A practical inference method with several implicative gradual rules and a fuzzy input: one and two dimensions

International audienceA general approach to practical inference with gradual implicative rules and fuzzy inputs is presented. Gradual rules represent constraints restricting outputs of a fuzzy system for each input. They are tailored for interpolative reasoning. Our approach to inference relies on the use of inferential independence. It is based on fuzzy output computation under an interval-valued input. A double decomposition of fuzzy inputs is done in terms of alpha-cuts and in terms of a partitioning of these cuts according to areas where only a few rules apply. The case of one and two dimensional inputs is consideredCet article présente une méthode d'inférence avec des règles implicatives graduelles pour une entrée floue. Les règles graduelles représentent des contraintes qui restreignent l'univers de sortie pour chacune des entrées. Elles sont conçues pour réaliser des interpolations. L'algorithme que nous proposons s'appuie sur le principe de indépendance inférentielle. Il met en oeuvre une double décomposition de l'ensemble flou d'entrée, par alpha-coupes et suivant le partitionnement de l'univers des variables d'entrée. Les cas étudiés correspondent à des systèmes à une et deux dimension

Uncertainty in the conjunctive approach to fuzzy inference

Fuzzy inference using the conjunctive approach is very popular in many practical applications. It is intuitive for engineers, simple to understand, and characterized by the lowest computational complexity. However, it leads to incorrect results in the cases when the relationship between a fact and a premise is undefined. This article analyses the problem thoroughly and provides several possible solutions. The drawbacks of uncertainty in the conjunctive approach are presented using fuzzy inference based on a fuzzy truth value, first introduced by Baldwin (1979c). The theory of inference is completed with a new truth function named 0-undefined for two-valued logic, which is further generalized into fuzzy logic as α-undefined. Eventually, the proposed modifications allow altering existing implementations of conjunctive fuzzy systems to interpret the undefined state, giving adequate results

