Recent work proposed the computation of so-called PIexplanations of Naive Bayes Classifiers (NBCs... more Recent work proposed the computation of so-called PIexplanations of Naive Bayes Classifiers (NBCs) [29]. PI-explanations are subset-minimal sets of feature-value pairs that are sufficient for the prediction, and have been computed with state-of-the-art exact algorithms that are worst-case exponential in time and space. In contrast, we show that the computation of one PI-explanation for an NBC can be achieved in log-linear time, and that the same result also applies to the more general class of linear classifiers. Furthermore, we show that the enumeration of PI-explanations can be obtained with polynomial delay. Experimental results demonstrate the performance gains of the new algorithms when compared with earlier work. The experimental results also investigate ways to measure the quality of heuristic explanations.
Proceedings of the International Conference on Automated Planning and Scheduling
This paper describes a polynomially-solvable sub-problem of temporal planning. Polynomiality foll... more This paper describes a polynomially-solvable sub-problem of temporal planning. Polynomiality follows from two assumptions. Firstly, by supposing that each sub-goal fluent can be established by at most one action, we can quickly determine which actions are necessary in any plan. Secondly, the monotonicity of sub-goal fluents allows us to express planning as an instance of STP≠ (Simple Temporal Problem, difference constraints). Our class includes temporally-expressive problems, which we illustrate with an example of chemical process planning.
International audienceDomain reduction is an essential tool for solving the constraint satisfacti... more International audienceDomain reduction is an essential tool for solving the constraint satisfaction problem (CSP). In the binary CSP, neighbourhood substitution consists in eliminating a value if there exists another value which can be substituted for it in each constraint. We show that the notion of neighbourhood substitution can be strengthened in two distinct ways without increasing time complexity. We also show the theoretical result that, unlike neighbourhood substitution, finding an optimal sequence of these new operations is NP-hard.La réduction de domaines est un outil essentiel dans la résoluion de problèmes de satisfaction de contraintes (CSP). Pour les CSP binaires, la substitution de voisinage constisteàéliminer une valeur s'il existe une autre valeur qui peut la remplacer dans chaque contrainte. Nous démontrons qu'il est possible de rendre plus forte la notion de substitution de voi-sinage de deux façons distinctes sans augmenter la complexité temporelle. Nous d...
Proceedings of the AAAI Conference on Artificial Intelligence
Compilation into propositional languages finds a growing number of practical uses, including in c... more Compilation into propositional languages finds a growing number of practical uses, including in constraint programming, diagnosis and machine learning (ML), among others. One concrete example is the use of propositional languages as classifiers, and one natural question is how to explain the predictions made. This paper shows that for classifiers represented with some of the best-known propositional languages, different kinds of explanations can be computed in polynomial time. These languages include deterministic decomposable negation normal form (d-DNNF), and so any propositional language that is strictly less succinct than d-DNNF. Furthermore, the paper describes optimizations, specific to Sentential Decision Diagrams (SDDs), which are shown to yield more efficient algorithms in practice.
Proceedings of the AAAI Conference on Artificial Intelligence
Constraint programming is a natural paradigm for many combinatorial optimisation problems. The co... more Constraint programming is a natural paradigm for many combinatorial optimisation problems. The complexity of constraint satisfaction for various forms of constraints has been widely-studied, both to inform the choice of appropriate algorithms, and to understand better the boundary between polynomial-time complexity and NP-hardness. In constraint programming it is well-known that any constraint satisfaction problem can be converted to an equivalent binary problem using the so-called dual encoding. Using this standard approach any fixed collection of constraints, of arbitrary arity, can be converted to an equivalent set of constraints of arity at most two. Here we show that this transformation, although it changes the domain of the constraints, preserves all the relevant algebraic properties that determine the complexity. Moreover, we show that the dual encoding preserves many of the key algorithmic properties of the original instance. We also show that this remains true for more gene...
