Tableau method for residuated logic

Residuated logic is a generalization of intuitionistic logic, which does not assume the idempotence of the conjunction operator. Such generalized conjunction operators have proved important in expert systems (in the area of Approximate Reasoning) and in some areas of Theoretical Computer Science. Here we generalize the intuitionistic tableau procedure and prove that this generalized tableau method is sound for the semantics (the class of residuated algebras) of residuated propositional calculus (RPC). Since the axioms of RPC are complete for the semantics we may conclude that whenever a formula 0 is tableau provable, it is deducible in RPC. We present two different approaches for constructing residuated algebras which give us countermodels for some formulas φ which are not tableau provable. The first uses the fact that the theory of residuated algebras is equational, to construct quotients of free algebras. The second uses finite algebras. We end by discussing a number of open questions.     

Feasible algorithms for approximate reasoning using fuzzy logic

Deductive reasoning with classical logic is hampered when imprecision is present in the variables, although human reasoning can cope quite adequately with vague concepts. A new approach to reasoning which allows imprecise conclusions to be drawn consistently from imprecise premises was introduced by Baldwin [2]. This method is economical in calculation as it avoids the high dimensionality that fuzzy set representations often involve.This paper briefly reviews the method from an operational viewpoint, isolating the individual processes that are used in the method. A feasible algorithm for computing each process is then presented.It is assumed that the reader is familiar with the concept of, and operations on, fuzzy sets introduced by Zadeh [14].     

Array approach to fuzzy logic

Promising results from applying an array-based approach to two-valued logic suggests its application to fuzzy logic. The idea is to limit the domain of truth-values to a discrete, finite domain, such that a logical relationship can be evaluated by an exhaustive test of all possible combinations of truth-values. The paper presents a study of the topic from an engineer's viewpoint. As an example 31 logical sentences valid in two-valued logic were tested in three-valued logic using the nested interactive array language, Nial. Out of these, 24 turned out to be valid in a three-valued extension based on the well-known S^* implication operator, also called “Gödel's implication operator”. Applications to automated approximate reasoning and fuzzy control are also illustrated.     

A parametric representation of linguistic hedges in Zadeh’s fuzzy logic

This paper proposes a model for the parametric representation of linguistic hedges in Zadeh’s fuzzy logic. In this model each linguistic truth-value, which is generated from a primary term of the linguistic truth variable, is identified by a real number r depending on the primary term. It is shown that the model yields a method of efficiently computing linguistic truth expressions accompanied with a rich algebraic structure of the linguistic truth domain, namely De Morgan algebra. Also, a fuzzy logic based on the parametric representation of linguistic truth-values is introduced.     

Fuzzy Horn logic II

The paper studies closure properties of classes of fuzzy structures defined by fuzzy implicational theories, i.e. theories whose formulas are implications between fuzzy identities. We present generalizations of results from the bivalent case. Namely, we characterize model classes of general implicational theories, finitary implicational theories, and Horn theories by means of closedness under suitable algebraic constructions.     

Fuzzy logic in measurements

Our main interest in this paper is to translate from “natural language” into “system theoretical language”. This is of course important since a statement in system theory can be analyzed mathematically or computationally. We assume that, in order to obtain a good translation, “system theoretical language” should have great power of expression. Thus we first propose a new frame of system theory, which includes the concepts of “measurement” as well as “state equation”. And we show that a certain statement in usual conversation, i.e., fuzzy modus ponens with the word “very”, can be translated into a statement in the new frame of system theory. Though our result is merely one example of the translation from “natural language” into “system theoretical language”, we believe that our method is fairly general.     

The case for an interval-based representation of linguistic truth

This paper develops an interval-based approach to the concept of linguistic truth. A special-purpose interval logic is defined, and it is argued that, for many applications, this logic provides a potentially useful alternative to the conventional fuzzy logic.The key idea is to interpret the numerical truth value v(p) of a proposition p as a degree of belief in the logical certainty of p, in which case p is regarded as true, for example, if v(p) falls within a certain range, say, the interval [0.7, 1]. This leads to a logic which, although being only a special case of fuzzy logic, appears to be no less linguistically correct and at the same time offers definite advantages in terms of mathematical simplicity and computational speed.It is also shown that this same interval logic can be generalized to a lattice-based logic having the capacity to accommodate propositions p which employ fuzzy predicates of type 2.     

Selection of parameters for a fuzzy logic controller

The fuzzy logic controller is reviewed and its parameter are explicitly identified. The problem of initial selection and subsequent adjustment of the parameters are discussed in detail by example.     

