may be their relatively small space complexity, compared with randomly generated instances. We provide two generation methods of k-SAT instances, called geometrical and the geo-regular, as generalizations of the uniform and regular k-CNF generators. Both are based on the use of a geometric probability distribution to select variables. We study the phase transition phenomena and the hardness of the generated instances as a function of the number of variables and the base of the geometric distribution. We prove that, with these two parameters we can adjust the difficulty of the problems in the phase transition point. We conjecture that this will allow us to generate random instances more similar to industrial instances, of interest for testing purposes.
The success of portfolio approaches in SAT solving relies on the observation that different SAT s... more 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 selfsimilar 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.
Nominal Logic is an extension of first-order logic with equality, name-binding, name-swapping, an... more Nominal Logic is an extension of first-order logic with equality, name-binding, name-swapping, and freshness of names. Contrarily to higher-order logic, bound variables are treated as atoms, and only free variables are proper unknowns in nominal unification. This allows "variable capture", breaking a fundamental principle of lambda-calculus. Despite this difference, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: higher-order patterns unification. This reduction proves that nominal unification can be decided in quadratic deterministic time. This research has been partially founded by the CICYT research project TIN2007-68005-C04-01/02/03.
Zenodo (CERN European Organization for Nuclear Research), Jul 8, 2023
Some applications of artificial intelligence make it desirable that logical formulae be converted... more Some applications of artificial intelligence make it desirable that logical formulae be converted computationally to comprehensible natural language sentences. As there are many logical equivalents to a given formula, finding the most suitable equivalent to be used as input for such a "logic-to-text" generation system is a difficult challenge. In this paper, we focus on the role of brevity: Are the shortest formulae the most suitable? We focus on propositional logic (PL), framing formula minimization (i.e., the problem of finding the shortest equivalent of a given formula) as a Quantified Boolean Formulae (QBFs) satisfiability problem. We experiment with several generators and selection strategies to prune the resulting candidates. We conduct exhaustive automatic and human evaluations of the comprehensibility and fluency of the generated texts. The results suggest that while, in many cases, minimization has a positive impact on the quality of the sentences generated, formula minimization may ultimately not be the best strategy.
UPCommons institutional repository (Universitat Politècnica de Catalunya), Jun 10, 2016
Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is ... more 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.
Nowadays, Conflict-Driven Clause Learning (CDCL) techniques are one of the key components of mode... more 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.
Society for Industrial and Applied Mathematics eBooks, 2021
Satisfiability is considered the canonical NP-complete problem and is used as a starting point fo... more 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 an 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. Our bound is linear in the total weight. This is in stark contrast to quadratic lower bounds for the worst case.
Representing some problems with XOR clauses (parity constraints) can allow to apply more efficien... more 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 recursi... more 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 garbagefree 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.
The minimization of propositional formulae is a classical problem in logic, whose first algorithm... more 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 Σ p 2-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.
Phase-transition in random SAT formulas is one of the properties best studied by theoretical SAT ... more 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.
Frontiers in artificial intelligence and applications, Oct 17, 2022
Obviously, we do not prove P = NP in this article. In fact, the title only refers to the first pa... more Obviously, we do not prove P = NP in this article. In fact, the title only refers to the first part, where the proof that we present contains an error that, to make reading more attractive, is only revealed in the second part. In the second part, we describe how the reduction of SAT to Max2XOR and the proof system presented in the first part-although they do not solve one of the Millennium Prize Problems-may trigger new complementary ways of solving the SAT problem.
We focus on the random generation of SAT instances that have properties similar to real-world ins... more 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=1−2β(1−β)2. We also p...
We investigate a problem that lies at the intersection of three research areas, namely Automated ... more 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.
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Papers by Jordi Levy