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Da Ruan - One of the best experts on this subject based on the ideXlab platform.

  • filter based Resolution Principle for lattice valued propositional logic lp x
    Information Sciences, 2007
    Co-Authors: Da Ruan
    Abstract:

    As one of most powerful approaches in automated reasoning, Resolution Principle has been introduced to non-classical logics, such as many-valued logic. However, most of the existing works are limited to the chain-type truth-value fields. Lattice-valued logic is a kind of important non-classical logic, which can be applied to describe and handle incomparability by the incomparable elements in its truth-value field. In this paper, a filter-based Resolution Principle for the lattice-valued propositional logic LP(X) based on lattice implication algebra is presented, where filter of the truth-value field being a lattice implication algebra is taken as the criterion for measuring the satisfiability of a lattice-valued logical formula. The notions and properties of lattice implication algebra, filter of lattice implication algebra, and the lattice-valued propositional logic LP(X) are given firstly. The definitions and structures of two kinds of lattice-valued logical formulae, i.e., the simple generalized clauses and complex generalized clauses, are presented then. Finally, the filter-based Resolution Principle is given and after that the soundness theorem and weak completeness theorems for the presented approach are proved.

  • lattice valued logic an alternative approach to treat fuzziness and incomparability
    2003
    Co-Authors: Da Ruan, K Y Qin, Jun Liu
    Abstract:

    I Introduction.- 1 Introduction.- 1.1 Major Methodologies in Artificial Intelligence.- 1.2 Basic Academic Ideas.- 1.3 Some Related Concepts.- 1.4 Many-Valued Logic and Lattice-Valued Logic.- 1.5 Uncertainty Inference.- 1.5.1 Probability-Based Uncertainty Reasoning.- 1.5.2 Fuzzy Set Based Uncertainty Reasoning.- 1.5.3 Non-Monotonic Logic Based Uncertainty Reasoning.- 1.6 Automated Reasoning in Many-Valued Logic.- II Lattice Implication Algebras.- 2 Concepts and Properties.- 2.1 Lattice Implication Algebras.- 2.1.1 Concepts and Examples.- 2.1.2 Basic Properties.- 2.2 Lattice H Implication Algebras.- 2.3 Lattice Properties.- 2.4 Homomorphisms.- 3 Filters.- 3.1 Filters and Implicative Filters.- 3.2 Generated Filters.- 3.3 Positive Implicative Filters and Associative Filters.- 3.4 Prime Filters and Ultra-Filters.- 3.5 I-Filters, Involution Filters and Obstinate Filters.- 3.6 Fuzzy Filters.- 4 LI-Ideals.- 4.1 LI-Ideals.- 4.2 Fuzzy LI-Ideals.- 4.3 Normal Fuzzy LI-Ideals.- 4.4 Intuitionistic Fuzzy LI-Ideals.- 5 Homomorphisms and Representations.- 5.1 Congruence Relations.- 5.1.1 Congruence Relations Induced by Filters.- 5.1.2 Congruences Relations Induced by LI-ideals.- 5.1.3 Congruence Relations Induced by Fuzzy Filters.- 5.1.4 Congruence Relations Induced by Fuzzy LI-ideals.- 5.2 Proper Lattice Implication Algebras.- 5.3 Representations.- 6 Topological Structure of Filter Spaces.- 6.1 Filter Spaces.- 6.1.1 Basic Concepts.- 6.1.2 Topological Properties.- 6.2 Product Topology and Quotient Topology.- 6.3 Lattice Topology.- 6.4 Prime Spaces.- 7 Connections with Related Algebras.- 7.1 Lattice Implication Algebras and BCK-Algebras.- 7.2 Lattice Implication Algebras and MV-Algebras.- 7.3 Lattice Implication Algebras and Related Algebras.- 8 Related Issues.- 8.1 Category of Lattice Implication Algebras.- 8.2 Category of Fuzzy Lattice Implication Algebras.- 8.3 Fuzzy Power Sets.- 8.4 Adjoint Semigroups.- 8.5 Logical Properties.- III Lattice-Valued Logic Systems.- 9 Lattice-Valued Propositional Logics.- 9.1 Lattice-Valued Propositional Logic LP(X).- 9.1.1 Language.- 9.1.2 Semantics.- 9.1.3 Syntax.- 9.1.4 Examples.- 9.2 Gradational Lattice-Valued Propositional Logic Lvpl.- 9.2.1 Language.- 9.2.2 Rules of Inference.- 9.2.3 Semantics.- 9.2.4 Syntax.- 9.2.5 Satisfiability and Consistency.- 9.2.6 Deduction Theorem.- 9.2.7 Compactness.- 9.2.8 Examples.- 10 Lattice-Valued First-Order Logics.- 10.1 Lattice-Valued First-Order Logic LF(X).- 10.1.1 Language.- 10.1.2 Interpretation.- 10.1.3 Semantics.- 10.1.4 Syntax.- 10.1.5 Properties of Model Theory.- 10.2 Gradational Lattice-Valued First-Order Logic Lvfl.- 10.2.1 Language.- 10.2.2 Interpretation.- 10.2.3 Semantics.- 10.2.4 Standardization of Formulae.- 10.2.5 Syntax.- 10.2.6 Soundness and Completeness.- 10.2.7 Satisfiability and Consistency.- 10.2.8 Deduction Theorem.- 10.2.9 Compactness.- 10.2.10Examples.- 11 Uncertainty and Automated Reasoning.- 11.1 Uncertainty Reasoning Based on LP(X).- 11.2 Uncertainty Reasoning Based on Lvpl.- 11.2.1 Another Kind of Interpretation of X ? Y.- 11.2.2 Basic Theory.- 11.2.3 Examples.- 11.2.4 Multi-Dimensional and Multiple Uncertainty Reasoning.- Models and Methods.- Semantical Interpretation and Syntactical Proof.- 11.3 ?-Resolution Principle Based on LP(X).- 11.3.1 ?-Resolution Principle.- 11.3.2 Soundness and Completeness.- 11.4 ?-Resolution Principle Based on LF(X).- 11.4.1 Interpretation of Formulae.- 11.4.2 ?-Resolution Principle.- References.