On modal expansions of t-norm based logics with rational constants

[eng] According to Zadeh, the term “fuzzy logic” has two different meanings: wide and narrow. In a narrow sense it is a logical system which aims a formalization of approximate reasoning, and so it can be considered an extension of many-valued logic. However, Zadeh also says that the agenda of fuzzy logic is quite different from that of traditional many-valued logic, as it addresses concepts like linguistic variable, fuzzy if-then rule, linguistic quantifiers etc. Hájek, in the preface of his foundational book Metamathematics of Fuzzy Logic, agrees with Zadeh’s distinction, but stressing that formal calculi of many-valued logics are the kernel of the so-called Basic Fuzzy logic (BL), having continuous triangular norms (t-norm) and their residua as semantics for the conjunction and implication respectively, and of its most prominent extensions, namely Lukasiewicz, Gödel and Product fuzzy logics. Taking advantage of the fact that a t-norm has residuum if, and only if, it is left-continuous, the logic of the left-continuous t-norms, called MTL, was soon after introduced. On the other hand, classical modal logic is an active field of mathematical logic, originally introduced at the beginning of the XXth century for philosophical purposes, that more recently has shown to be very successful in many other areas, specially in computer science. That are the most well-known semantics for classical modal logics. Modal expansions of non-classical logics, in particular of many-valued logics, have also been studied in the literature. In this thesis we focus on the study of some modal logics over MTL, using natural generalizations of the classical Kripke relational structures where propositions at possible words can be many-valued, but keeping classical accessibility relations. In more detail, the main goal of this thesis has been to study modal expansions of the logic of a left-continuous t-norm, defined over the language of MTL expanded with rational truth-constants and the Monteiro-Baaz Delta-operator, whose intended (standard) semantics is given by Kripke models with crisp accessibility relations and taking the unit real interval [0, 1] as set of truth-values. To get complete axiomatizations, already known techniques based on the canonical model construction are uses, but this requires to ensure that the underlying (propositional) fuzzy logic is strongly standard complete. This constraint leads us to consider axiomatic systems with infinitary inference rules, already at the propositional level. A second goal of the thesis has been to also develop and automated reasoning software tool to solve satisfiability and logical consequence problems for some of the fuzzy logic modal logics considered. This dissertation is structured in four parts. After a gentle introduction, Part I contains the needed preliminaries for the thesis be as self-contained as possible. Most of the theoretical results are developed in Parts II and III. Part II focuses on solving some problems concerning the strong standard completeness of underlying non-modal expansions. We first present and axiomatic system for the non-nodal propositional logic of a left-continuous t-norm who makes use of a unique infinitary inference rule, the “density rule”, that solves several problems pointed out in the literature. We further expand this axiomatic system in order to also characterize arbitrary operations over [0, 1] satisfying certain regularity conditions. However, since this axiomatic system turn out to be not well-behaved for the modal expansion, we search for alternative axiomatizations with some particular kind of inference rules (that will be called conjunctive). Unfortunately, this kind of axiomatization does not necessarily exist for all left-continuous t-norms (in particular, it does not exist for the Gödel logic case), but we identify a wide class of t-norms for which it works. This “well-behaved” t-norms include all ordinal sums of Lukasiewiczand Product t-norms. Part III focuses on the modal expansion of the logics presented before. We propose axiomatic systems (which are, as expected, modal expansions of the ones given in the previous part) respectively strongly complete with respect to local and global Kripke semantics defined over frames with crisp accessibility relations and worlds evaluated over a “well-behaved” left-continuous t-norm. We also study some properties and extensions of these logics and also show how to use it for axiomatizing the possibilistic logic over the very same t-norm. Later on, we characterize the algebraic companion of these modal logics, provide some algebraic completeness results and study the relation between their Kripke and algebraic semantics. Finally, Part IV of the thesis is devoted to a software application, mNiB-LoS, who uses Satisfability Modulo Theories in order to build an automated reasoning system to reason over modal logics evaluated over BL algebras. The acronym of this applications stands for a modal Nice BL-logics Solver. The use of BL logics along this part is motivated by the fact that continuous t-norms can be represented as ordinal sums of three particular t-norms: Gödel, Lukasiewicz and Product ones. It is then possible to show that these t-norms have alternative characterizations that, although equivalent from the point of view of the logic, have strong differences for what concerns the design, implementation and efficiency of the application. For practical reasons, the modal structures included in the solver are limited to the finite ones (with no bound on the cardinality)

Knowledge-based approach to risk analysis in the customs domain

The aim of this PhD project is to develop a fuzzy knowledge-based approach in support of risk analysis in the Customs domain. Focusing upon risk management and risk analysis in the Customs domain, this thesis explores the relationship of risk with uncertainty, fuzziness, vagueness, and imprecise knowledge and it analyses state of the art detection techniques for fraud and risk. Special focus is given to fuzzy logic, ontological engineering, and semantic modelling considering aspects such as the importance of human knowledge and semantic knowledge in the context of risk analysis for the Customs domain. An approach is presented combining the fuzzy modelling and reasoning with semantic modelling and ontologies. Fuzzy modelling and reasoning is explored in the context of risk analysis and detection in order to examine approximate human reasoning based on human knowledge. Ontologies and semantic modelling are explored as an approach to represent domain knowledge and concepts. The purpose is to enable easier communication and understanding as well as interoperability. Risk management is broader, multi-dimensional process involving a number of task, activities, and practises. The presented approach is focused on examining the analysis and detection of the risk, based on the outputs of the risk management process with the use of ontologies and fuzzy rule-based reasoning. An ontological architecture is developed in the context of the presented approach. It is considered that such architecture is possible to enable modularity, maintainability, re-usability, and extensibility and can also be extended or integrated with other ontologies. In addition, examples are discussed to illustrate representation of concepts at various levels (generic or specific) and the modelling of various semantics. Furthermore, fuzzy modelling and reasoning are investigated. This investigation consists of literature research and the use of a generic research prototype (examination of Mamdani and Sugeno model types). From theoretical research, fuzzy logic enables the expression of human knowledge with linguistic terms and it could simulate human reasoning in the context of risk analysis and detection. In addition, Hierarchical Fuzzy Systems (HFS) or Hybrid Hierarchical Fuzzy Controllers (HHFC) approaches can be used to manage complexity especially for complex domains. Linguistic fuzzy modelling (LFM) is an aspect that should be considered during fuzzy modelling. From the generic research prototype, fuzzy modelling with the use of ontologies is demonstrated together with their integration in the context of fuzzy rule-based reasoning. It is also considered that Mamdani type of fuzzy models is easier to express human knowledge since the output can be expressed with linguistic terms. However, Sugeno type of fuzzy model could be used from adaptive techniques for optimisation purposes

