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Didier Dubois - One of the best experts on this subject based on the ideXlab platform.

  • symbolic possibilistic logic completeness and inference methods
    Journal of Logic and Computation, 2018
    Co-Authors: Claudette Cayrol, Didier Dubois, Faycal Touazi
    Abstract:

    This paper studies the extension of possibilistic logic to the case when weights attached to formulas are symbolic and stand for variables that lie in a Totally Ordered Scale, and only partial knowledge is available on the relative strength of these weights. A proof of the soundness and the completeness of this logic according to the relative certainty semantics in the sense of necessity measures is provided. Based on this result, two syntactic inference methods are presented. The first one calculates the necessity degree of a possibilistic formula using the notion of minimal inconsistent sub-base. A second method is proposed that takes inspiration from the concept of ATMS. Notions introduced in that area, such as nogoods and labels, are used to calculate the necessity degree of a possibilistic formula. A comparison of the two methods is provided, as well as a comparison with the original version of symbolic possibilistic logic.

  • Representing qualitative capacities as families of possibility measures
    International Journal of Approximate Reasoning, 2015
    Co-Authors: Didier Dubois, Henri Prade, Agnés Rico
    Abstract:

    This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite Totally Ordered Scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multiple-criteria decision-making. We show that any capacity or fuzzy measure ranging on a qualitative Scale can be viewed both as the lower bound of a set of possibility measures and the upper bound of a set of necessity measures (a situation somewhat similar to the one of quantitative capacities with respect to imprecise probability theory). We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of them is characteristic of its complexity. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Möbius transform, and the use of selection functions of the corresponding focal sets. We provide the connection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular modal logics and their neighborhood semantics can be described in terms of qualitative Möbius transforms.

  • Qualitative Capacities as Imprecise Possibilities.
    2013
    Co-Authors: Didier Dubois, Henri Prade, Agnés Rico
    Abstract:

    This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite Totally Ordered Scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multicriteria decision-making. More specifically, we investigate the question whether these qualitative set-functions can be viewed as classes of simpler set-functions, typically possibility measures, paralleling the situation of quantitative capacities with respect to imprecise probability theory. We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and their number is characteristic of its complexity. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular multi-source modal logics and their neighborhood semantics can be described in terms of qualitative Moebius transforms.

  • Multiple agent possibilistic logic
    Journal of Applied Non-Classical Logics, 2013
    Co-Authors: Asma Belhadi, Didier Dubois, Faiza Khellaf, Henri Prade
    Abstract:

    The paper presents a ‘multiple agent’ logic where formulas are pairs of the form (a, A), made of a proposition a and a subset of agents A. The formula (a, A) is intended to mean ‘(at least) all agents in A believe that a is true’. The formal similarity of such formulas with those of possibilistic logic, where propositions are associated with certainty levels, is emphasised. However, the subsets of agents are organised in a Boolean lattice, while certainty levels belong to a Totally Ordered Scale. The semantics of a set of ‘multiple agent’ logic formulas is expressed by a mapping which associates a subset of agents with each interpretation (intuitively, the maximal subset of agents for whom this interpretation is possibly true). Soundness and completeness results are established. Then a joint extension of the multiple agent logic and possibilistic logic is outlined. In this extended logic, propositions are then associated with both sets of agents and certainty levels. A formula then expresses that ‘all agents in set A believe that a is true at least at some level’. The semantics is then given in terms of fuzzy sets of agents that find an interpretation more or less possible. A specific feature of possibilistic logic is that the inconsistency of a knowledge base is a matter of degree. The proposed setting enables us to distinguish between the global consistency of a set of agents and their individual consistency (where both can be a matter of degree). In particular, given a set of multiple agent possibilistic formulas, one can compute the subset of agents that are individually consistent to some degree.

Henri Prade - One of the best experts on this subject based on the ideXlab platform.