Proceedings of the AAAI Conference on Artificial Intelligence
Novel tractable classes of the binary CSP (constraint satisfaction problem) have recently been di... more Novel tractable classes of the binary CSP (constraint satisfaction problem) have recently been discovered by studying classes of instances defined by excluding subproblems described by patterns. The complete characterisation of all tractable classes defined by forbidden patterns is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of two constraints.
International audienceAs an extension of constraint networks, valued constraint networks (or valu... more International audienceAs an extension of constraint networks, valued constraint networks (or valued CSPs) define a unifying framework for modelling optimisation problems over finite domains in which the cost domain (also denoted as the valuation structure) can be symbolic or numeric but is totally ordered. As is the case for constraint networks, valued CSPs can be viewed as graphical models, defined by a set of variables with an associated finite domain and by a set of cost functions, each involving a subset of the variables. Before reading this chapter, we strongly advise the reader to finish reading chapter “Constraint Reasoning” of this Volume, dedicated to the Constraint Satisfaction Problem
OATAO is an open access repository that collects the work of Toulouse researchers and makes it fr... more OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 15300 The contribution was presented at JFPC 2015: Résumé L'´ etude des classes polynomiales constitue une question importante en intelligence artificielle, en particulier au niveau desprobì emes de satisfaction de contraintes. Dans ce contexte, la propriété BTP fournit une classe importante de l'´ etat de l'art. Dans cet article, nous pro-posons d'´ etendre et de généraliser cette classe en intro-duisant la propriété k-BTP (et la classe des instances satisfaisant cette propriété) o` u le paramètre k est une constante donnée. Ainsi, nous avons 2-BTP = BTP, et pour k > 2, k-BTP est une relaxation de BTP au sens o` u k-BT P (k + 1)-BT P. En outre, nous montrons que si k-TW est la classe d'instances ayant une largeur arborescente b...
OATAO is an open access repository that collects the work of Toulouse researchers and makes it fr... more OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 16981 The contribution was presented at IJCAI 2016 : Abstract A binary CSP instance satisfying the broken-triangle property (BTP) can be solved in polynomial time. Unfortunately, in practice, few instances satisfy the BTP. We show that a local version of the BTP allows the merging of domain values in binary CSPs, thus providing a novel polynomial-time reduction operation. Experimental trials on benchmark instances demonstrate a significant decrease in instance size for certain classes of problems. We show that BTP-merging can be generalised to instances with constraints of arbitrary arity. A directional version of the general-arity BTP then allows us to extend the BTP tractable class previously defined only for binary CSP.
Trouver une solutionà un problème de satisfaction de contraintes est en général un problème NP-di... more Trouver une solutionà un problème de satisfaction de contraintes est en général un problème NP-difficile. Ainsi, identifier une nouvelle classe polynomiale de CSP, en d'autres termes, une classe de problèmes de satisfaction de contraintes pour lesquelles il existe une méthode de reconnaissance et de résolution de complexité polynomiale, constitue un axe de recherche fondamental dans l'étude des CSPs. Dans cet article, nous présentons une nouvelle classe polynomiale relationnelle pour les CSPs binaires. Tout d'abord nous présentons un nouvel opérateur ternaire que nous appelons mjx. Nous décrivons ensuite les relations fermées par cet opérateur en proposant un algorithme d'identification de ces relations. Afin de réduire la complexité spatiale et temporelle des traitements, nous définissons une nouvelle méthode de stockage de ces relations qui permet de réduire la complexité de chemin consistance, ce dernier niveau de consistanceétant suffisant pour résoudre les instances de la classe proposée.
Although the CSP (constraint satisfaction problem) is NP-complete, even in the case when all cons... more Although the CSP (constraint satisfaction problem) is NP-complete, even in the case when all constraints are binary, certain classes of instances are tractable. We study classes of instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes. The complete characterisation of all tractable classes defined by forbidding patterns (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of either one or two constraints. This has allowed us to discover new tractable classes including, for example, a novel generalisation of 2SAT.