Multispace & Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of Sciences), Vol. IV

The fourth volume, in my book series of “Collected Papers”, includes 100 published and unpublished articles, notes, (preliminary) drafts containing just ideas to be further investigated, scientific souvenirs, scientific blogs, project proposals, small experiments, solved and unsolved problems and conjectures, updated or alternative versions of previous papers, short or long humanistic essays, letters to the editors - all collected in the previous three decades (1980-2010) – but most of them are from the last decade (2000-2010), some of them being lost and found, yet others are extended, diversified, improved versions. This is an eclectic tome of 800 pages with papers in various fields of sciences, alphabetically listed, such as: astronomy, biology, calculus, chemistry, computer programming codification, economics and business and politics, education and administration, game theory, geometry, graph theory, information fusion, neutrosophic logic and set, non-Euclidean geometry, number theory, paradoxes, philosophy of science, psychology, quantum physics, scientific research methods, and statistics. It was my preoccupation and collaboration as author, co-author, translator, or cotranslator, and editor with many scientists from around the world for long time. Many topics from this book are incipient and need to be expanded in future explorations

A predicated network formalism for commonsense reasoning.

Chiu, Yiu Man Edmund.Thesis submitted in: December 1999.Thesis (M.Phil.)--Chinese University of Hong Kong, 2000.Includes bibliographical references (leaves 269-248).Abstracts in English and Chinese.Abstract --- p.iAcknowledgments --- p.iiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- The Beginning Story --- p.2Chapter 1.2 --- Background --- p.3Chapter 1.2.1 --- History of Nonmonotonic Reasoning --- p.3Chapter 1.2.2 --- Formalizations of Nonmonotonic Reasoning --- p.6Chapter 1.2.3 --- Belief Revision --- p.13Chapter 1.2.4 --- Network Representation of Knowledge --- p.17Chapter 1.2.5 --- Reference from Logic Programming --- p.21Chapter 1.2.6 --- Recent Work on Network-type Automatic Reasoning Sys- tems --- p.22Chapter 1.3 --- A Novel Inference Network Approach --- p.23Chapter 1.4 --- Objectives --- p.23Chapter 1.5 --- Organization of the Thesis --- p.24Chapter 2 --- The Predicate Inference Network PIN --- p.25Chapter 2.1 --- Preliminary Terms --- p.26Chapter 2.2 --- Overall Structure --- p.27Chapter 2.3 --- Object Layer --- p.29Chapter 2.3.1 --- Virtual Object --- p.31Chapter 2.4 --- Predicate Layer --- p.33Chapter 2.4.1 --- Node Values --- p.34Chapter 2.4.2 --- Information Source --- p.35Chapter 2.4.3 --- Belief State --- p.36Chapter 2.4.4 --- Predicates --- p.37Chapter 2.4.5 --- Prototypical Predicates --- p.37Chapter 2.4.6 --- Multiple Inputs for a Single Belief --- p.39Chapter 2.4.7 --- External Program Call --- p.39Chapter 2.5 --- Variable Layer --- p.40Chapter 2.6 --- Inter-Layer Links --- p.42Chapter 2.7 --- Chapter Summary --- p.43Chapter 3 --- Computation for PIN --- p.44Chapter 3.1 --- Computation Functions for Propagation --- p.45Chapter 3.1.1 --- Computational Functions for Combinative Links --- p.45Chapter 3.1.2 --- Computational Functions for Alternative Links --- p.49Chapter 