A combined genetic algorithm-fuzzy logic controller (GA–FLC) in nonlinear programming

This paper presents a combined genetic algorithm-fuzzy logic controller (GA–FLC) technique for constrained nonlinear programming problems. In the standard Genetic algorithms, the upper and lower limits of the search regions should be given by the decision maker in advance to the optimization process. In general a needlessly large search region is used in fear of missing the global optimum outside the search region. Therefore, if the search region is able to adapt toward a promising area during the optimization process, the performance of GA will be enhanced greatly. Thus in this work we tried to investigate the influence of the bounding intervals on the final result. The proposed algorithm is made of classical GA coupled with FLC. This controller monitors the variation of the decision variables during process of the algorithm and modifies the boundary intervals to restart the next round of the algorithm. These characteristics make this approach well suited for finding optimal solutions to the highly NLP problems. Compared to previous works on NLP, our method proved to be more efficient in computation time and accuracy of the final solution.     

Applications of fuzzy logic in the control of robotic manipulators

Fuzzy logic has been utilized at several hierarchical levels of a typical robotic control system. Four broad levels of application may be identified - task design, system monitoring (including self-tuning and self-organization), information filtering and preprocessing, and in-loop direct control. Even though the need for fuzzy logic is felt mostly at upper levels of the control system, the present applications are mainly concentrated within the lowest level, perhaps driven by convenience rather than necessity. This paper surveys several applications of fuzzy logic in the control of robotic manipulators. Applications are grouped into four hierarchical categories, broadly corresponding to an existing architecture of a robotic control system. Such a classification can be beneficial in ascertaining the appropriateness of fuzzy logic for the specific control task.     

Local computation in linear logic

This work deals with the exponential fragment of Girard's linear logic ([3]) without the contraction rule, a logical system which has a natural relation with the direct logic ([10], [7]). A new sequent calculus for this logic is presented in order to remove the weakening rule and recover its behavior via a special treatment of the propositional constants, so that the process of cut-elimination can be performed using only “local” reductions. Hence a typed calculus, which admits only local rewriting rules, can be introduced in a natural manner. Its main properties — normalizability and confluence — has been investigated; moreover this calculus has been proved to satisfy a Curry-Howard isomorphism ([6]) with respect to the logical system in question. MSC: 03B40, 03F05.     

A computer program for fuzzy reasoning

An interactive computer program is described which implements the procedure proposed in “A Formal System for Fuzzy Reasoning” [1]. The problem in question is that of deciding what conclusions may be drawn in the presence of (posibly conflicting) evidence provided, generally with associated partial degrees of belief, by several sources of differing reliability. In using the program, each piece of evidence is entered as a sentence (using the terms NOT, AND, OR, IMPLIES as necessary), with an associated ‘degree of belief’ and ‘weight’; followed by a tentative conclusion. The system returns the degree(s) of belief and weight(s) which may rationally be attached to the conclusion. Copies of the program, written in FORTRAN IV (870 lines) have been lodged with the program libraries CUBE, DECUS, and SHARE, or may be obtained by writing to the author.     

Advances in integrative statistics for logic programming

We present recent developments on the syntax of Real, a library for interfacing two Prolog systems to the statistical language R. We focus on the changes in Prolog syntax within SWI-Prolog that accommodate greater syntactic integration, enhanced user experience and improved features for web-services. We recount the full syntax and functionality of Real as well as presenting a full application and sister packages which include Prolog code interfacing a number of common and useful tasks that can be delegated to R. We argue that Real is a powerful extension to logic programming, providing access to a popular statistical system that has complementary strengths in areas such as machine learning, statistical inference and visualisation. Furthermore, Real has a central role to play in the uptake of semantic web, computational biology and bioinformatics as application areas for research in logic programming.     

Design of fuzzy logic controllers based on generalized T-operators

Since Zadeh first proposed the basic principle of fuzzy logic controllers in 1968, the and operators have been popular in the design of fuzzy logic controllers. In this paper, the general concept of T-operators is introduced into the conventional design methods for fuzzy logic controllers so that a general and flexible methodology for the design of these fuzzy logic controllers is available. Then, by computer simulations, studies are made so as to determine the relations between the various T-operators and the performance of a fuzzy logic controller. It is concluded that the performance of the fuzzy logic controller for a given class of plants very much depends upon the choice of the T-operators.     

Modeling holistic fuzzy implication using co-copulas

Two related aggregation operators called copulas and co-copulas are introduced and various properties are described. The relationship, of these operators to t-norms and t-conorms is noted. Generalizations of these, respectively, called conjunctors and disjunctors, are introduced. We suggest the use of disjunctor operators for modeling the multi-valued implication operator in fuzzy logic. We point out that the selection of operators used in fuzzy logic, in addition to having appropriate pointwise properties, should be holistic, this requires consideration of the nature of the resulting fuzzy set as a whole. Focusing on the protoform of fuzzy modus ponens and looking at the information contained in the inferred fuzzy set we show that the use of co-copulas has some desirable properties. Taking advantage of the fact that the weighted sum of co-copulas is a co-copula we consider the problem of constructing customized implication operators.     

Distributions in fuzzy logic

Probability distributions associated with several ‘ply’-operators are discussed. These exact distributions are compared with relevant Gaussian approximations.