  • a Resolution Principle based on first order lattice valued logic lf x
    Information Sciences, 2001
    Co-Authors: Da Ruan, Etienne Kerre, Jun Liu
    Abstract:

    Abstract In the present paper, as a continuous work about α-Resolution Principle based on lattice-valued propositional logic LP(X) (Information Sciences 130 (2000) 1–29) whose algebra of truth-values is a relatively general lattice – lattice implication algebra (LIA), the lattice-valued Resolution Principle for the corresponding first-order lattice-valued logic system LF(X) is focused. Firstly, some concepts about lattice-valued Resolution Principle for LF(X) are introduced and the Herbrand theorem for LF(X) is proved. Then, an α-Resolution Principle, which can be used to judge if a first-order lattice-valued logical formula in LF(X) is false at a truth-valued level α (i.e., α-false), is established. Finally, the completeness theorem of this α-Resolution Principle and the soundness theorem for the strong α-Resolution are also proved. It is hoped that the current work would serve as a foundation for constructing Resolution-based automated reasoning methods for lattice-valued logic capable of dealing with both comparable and incomparable uncertain information.

  • a Resolution Principle based on lattice valued propositional logic lp x
    Information Sciences, 2000
    Co-Authors: Da Ruan, Etienne Kerre, Jun Liu
    Abstract:

    Abstract In the present paper, Resolution-based automated reasoning theory in an L-type fuzzy logic is focused. Concretely, the α-Resolution Principle, which is based on lattice-valued propositional logic LP(X) with truth-value in a logical algebra – lattice implication algebra, is investigated. Finally, an α-Resolution Principle that can be used to judge if a lattice-valued logical formula in LP(X) is always false at a truth-valued level α (i.e., α-false), is established, and the theorems of both soundness and completeness of this α-Resolution Principle are also proved. This will become the theoretical foundation for automated reasoning based on lattice-valued logical LP(X).