3.2 --- Applying the Computation Functions --- p.52Chapter 3.3 --- Relations Represented in PIN --- p.55Chapter 3.3.1 --- Relations Represented by Combinative Links --- p.56Chapter 3.3.2 --- Relations Represented by Alternative Links --- p.59Chapter 3.4 --- Chapter Summary --- p.61Chapter 4 --- Dynamic Knowledge Update --- p.62Chapter 4.1 --- Operations for Knowledge Update --- p.63Chapter 4.2 --- Logical Expression --- p.63Chapter 4.3 --- Applicability of Operators --- p.64Chapter 4.4 --- Add Operation --- p.65Chapter 4.4.1 --- Add a fully instantiated single predicate proposition with no virtual object --- p.66Chapter 4.4.2 --- Add a fully instantiated pure disjunction --- p.68Chapter 4.4.3 --- Add a fully instantiated expression which is a conjunction --- p.71Chapter 4.4.4 --- Add a human biased relation --- p.74Chapter 4.4.5 --- Add a single predicate expression with virtual objects --- p.76Chapter 4.4.6 --- Add a IF-THEN rule --- p.80Chapter 4.5 --- Remove Operation --- p.88Chapter 4.5.1 --- Remove a Belief --- p.88Chapter 4.5.2 --- Remove a Rule --- p.91Chapter 4.6 --- Revise Operation --- p.94Chapter 4.6.1 --- Revise a Belief --- p.94Chapter 4.6.2 --- Revise a Rule --- p.96Chapter 4.7 --- Consistency Maintenance --- p.97Chapter 4.7.1 --- Logical Suppression --- p.98Chapter 4.7.2 --- Example on Handling Inconsistent Information --- p.99Chapter 4.8 --- Chapter Summary --- p.102Chapter 5 --- Knowledge Query --- p.103Chapter 5.1 --- Domains of Quantification --- p.104Chapter 5.2 --- Reasoning through Recursive Rules --- p.109Chapter 5.2.1 --- Infinite Looping Control --- p.110Chapter 5.2.2 --- Proof of the finite termination of recursive rules --- p.111Chapter 5.3 --- Query Functions --- p.117Chapter 5.4 --- Type I Queries --- p.119Chapter 5.4.1 --- Querying a Simple Single Predicate Proposition (Type I) --- p.122Chapter 5.4.2 --- Querying a Belief with Logical Connective(s) (Type I) --- p.128Chapter 5.5 --- Type II Queries --- p.132Chapter 5.5.1 --- Querying Single Predicate Expressions (Type II) --- p.134Chapter 5.5.2 --- Querying an Expression with Logical Connectives (Type II) --- p.143Chapter 5.6 --- Querying an Expression with Virtual Objects --- p.152Chapter 5.6.1 --- Type I Queries Involving Virtual Object --- p.152Chapter 5.6.2 --- Type II Queries involving Virtual Objects --- p.156Chapter 5.7 --- Chapter Summary --- p.157Chapter 6 --- Uniqueness and Finite Termination --- p.159Chapter 6.1 --- Proof Structure --- p.160Chapter 6.2 --- Proof for Completeness and Finite Termination of Domain Search- ing Procedure --- p.161Chapter 6.3 --- Proofs for Type I Queries --- p.167Chapter 6.3.1 --- Proof for Single Predicate Expressions --- p.167Chapter 6.3.2 --- Proof of Type I Queries on Expressions with Logical Con- nectives --- p.172Chapter 6.3.3 --- General Proof for Type I Queries --- p.174Chapter 6.4 --- Proofs for Type II Queries --- p.175Chapter 6.4.1 --- Proof for Type II Queries on Single Predicate Expressions --- p.176Chapter 6.4.2 --- Proof for Type II Queries on Disjunctions --- p.178Chapter 6.4.3 --- Proof for Type II Queries on Conjunctions --- p.179Chapter 6.4.4 --- General Proof for Type II Queries --- p.181Chapter 6.5 --- Proof for Queries Involving Virtual Objects --- p.182Chapter 6.6 --- Uniqueness and Finite Termination of PIN Queries --- p.183Chapter 