A realistic model of multi-agent planning must allow us to formalize notions which are absent in ... more A realistic model of multi-agent planning must allow us to formalize notions which are absent in classical planning, such as communication and knowledge.We investigate multi-agent planning based on a simple logic of knowledge that is grounded on the visibility of propositional variables. Using such a formal logic allows us to prove the existence of a plan given the description of the individual actions. We present an encoding of multi-agent planning problems expressed in this logic into the classical planning language PDDL. The solvability of a planning task is reduced to a model checking problem in a dynamic extension of our logic, proving its complexity. Feeding the resulting problem into a PDDL planner provides a provably correct plan for the original multi-agent planning problem. We apply our method on several examples such as the gossip problem.
Domain reduction is an essential tool for solving the constraint satisfaction problem (CSP). In t... more Domain reduction is an essential tool for solving the constraint satisfaction problem (CSP). In the binary CSP, neighbourhood substitution consists in eliminating a value if there exists another value which can be substituted for it in each constraint. We show that the notion of neighbourhood substitution can be strengthened in two distinct ways without increasing time complexity. We also show the theoretical result that, unlike neighbourhood substitution, finding an optimal sequence of these new operations is NP-hard. This work was partially funded by ANITI, funded by the French program "Investing for the Future-PIA3" under Grant agreement n o ANR-19-PI3A-0004.
Recent work proposed δ-relevant inputs (or sets) as a probabilistic explanation for the predictio... more Recent work proposed δ-relevant inputs (or sets) as a probabilistic explanation for the predictions made by a classifier on a given input. δ-relevant sets are significant because they serve to relate (model-agnostic) Anchors with (model-accurate) PI-explanations, among other explanation approaches. Unfortunately, the computation of smallest size δ-relevant sets is complete for NP, rendering their computation largely infeasible in practice. This paper investigates solutions for tackling the practical limitations of δ-relevant sets. First, the paper alternatively considers the computation of subset-minimal sets. Second, the paper studies concrete families of classifiers, including decision trees among others. For these cases, the paper shows that the computation of subset-minimal δ-relevant sets is in NP, and can be solved with a polynomial number of calls to an NP oracle. The experimental evaluation compares the proposed approach with heuristic explainers for the concrete case of the...
A pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the co... more A pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the compatibility of certain pairs of variable-value assignments may be unspecified. The notion of forbidden pattern has led to the discovery of several novel tractable classes for the CSP. However, for this field to come of age it is time for a theoretical study of the algebra of patterns. We present a Galois connection between lattices composed of sets of forbidden patterns and sets of generic instances, and investigate its consequences. We then extend patterns to augmented patterns and exhibit a similar Galois connection. Augmented patterns are a more powerful language than flat (i.e. non-augmented) patterns, as we demonstrate by showing that, for any k ≥ 1, instances with tree-width bounded by k cannot be specified by forbidding a finite set of flat patterns but can be specified by a finite set of augmented patterns. A single finite set of augmented patterns can also describe the class of ...
Les triangles casses representent un concept important, non seulement pour la resolution en temps... more Les triangles casses representent un concept important, non seulement pour la resolution en temps polynomial des problemes de satisfaction de contraintes, mais aussi pour l'elimination de variables ou encore la reduction de la taille des domaines par la fusion des valeurs. Plus precisement, pour une variable donnee d'un CSP binaire arc-coherent, si aucun triangle casse ne se produit sur aucun couple de valeurs, alors cette variable peut etre eliminee tout en preservant la satisfiabilite. Plus recemment, il a ete demontre qu'en cas de non applicabilite de cette regle a cause de l'existence d'au moins un triangle casse, il se peut qu'il existe deux valeurs de cette meme variable sur lesquelles aucun triangle casse ne s'est produit. Dans ce cas, ces deux valeurs peuvent etre fusionnees en une seule tout en preservant la satisfiabilite. Dans ce papier, nous montrons que sous certaines conditions, et meme en cas d'existence de quelques triangles casses sur...