Jun Liu - One of the best experts on this subject based on the ideXlab platform.

  • α minimal Resolution Principle for a lattice valued logic
    International Journal of Computational Intelligence Systems, 2015
    Co-Authors: Hairui Jia, Yi Liu, Jun Liu
    Abstract:

    Based on the academic ideas of Resolution-based automated reasoning and the previously established research work on binary α-Resolution based automated reasoning schemes in the framework of lattice...

  • general form of α Resolution Principle for linguistic truth valued lattice valued logic
    Soft Computing, 2012
    Co-Authors: Xiaomei Zhong, Jun Liu, Shuwei Chen
    Abstract:

    This paper is focused on Resolution-based automated reasoning theory in linguistic truth-valued lattice-valued logic based on linguistic truth-valued lattice implication algebra. Concretely, the general form of ?-Resolution Principle based on the above lattice-valued logic is equivalently transformed into another simpler lattice-valued logic system. Firstly, the general form of ?-Resolution Principle for lattice-valued propositional logic $$ ({\fancyscript{L}}_{n} \times {\fancyscript{L}}_{2}){\text{P(X)}} $$ is equivalently transformed into that for lattice-valued propositional logic $$ \fancyscript{L}_{n} $$ P(X). A similar conclusion is obtained between the general form of ?-Resolution Principle for linguistic truth-valued lattice-valued propositional logic $${\fancyscript{L}}_{V(n \times 2)}$$ P(X) and that for lattice-valued propositional logic $${\fancyscript{L}}_{Vn} $$ P(X). Secondly, the general form of ?-Resolution Principle for lattice-valued first-order logic $$ ({\fancyscript{L}}_{n} \times {\fancyscript{L}}_{2}) $$ F(X) is equivalently transformed into that for $${\fancyscript{L}}_{n} $$ P(X). Similarly, this conclusion also holds for linguistic truth-valued lattice-valued first-order $${\fancyscript{L}}_{V(n \times 2)} $$ F(X) and $${\fancyscript{L}}_{Vn} $$ P(X). The presented work provides a key theoretical support for automated reasoning approaches and algorithms in linguistic truth-valued logic, which can further support linguistic information processing for decision making, i.e., reasoning with words.

  • lattice valued logic an alternative approach to treat fuzziness and incomparability
    2003
    Co-Authors: Da Ruan, K Y Qin, Jun Liu
    Abstract:

    I Introduction.- 1 Introduction.- 1.1 Major Methodologies in Artificial Intelligence.- 1.2 Basic Academic Ideas.- 1.3 Some Related Concepts.- 1.4 Many-Valued Logic and Lattice-Valued Logic.- 1.5 Uncertainty Inference.- 1.5.1 Probability-Based Uncertainty Reasoning.- 1.5.2 Fuzzy Set Based Uncertainty Reasoning.- 1.5.3 Non-Monotonic Logic Based Uncertainty Reasoning.- 1.6 Automated Reasoning in Many-Valued Logic.- II Lattice Implication Algebras.- 2 Concepts and Properties.- 2.1 Lattice Implication Algebras.- 2.1.1 Concepts and Examples.- 2.1.2 Basic Properties.- 2.2 Lattice H Implication Algebras.- 2.3 Lattice Properties.- 2.4 Homomorphisms.- 3 Filters.- 3.1 Filters and Implicative Filters.- 3.2 Generated Filters.- 3.3 Positive Implicative Filters and Associative Filters.- 3.4 Prime Filters and Ultra-Filters.- 3.5 I-Filters, Involution Filters and Obstinate Filters.- 3.6 Fuzzy Filters.- 4 LI-Ideals.- 4.1 LI-Ideals.- 4.2 Fuzzy LI-Ideals.- 4.3 Normal Fuzzy LI-Ideals.- 4.4 Intuitionistic Fuzzy LI-Ideals.- 5 Homomorphisms and Representations.- 5.1 Congruence Relations.- 5.1.1 Congruence Relations Induced by Filters.- 5.1.2 Congruences Relations Induced by LI-ideals.- 5.1.3 Congruence Relations Induced by Fuzzy Filters.- 5.1.4 Congruence Relations Induced by Fuzzy LI-ideals.- 5.2 Proper Lattice Implication Algebras.- 5.3 Representations.- 6 Topological Structure of Filter Spaces.- 6.1 Filter Spaces.- 6.1.1 Basic Concepts.- 6.1.2 Topological Properties.- 6.2 Product Topology and Quotient Topology.- 6.3 Lattice Topology.- 6.4 Prime Spaces.- 7 Connections with Related Algebras.- 7.1 Lattice Implication Algebras and BCK-Algebras.- 7.2 Lattice Implication Algebras and MV-Algebras.- 7.3 Lattice Implication Algebras and Related Algebras.- 8 Related Issues.- 8.1 Category of Lattice Implication Algebras.- 8.2 Category of Fuzzy Lattice Implication Algebras.- 8.3 Fuzzy Power Sets.- 8.4 Adjoint Semigroups.- 8.5 Logical Properties.- III Lattice-Valued Logic Systems.- 9 Lattice-Valued Propositional Logics.- 9.1 Lattice-Valued Propositional Logic LP(X).- 9.1.1 Language.- 9.1.2 Semantics.- 9.1.3 Syntax.- 9.1.4 Examples.- 9.2 Gradational Lattice-Valued Propositional Logic Lvpl.- 9.2.1 Language.- 9.2.2 Rules of Inference.- 9.2.3 Semantics.- 9.2.4 Syntax.- 9.2.5 Satisfiability and Consistency.- 9.2.6 Deduction Theorem.- 9.2.7 Compactness.- 9.2.8 Examples.- 10 Lattice-Valued First-Order Logics.- 10.1 Lattice-Valued First-Order Logic LF(X).- 10.1.1 Language.- 10.1.2 Interpretation.- 10.1.3 Semantics.- 10.1.4 Syntax.- 10.1.5 Properties of Model Theory.- 10.2 Gradational Lattice-Valued First-Order Logic Lvfl.- 10.2.1 Language.- 10.2.2 Interpretation.- 10.2.3 Semantics.- 10.2.4 Standardization of Formulae.- 10.2.5 Syntax.- 10.2.6 Soundness and Completeness.- 10.2.7 Satisfiability and Consistency.- 10.2.8 Deduction Theorem.- 10.2.9 Compactness.- 10.2.10Examples.- 11 Uncertainty and Automated Reasoning.- 11.1 Uncertainty Reasoning Based on LP(X).- 11.2 Uncertainty Reasoning Based on Lvpl.- 11.2.1 Another Kind of Interpretation of X ? Y.- 11.2.2 Basic Theory.- 11.2.3 Examples.- 11.2.4 Multi-Dimensional and Multiple Uncertainty Reasoning.- Models and Methods.- Semantical Interpretation and Syntactical Proof.- 11.3 ?-Resolution Principle Based on LP(X).- 11.3.1 ?-Resolution Principle.- 11.3.2 Soundness and Completeness.- 11.4 ?-Resolution Principle Based on LF(X).- 11.4.1 Interpretation of Formulae.- 11.4.2 ?-Resolution Principle.- References.