6.7 --- Chapter Summary --- p.184Chapter 7 --- Lifschitz's Benchmark Problems --- p.185Chapter 7.1 --- Structure --- p.186Chapter 7.2 --- Default Reasoning --- p.186Chapter 7.2.1 --- Basic Default Reasoning --- p.186Chapter 7.2.2 --- Default Reasoning with Irrelevant Information --- p.187Chapter 7.2.3 --- Default Reasoning with Several Defaults --- p.188Chapter 7.2.4 --- Default Reasoning with a Disabled Default --- p.190Chapter 7.2.5 --- Default Reasoning in Open Domain --- p.191Chapter 7.2.6 --- Reasoning about Unknown Exceptions I --- p.193Chapter 7.2.7 --- Reasoning about Unknown Exceptions II --- p.194Chapter 7.2.8 --- Reasoning about Unknown Exceptions III --- p.196Chapter 7.2.9 --- Priorities between Defaults --- p.198Chapter 7.2.10 --- Priorities between Instances of a Default --- p.199Chapter 7.2.11 --- Reasoning about Priorities --- p.199Chapter 7.3 --- Inheritance --- p.200Chapter 7.3.1 --- Linear Inheritance --- p.200Chapter 7.3.2 --- Tree-Structured Inheritance --- p.202Chapter 7.3.3 --- One-Step Multiple Inheritance --- p.203Chapter 7.3.4 --- Multiple Inheritance --- p.204Chapter 7.4 --- Uniqueness of Names --- p.205Chapter 7.4.1 --- Unique Names Hypothesis for Objects --- p.205Chapter 7.4.2 --- Unique Names Hypothesis for Functions --- p.206Chapter 7.5 --- Reasoning about Action --- p.206Chapter 7.6 --- Autoepistemic Reasoning --- p.206Chapter 7.6.1 --- Basic Autoepistemic Reasoning --- p.206Chapter 7.6.2 --- Autoepistemic Reasoning with Incomplete Information --- p.207Chapter 7.6.3 --- Autoepistemic Reasoning with Open Domain --- p.207Chapter 7.6.4 --- Autoepistemic Default Reasoning --- p.208Chapter 8 --- Comparison with PROLOG --- p.214Chapter 8.1 --- Introduction of PROLOG --- p.215Chapter 8.1.1 --- Brief History --- p.215Chapter 8.1.2 --- Structure and Inference --- p.215Chapter 8.1.3 --- Why Compare PIN with Prolog --- p.216Chapter 8.2 --- Representation Power --- p.216Chapter 8.2.1 --- Close World Assumption and Negation as Failure --- p.216Chapter 8.2.2 --- Horn Clauses --- p.217Chapter 8.2.3 --- Quantification --- p.218Chapter 8.2.4 --- Build-in Functions --- p.219Chapter 8.2.5 --- Other Representation Issues --- p.220Chapter 8.3 --- Inference and Query Processing --- p.220Chapter 8.3.1 --- Unification --- p.221Chapter 8.3.2 --- Resolution --- p.222Chapter 8.3.3 --- Computation Efficiency --- p.225Chapter 8.4 --- Knowledge Updating and Consistency Issues --- p.227Chapter 8.4.1 --- PIN and AGM Logic --- p.228Chapter 8.4.2 --- Knowledge Merging --- p.229Chapter 8.5 --- Chapter Summary --- p.229Chapter 9 --- Conclusion and Discussion --- p.230Chapter 9.1 --- Conclusion --- p.231Chapter 9.1.1 --- General Structure --- p.231Chapter 9.1.2 --- Representation Power --- p.231Chapter 9.1.3 --- Inference --- p.232Chapter 9.1.4 --- Dynamic Update and Consistency --- p.233Chapter 9.1.5 --- Soundness and Completeness Versus Efficiency --- p.233Chapter 9.2 --- Discussion --- p.234Chapter 9.2.1 --- Different Selection Criteria --- p.234Chapter 9.2.2 --- Link Order --- p.235Chapter 9.2.3 --- Inheritance Reasoning --- p.236Chapter 9.3 --- Future Work --- p.237Chapter 9.3.1 --- Implementation --- p.237Chapter 9.3.2 --- Application --- p.237Chapter 9.3.3 --- Probabilistic and Fuzzy PIN --- p.238Chapter 9.3.4 --- Temporal Reasoning --- p.238Bibliography --- p.23