Recent work proposed the computation of so-called PIexplanations of Naive Bayes Classifiers (NBCs... more Recent work proposed the computation of so-called PIexplanations of Naive Bayes Classifiers (NBCs) [29]. PI-explanations are subset-minimal sets of feature-value pairs that are sufficient for the prediction, and have been computed with state-of-the-art exact algorithms that are worst-case exponential in time and space. In contrast, we show that the computation of one PI-explanation for an NBC can be achieved in log-linear time, and that the same result also applies to the more general class of linear classifiers. Furthermore, we show that the enumeration of PI-explanations can be obtained with polynomial delay. Experimental results demonstrate the performance gains of the new algorithms when compared with earlier work. The experimental results also investigate ways to measure the quality of heuristic explanations.
Proceedings of the International Conference on Automated Planning and Scheduling
This paper describes a polynomially-solvable sub-problem of temporal planning. Polynomiality foll... more This paper describes a polynomially-solvable sub-problem of temporal planning. Polynomiality follows from two assumptions. Firstly, by supposing that each sub-goal fluent can be established by at most one action, we can quickly determine which actions are necessary in any plan. Secondly, the monotonicity of sub-goal fluents allows us to express planning as an instance of STP≠ (Simple Temporal Problem, difference constraints). Our class includes temporally-expressive problems, which we illustrate with an example of chemical process planning.
International audienceDomain reduction is an essential tool for solving the constraint satisfacti... more International audienceDomain reduction is an essential tool for solving the constraint satisfaction problem (CSP). In the binary CSP, neighbourhood substitution consists in eliminating a value if there exists another value which can be substituted for it in each constraint. We show that the notion of neighbourhood substitution can be strengthened in two distinct ways without increasing time complexity. We also show the theoretical result that, unlike neighbourhood substitution, finding an optimal sequence of these new operations is NP-hard.La réduction de domaines est un outil essentiel dans la résoluion de problèmes de satisfaction de contraintes (CSP). Pour les CSP binaires, la substitution de voisinage constisteàéliminer une valeur s'il existe une autre valeur qui peut la remplacer dans chaque contrainte. Nous démontrons qu'il est possible de rendre plus forte la notion de substitution de voi-sinage de deux façons distinctes sans augmenter la complexité temporelle. Nous d...
Proceedings of the AAAI Conference on Artificial Intelligence
Compilation into propositional languages finds a growing number of practical uses, including in c... more Compilation into propositional languages finds a growing number of practical uses, including in constraint programming, diagnosis and machine learning (ML), among others. One concrete example is the use of propositional languages as classifiers, and one natural question is how to explain the predictions made. This paper shows that for classifiers represented with some of the best-known propositional languages, different kinds of explanations can be computed in polynomial time. These languages include deterministic decomposable negation normal form (d-DNNF), and so any propositional language that is strictly less succinct than d-DNNF. Furthermore, the paper describes optimizations, specific to Sentential Decision Diagrams (SDDs), which are shown to yield more efficient algorithms in practice.
Proceedings of the AAAI Conference on Artificial Intelligence
Constraint programming is a natural paradigm for many combinatorial optimisation problems. The co... more Constraint programming is a natural paradigm for many combinatorial optimisation problems. The complexity of constraint satisfaction for various forms of constraints has been widely-studied, both to inform the choice of appropriate algorithms, and to understand better the boundary between polynomial-time complexity and NP-hardness. In constraint programming it is well-known that any constraint satisfaction problem can be converted to an equivalent binary problem using the so-called dual encoding. Using this standard approach any fixed collection of constraints, of arbitrary arity, can be converted to an equivalent set of constraints of arity at most two. Here we show that this transformation, although it changes the domain of the constraints, preserves all the relevant algebraic properties that determine the complexity. Moreover, we show that the dual encoding preserves many of the key algorithmic properties of the original instance. We also show that this remains true for more gene...