  • a Resolution Principle based on first order lattice valued logic lf x
    Information Sciences, 2001
    Co-Authors: Da Ruan, Etienne Kerre, Jun Liu
    Abstract:

    Abstract In the present paper, as a continuous work about α-Resolution Principle based on lattice-valued propositional logic LP(X) (Information Sciences 130 (2000) 1–29) whose algebra of truth-values is a relatively general lattice – lattice implication algebra (LIA), the lattice-valued Resolution Principle for the corresponding first-order lattice-valued logic system LF(X) is focused. Firstly, some concepts about lattice-valued Resolution Principle for LF(X) are introduced and the Herbrand theorem for LF(X) is proved. Then, an α-Resolution Principle, which can be used to judge if a first-order lattice-valued logical formula in LF(X) is false at a truth-valued level α (i.e., α-false), is established. Finally, the completeness theorem of this α-Resolution Principle and the soundness theorem for the strong α-Resolution are also proved. It is hoped that the current work would serve as a foundation for constructing Resolution-based automated reasoning methods for lattice-valued logic capable of dealing with both comparable and incomparable uncertain information.

  • a Resolution Principle based on lattice valued propositional logic lp x
    Information Sciences, 2000
    Co-Authors: Da Ruan, Etienne Kerre, Jun Liu
    Abstract:

    Abstract In the present paper, Resolution-based automated reasoning theory in an L-type fuzzy logic is focused. Concretely, the α-Resolution Principle, which is based on lattice-valued propositional logic LP(X) with truth-value in a logical algebra – lattice implication algebra, is investigated. Finally, an α-Resolution Principle that can be used to judge if a lattice-valued logical formula in LP(X) is always false at a truth-valued level α (i.e., α-false), is established, and the theorems of both soundness and completeness of this α-Resolution Principle are also proved. This will become the theoretical foundation for automated reasoning based on lattice-valued logical LP(X).

Xiaomei Zhong - One of the best experts on this subject based on the ideXlab platform.

  • Multiary α-Resolution Principle for a Lattice-Valued Logic
    IEEE Transactions on Fuzzy Systems, 2013
    Co-Authors: Yang Xu, Xiaomei Zhong, Shuwei Chen
    Abstract:

    This paper focuses on Resolution-based automated reasoning theory in a lattice-valued logic system with truth values that are defined in a lattice-valued logical algebraic structure-lattice implication algebras (LIAs) - which essentially aims to extend the classical logic to handle automated deduction under an uncertain environment. Concretely, we investigate a generalization of the known conjunctive normal form (CNF) in the classical logic representation that we call the generalized conjunctive normal form (GCNF), which aims to characterize the constants and implication connectives that are essentially different from the ones in classical logic. We then extend the established Resolution Principle at a certain truth-value level α (called α-Resolution) in this lattice-valued logic to a more general form, i.e., from binary α -Resolution to multiary α-Resolution. The extension to multiary α-Resolution starts from lattice-valued propositional logic LP(X), while its theorems of both soundness and completeness are proved. Multiary α-Resolution Principle is then further established in the corresponding lattice-valued first-order logic LF(X), along with its soundness theorem, lifting lemma, and completeness theorem. Meanwhile, an important result that multiary α-Resolution Principle in LF(X) can be equivalently transformed into that in LP(X) to some extent is obtained. All these works will theoretically support the establishment of an automated reasoning algorithm and its implementation with further applications into automated deduction and decision-making problems under uncertainty.

  • general form of α Resolution Principle for linguistic truth valued lattice valued logic
    Soft Computing, 2012
    Co-Authors: Xiaomei Zhong, Jun Liu, Shuwei Chen
    Abstract:

    This paper is focused on Resolution-based automated reasoning theory in linguistic truth-valued lattice-valued logic based on linguistic truth-valued lattice implication algebra. Concretely, the general form of ?-Resolution Principle based on the above lattice-valued logic is equivalently transformed into another simpler lattice-valued logic system. Firstly, the general form of ?-Resolution Principle for lattice-valued propositional logic $$ ({\fancyscript{L}}_{n} \times {\fancyscript{L}}_{2}){\text{P(X)}} $$ is equivalently transformed into that for lattice-valued propositional logic $$ \fancyscript{L}_{n} $$ P(X). A similar conclusion is obtained between the general form of ?-Resolution Principle for linguistic truth-valued lattice-valued propositional logic $${\fancyscript{L}}_{V(n \times 2)}$$ P(X) and that for lattice-valued propositional logic $${\fancyscript{L}}_{Vn} $$ P(X). Secondly, the general form of ?-Resolution Principle for lattice-valued first-order logic $$ ({\fancyscript{L}}_{n} \times {\fancyscript{L}}_{2}) $$ F(X) is equivalently transformed into that for $${\fancyscript{L}}_{n} $$ P(X). Similarly, this conclusion also holds for linguistic truth-valued lattice-valued first-order $${\fancyscript{L}}_{V(n \times 2)} $$ F(X) and $${\fancyscript{L}}_{Vn} $$ P(X). The presented work provides a key theoretical support for automated reasoning approaches and algorithms in linguistic truth-valued logic, which can further support linguistic information processing for decision making, i.e., reasoning with words.

  • ideal based Resolution Principle for lattice valued propositional logic lp x
    IEEE International Conference on Intelligent Systems and Knowledge Engineering, 2009
    Co-Authors: Wenhong Deng, Xiaomei Zhong
    Abstract:

    In the present paper, an ideal-based Resolution Principle for the lattice-valued propositional logic LP(X) based on lattice implication algebra is focused A LI-ideal of lattice implication algebra is taken as the criterion for measuring the unsatisfiability of a lattice-valued logical formula The ideal-based Resolution Principle for lattice-valued propositional logic LP(X) is established The soundness and weak completeness theorems of A Resolution Principle based on LP(X) are established Finally, the properties of A Resolution are discussed

Antonio Benedetto - One of the best experts on this subject based on the ideXlab platform.

Etienne Kerre - One of the best experts on this subject based on the ideXlab platform.

  • a Resolution Principle based on first order lattice valued logic lf x
    Information Sciences, 2001
    Co-Authors: Da Ruan, Etienne Kerre, Jun Liu
    Abstract:

    Abstract In the present paper, as a continuous work about α-Resolution Principle based on lattice-valued propositional logic LP(X) (Information Sciences 130 (2000) 1–29) whose algebra of truth-values is a relatively general lattice – lattice implication algebra (LIA), the lattice-valued Resolution Principle for the corresponding first-order lattice-valued logic system LF(X) is focused. Firstly, some concepts about lattice-valued Resolution Principle for LF(X) are introduced and the Herbrand theorem for LF(X) is proved. Then, an α-Resolution Principle, which can be used to judge if a first-order lattice-valued logical formula in LF(X) is false at a truth-valued level α (i.e., α-false), is established. Finally, the completeness theorem of this α-Resolution Principle and the soundness theorem for the strong α-Resolution are also proved. It is hoped that the current work would serve as a foundation for constructing Resolution-based automated reasoning methods for lattice-valued logic capable of dealing with both comparable and incomparable uncertain information.

  • a Resolution Principle based on lattice valued propositional logic lp x
    Information Sciences, 2000
    Co-Authors: Da Ruan, Etienne Kerre, Jun Liu
    Abstract:

    Abstract In the present paper, Resolution-based automated reasoning theory in an L-type fuzzy logic is focused. Concretely, the α-Resolution Principle, which is based on lattice-valued propositional logic LP(X) with truth-value in a logical algebra – lattice implication algebra, is investigated. Finally, an α-Resolution Principle that can be used to judge if a lattice-valued logical formula in LP(X) is always false at a truth-valued level α (i.e., α-false), is established, and the theorems of both soundness and completeness of this α-Resolution Principle are also proved. This will become the theoretical foundation for automated reasoning based on lattice-valued logical LP(X).