  • Representing qualitative capacities as families of possibility measures
    International Journal of Approximate Reasoning, 2015
    Co-Authors: Didier Dubois, Henri Prade, Agnés Rico
    Abstract:

    This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite Totally Ordered Scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multiple-criteria decision-making. We show that any capacity or fuzzy measure ranging on a qualitative Scale can be viewed both as the lower bound of a set of possibility measures and the upper bound of a set of necessity measures (a situation somewhat similar to the one of quantitative capacities with respect to imprecise probability theory). We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of them is characteristic of its complexity. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Möbius transform, and the use of selection functions of the corresponding focal sets. We provide the connection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular modal logics and their neighborhood semantics can be described in terms of qualitative Möbius transforms.

  • Qualitative Capacities as Imprecise Possibilities.
    2013
    Co-Authors: Didier Dubois, Henri Prade, Agnés Rico
    Abstract:

    This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite Totally Ordered Scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multicriteria decision-making. More specifically, we investigate the question whether these qualitative set-functions can be viewed as classes of simpler set-functions, typically possibility measures, paralleling the situation of quantitative capacities with respect to imprecise probability theory. We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and their number is characteristic of its complexity. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular multi-source modal logics and their neighborhood semantics can be described in terms of qualitative Moebius transforms.

  • Multiple agent possibilistic logic
    Journal of Applied Non-Classical Logics, 2013
    Co-Authors: Asma Belhadi, Didier Dubois, Faiza Khellaf, Henri Prade
    Abstract:

    The paper presents a ‘multiple agent’ logic where formulas are pairs of the form (a, A), made of a proposition a and a subset of agents A. The formula (a, A) is intended to mean ‘(at least) all agents in A believe that a is true’. The formal similarity of such formulas with those of possibilistic logic, where propositions are associated with certainty levels, is emphasised. However, the subsets of agents are organised in a Boolean lattice, while certainty levels belong to a Totally Ordered Scale. The semantics of a set of ‘multiple agent’ logic formulas is expressed by a mapping which associates a subset of agents with each interpretation (intuitively, the maximal subset of agents for whom this interpretation is possibly true). Soundness and completeness results are established. Then a joint extension of the multiple agent logic and possibilistic logic is outlined. In this extended logic, propositions are then associated with both sets of agents and certainty levels. A formula then expresses that ‘all agents in set A believe that a is true at least at some level’. The semantics is then given in terms of fuzzy sets of agents that find an interpretation more or less possible. A specific feature of possibilistic logic is that the inconsistency of a knowledge base is a matter of degree. The proposed setting enables us to distinguish between the global consistency of a set of agents and their individual consistency (where both can be a matter of degree). In particular, given a set of multiple agent possibilistic formulas, one can compute the subset of agents that are individually consistent to some degree.

Agnés Rico - One of the best experts on this subject based on the ideXlab platform.

  • Representing qualitative capacities as families of possibility measures
    International Journal of Approximate Reasoning, 2015
    Co-Authors: Didier Dubois, Henri Prade, Agnés Rico
    Abstract:

    This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite Totally Ordered Scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multiple-criteria decision-making. We show that any capacity or fuzzy measure ranging on a qualitative Scale can be viewed both as the lower bound of a set of possibility measures and the upper bound of a set of necessity measures (a situation somewhat similar to the one of quantitative capacities with respect to imprecise probability theory). We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of them is characteristic of its complexity. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Möbius transform, and the use of selection functions of the corresponding focal sets. We provide the connection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular modal logics and their neighborhood semantics can be described in terms of qualitative Möbius transforms.

  • Qualitative Capacities as Imprecise Possibilities.
    2013
    Co-Authors: Didier Dubois, Henri Prade, Agnés Rico
    Abstract:

    This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite Totally Ordered Scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multicriteria decision-making. More specifically, we investigate the question whether these qualitative set-functions can be viewed as classes of simpler set-functions, typically possibility measures, paralleling the situation of quantitative capacities with respect to imprecise probability theory. We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and their number is characteristic of its complexity. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular multi-source modal logics and their neighborhood semantics can be described in terms of qualitative Moebius transforms.

Dubois Didier - One of the best experts on this subject based on the ideXlab platform.