Proceedings of the AAAI Conference on Artificial Intelligence
Novel tractable classes of the binary CSP (constraint satisfaction problem) have recently been di... more Novel tractable classes of the binary CSP (constraint satisfaction problem) have recently been discovered by studying classes of instances defined by excluding subproblems described by patterns. The complete characterisation of all tractable classes defined by forbidden patterns is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of two constraints.
International audienceAs an extension of constraint networks, valued constraint networks (or valu... more International audienceAs an extension of constraint networks, valued constraint networks (or valued CSPs) define a unifying framework for modelling optimisation problems over finite domains in which the cost domain (also denoted as the valuation structure) can be symbolic or numeric but is totally ordered. As is the case for constraint networks, valued CSPs can be viewed as graphical models, defined by a set of variables with an associated finite domain and by a set of cost functions, each involving a subset of the variables. Before reading this chapter, we strongly advise the reader to finish reading chapter “Constraint Reasoning” of this Volume, dedicated to the Constraint Satisfaction Problem
OATAO is an open access repository that collects the work of Toulouse researchers and makes it fr... more OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 15300 The contribution was presented at JFPC 2015: Résumé L'´ etude des classes polynomiales constitue une question importante en intelligence artificielle, en particulier au niveau desprobì emes de satisfaction de contraintes. Dans ce contexte, la propriété BTP fournit une classe importante de l'´ etat de l'art. Dans cet article, nous pro-posons d'´ etendre et de généraliser cette classe en intro-duisant la propriété k-BTP (et la classe des instances satisfaisant cette propriété) o` u le paramètre k est une constante donnée. Ainsi, nous avons 2-BTP = BTP, et pour k > 2, k-BTP est une relaxation de BTP au sens o` u k-BT P (k + 1)-BT P. En outre, nous montrons que si k-TW est la classe d'instances ayant une largeur arborescente b...
OATAO is an open access repository that collects the work of Toulouse researchers and makes it fr... more OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 16981 The contribution was presented at IJCAI 2016 : Abstract A binary CSP instance satisfying the broken-triangle property (BTP) can be solved in polynomial time. Unfortunately, in practice, few instances satisfy the BTP. We show that a local version of the BTP allows the merging of domain values in binary CSPs, thus providing a novel polynomial-time reduction operation. Experimental trials on benchmark instances demonstrate a significant decrease in instance size for certain classes of problems. We show that BTP-merging can be generalised to instances with constraints of arbitrary arity. A directional version of the general-arity BTP then allows us to extend the BTP tractable class previously defined only for binary CSP.
Trouver une solutionà un problème de satisfaction de contraintes est en général un problème NP-di... more Trouver une solutionà un problème de satisfaction de contraintes est en général un problème NP-difficile. Ainsi, identifier une nouvelle classe polynomiale de CSP, en d'autres termes, une classe de problèmes de satisfaction de contraintes pour lesquelles il existe une méthode de reconnaissance et de résolution de complexité polynomiale, constitue un axe de recherche fondamental dans l'étude des CSPs. Dans cet article, nous présentons une nouvelle classe polynomiale relationnelle pour les CSPs binaires. Tout d'abord nous présentons un nouvel opérateur ternaire que nous appelons mjx. Nous décrivons ensuite les relations fermées par cet opérateur en proposant un algorithme d'identification de ces relations. Afin de réduire la complexité spatiale et temporelle des traitements, nous définissons une nouvelle méthode de stockage de ces relations qui permet de réduire la complexité de chemin consistance, ce dernier niveau de consistanceétant suffisant pour résoudre les instances de la classe proposée.
Although the CSP (constraint satisfaction problem) is NP-complete, even in the case when all cons... more Although the CSP (constraint satisfaction problem) is NP-complete, even in the case when all constraints are binary, certain classes of instances are tractable. We study classes of instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes. The complete characterisation of all tractable classes defined by forbidding patterns (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of either one or two constraints. This has allowed us to discover new tractable classes including, for example, a novel generalisation of 2SAT.
A realistic model of multi-agent planning must allow us to formalize notions which are absent in ... more A realistic model of multi-agent planning must allow us to formalize notions which are absent in classical planning, such as communication and knowledge.We investigate multi-agent planning based on a simple logic of knowledge that is grounded on the visibility of propositional variables. Using such a formal logic allows us to prove the existence of a plan given the description of the individual actions. We present an encoding of multi-agent planning problems expressed in this logic into the classical planning language PDDL. The solvability of a planning task is reduced to a model checking problem in a dynamic extension of our logic, proving its complexity. Feeding the resulting problem into a PDDL planner provides a provably correct plan for the original multi-agent planning problem. We apply our method on several examples such as the gossip problem.
Domain reduction is an essential tool for solving the constraint satisfaction problem (CSP). In t... more Domain reduction is an essential tool for solving the constraint satisfaction problem (CSP). In the binary CSP, neighbourhood substitution consists in eliminating a value if there exists another value which can be substituted for it in each constraint. We show that the notion of neighbourhood substitution can be strengthened in two distinct ways without increasing time complexity. We also show the theoretical result that, unlike neighbourhood substitution, finding an optimal sequence of these new operations is NP-hard. This work was partially funded by ANITI, funded by the French program "Investing for the Future-PIA3" under Grant agreement n o ANR-19-PI3A-0004.
Recent work proposed δ-relevant inputs (or sets) as a probabilistic explanation for the predictio... more Recent work proposed δ-relevant inputs (or sets) as a probabilistic explanation for the predictions made by a classifier on a given input. δ-relevant sets are significant because they serve to relate (model-agnostic) Anchors with (model-accurate) PI-explanations, among other explanation approaches. Unfortunately, the computation of smallest size δ-relevant sets is complete for NP, rendering their computation largely infeasible in practice. This paper investigates solutions for tackling the practical limitations of δ-relevant sets. First, the paper alternatively considers the computation of subset-minimal sets. Second, the paper studies concrete families of classifiers, including decision trees among others. For these cases, the paper shows that the computation of subset-minimal δ-relevant sets is in NP, and can be solved with a polynomial number of calls to an NP oracle. The experimental evaluation compares the proposed approach with heuristic explainers for the concrete case of the...
A pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the co... more A pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the compatibility of certain pairs of variable-value assignments may be unspecified. The notion of forbidden pattern has led to the discovery of several novel tractable classes for the CSP. However, for this field to come of age it is time for a theoretical study of the algebra of patterns. We present a Galois connection between lattices composed of sets of forbidden patterns and sets of generic instances, and investigate its consequences. We then extend patterns to augmented patterns and exhibit a similar Galois connection. Augmented patterns are a more powerful language than flat (i.e. non-augmented) patterns, as we demonstrate by showing that, for any k ≥ 1, instances with tree-width bounded by k cannot be specified by forbidding a finite set of flat patterns but can be specified by a finite set of augmented patterns. A single finite set of augmented patterns can also describe the class of ...
Les triangles casses representent un concept important, non seulement pour la resolution en temps... more Les triangles casses representent un concept important, non seulement pour la resolution en temps polynomial des problemes de satisfaction de contraintes, mais aussi pour l'elimination de variables ou encore la reduction de la taille des domaines par la fusion des valeurs. Plus precisement, pour une variable donnee d'un CSP binaire arc-coherent, si aucun triangle casse ne se produit sur aucun couple de valeurs, alors cette variable peut etre eliminee tout en preservant la satisfiabilite. Plus recemment, il a ete demontre qu'en cas de non applicabilite de cette regle a cause de l'existence d'au moins un triangle casse, il se peut qu'il existe deux valeurs de cette meme variable sur lesquelles aucun triangle casse ne s'est produit. Dans ce cas, ces deux valeurs peuvent etre fusionnees en une seule tout en preservant la satisfiabilite. Dans ce papier, nous montrons que sous certaines conditions, et meme en cas d'existence de quelques triangles casses sur...
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