  • Learning rule sets and Sugeno integrals for monotonic classification problems
    'Elsevier BV', 2020
    Co-Authors: Brabant Quentin, Couceiro Miguel, Dubois Didier, Prade Henri, Rico Agnès
    Abstract:

    International audienceIn some variants of the supervised classification setting, the domains of the attributes and the set of classes are Totally Ordered sets. The task of learning a classifier that is nondecreasing w.r.t. each attribute is called monotonic classification. Several kinds of models can be used in this task; in this paper , we focus on decision rules. We propose a method for learning a set of decision rules that optimally fits the training data while favoring short rules over long ones. We give new results on the representation of sets of if-then rules by extensions of Sugeno integrals to distinct attribute domains, where local utility functions are used to map attribute domains to a common Totally Ordered Scale. We study whether such qualitative extensions of Sugeno integral provide compact representations of large sets of decision rules

  • Symbolic possibilistic logic: completeness and inference methods
    'Oxford University Press (OUP)', 2018
    Co-Authors: Cayrol Claudette, Dubois Didier, Touazi Fayçal
    Abstract:

    International audienceThis paper studies the extension of possibilistic logic to the case when weights attached to formulas are symbolic. These weights then stand for variables that lie in a Totally Ordered Scale, and only partial knowledge is available on the relative strength of these weights in the form of inequality constraints. Reasoning in symbolic possibilistic logic means solving two problems. One is to compute symbolic expressions representing the weights of conclusions of a possibilistic knowledge base. The other problem is that of comparing the relative strength of derived weights, so as to find out if one formula is more certain than another one. Regarding the first problem, a proof of the soundness and the completeness of this logic according to the relative certainty semantics in the sense of necessity measures is provided. Based on this result, two syntactic inference methods are suggested. The first one shows how to use the notion of minimal inconsistent subsets and known techniques that compute them, so as to obtain the symbolic expression representing the necessity degree of a possibilistic formula. A second family of methods computes prime implicates and takes inspiration from the concept of assumption-based theory. It enables symbolic weights attached to consequences to be simplified in the course of their computation, taking inequality constraints into account. Finally, an algorithm is proposed to find if a consequence is more certain than another one. A comparison with the original version of symbolic possibilistic logic introduced by Benferhat and Prade in 2005 is provided

  • Symbolic possibilistic logic: completeness and inference methods
    'Oxford University Press (OUP)', 2018
    Co-Authors: Cayrol Claudette, Dubois Didier, Touazi Fayçal
    Abstract:

    This paper studies the extension of possibilistic logic to the case when weights attached to formulas are symbolic. These weights then stand for variables that lie in a Totally Ordered Scale, and only partial knowledge is available on the relative strength of these weights in the form of inequality constraints. Reasoning in symbolic possibilistic logic means solving two problems. One is to compute symbolic expressions representing the weights of conclusions of a possibilistic knowledge base. The other problem is that of comparing the relative strength of derived weights, so as to find out if one formula is more certain than another one. Regarding the first problem, a proof of the soundness and the completeness of this logic according to the relative certainty semantics in the sense of necessity measures is provided. Based on this result, two syntactic inference methods are suggested. The first one shows how to use the notion of minimal inconsistent subsets and known techniques that compute them, so as to obtain the symbolic expression representing the necessity degree of a possibilistic formula. A second family of methods computes prime implicates and takes inspiration from the concept of assumption-based theory. It enables symbolic weights attached to consequences to be simplified in the course of their computation, taking inequality constraints into account. Finally, an algorithm is proposed to find if a consequence is more certain than another one. A comparison with the original version of symbolic possibilistic logic introduced by Benferhat and Prade in 2005 is provided

  • Symbolic Possibilistic Logic: Completeness and Inference Methods
    HAL CCSD, 2015
    Co-Authors: Cayrol Claudette, Dubois Didier, Touazi Fayçal
    Abstract:

    International audienceThis paper studies the extension of possibilistic logic to the case when weights attached to formulas are symbolic and stand for variables that lie in a Totally Ordered Scale, and only partial knowledge is available on the relative strength of these weights. A proof of the soundness and the completeness of this logic according to the relative certainty semantics in the sense of necessity measures is provided. Based on this result, two syntactic inference methods are presented. The first one calculates the necessity degree of a possibilistic formula using the notion of minimal inconsistent sub-base. A second method is proposed that takes inspiration from the concept of ATMS. Notions introduced in that area, such as nogoods and labels, are used to calculate the necessity degree of a possibilistic formula. A comparison of the two methods is provided, as well as a comparison with the original version of symbolic possibilistic logic

  • Qualitative Capacities as Imprecise Possibilities.
    HAL CCSD, 2013
    Co-Authors: Dubois Didier, Prade Henri, Rico Agnès
    Abstract:

    National audienceThis paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite Totally Ordered Scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multicriteria decision-making. More specifically, we investigate the question whether these qualitative set-functions can be viewed as classes of simpler set-functions, typically possibility measures, paralleling the situation of quantitative capacities with respect to imprecise probability theory. We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and their number is characteristic of its complexity. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular multi-source modal logics and their neighborhood semantics can be described in terms of qualitative Moebius transforms

Prade Henri - One of the best experts on this subject based on the ideXlab platform.

  • Learning rule sets and Sugeno integrals for monotonic classification problems
    'Elsevier BV', 2020
    Co-Authors: Brabant Quentin, Couceiro Miguel, Dubois Didier, Prade Henri, Rico Agnès
    Abstract:

    International audienceIn some variants of the supervised classification setting, the domains of the attributes and the set of classes are Totally Ordered sets. The task of learning a classifier that is nondecreasing w.r.t. each attribute is called monotonic classification. Several kinds of models can be used in this task; in this paper , we focus on decision rules. We propose a method for learning a set of decision rules that optimally fits the training data while favoring short rules over long ones. We give new results on the representation of sets of if-then rules by extensions of Sugeno integrals to distinct attribute domains, where local utility functions are used to map attribute domains to a common Totally Ordered Scale. We study whether such qualitative extensions of Sugeno integral provide compact representations of large sets of decision rules

  • Qualitative Capacities as Imprecise Possibilities.
    HAL CCSD, 2013
    Co-Authors: Dubois Didier, Prade Henri, Rico Agnès
    Abstract:

    National audienceThis paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite Totally Ordered Scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multicriteria decision-making. More specifically, we investigate the question whether these qualitative set-functions can be viewed as classes of simpler set-functions, typically possibility measures, paralleling the situation of quantitative capacities with respect to imprecise probability theory. We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and their number is characteristic of its complexity. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular multi-source modal logics and their neighborhood semantics can be described in terms of qualitative Moebius transforms

  • Multiple agent possibilistic logic
    'Informa UK Limited', 2013
    Co-Authors: Belhadi Asma, Dubois Didier, Khellaf Faiza, Prade Henri
    Abstract:

    International audienceThe paper presents a ‘multiple agent’ logic where formulas are pairs of the form (a, A), made of a proposition a and a subset of agents A. The formula (a, A) is intended to mean ‘(at least) all agents in A believe that a is true’. The formal similarity of such formulas with those of possibilistic logic, where propositions are associated with certainty levels, is emphasised. However, the subsets of agents are organised in a Boolean lattice, while certainty levels belong to a Totally Ordered Scale. The semantics of a set of ‘multiple agent’ logic formulas is expressed by a mapping which associates a subset of agents with each interpretation (intuitively, the maximal subset of agents for whom this interpretation is possibly true). Soundness and completeness results are established. Then a joint extension of the multiple agent logic and possibilistic logic is outlined. In this extended logic, propositions are then associated with both sets of agents and certainty levels. A formula then expresses that ‘all agents in set A believe that a is true at least at some level’. The semantics is then given in terms of fuzzy sets of agents that find an interpretation more or less possible. A specific feature of possibilistic logic is that the inconsistency of a knowledge base is a matter of degree. The proposed setting enables us to distinguish between the global consistency of a set of agents and their individual consistency (where both can be a matter of degree). In particular, given a set of multiple agent possibilistic formulas, one can compute the subset of agents that are individually consistent to some degree