Truth Space Research Articles

Many-Valued Modal Non-Monotonic Reasoning: Sequential Stable Sets and Logics with Linear Truth Spaces

A family of many-valued modal logics which correspond to possible-worlds models with many-valued accessibility relations, has been recently proposed by M. Fitting [7, 8]. Non-monotonic extensions of these logics are introduced with a fixpoint construction a la McDermott & Doyle and employ sequential belief sets as epistemic states [9]. In this paper we take a logical investigation of many-valued modal non-monotonic reasoning in Fitting's formal framework. We examine the notion of MV-stable sets which emerges as a sequential many-valued analog of Stalnaker-Moore stable sets and prove that several attractive epistemic properties are essentially retained in the many-valued setting, esp. when focusing on a syntactically simple epistemic fragment of MV-stable sets. We show that MV-stable sets are always closed under S4 consequence and identify three sufficient conditions for capturing axioms of negative introspection. Also, the relation of MV-stable sets to many-valued analogs of classical S5 models and to many-valued extensions of universal models is discussed. Finally, we pay special attention to the subclass of logics built on linear Heyting algebras and show that inside this subclass, the situation is very similar - in many respects - to the machinery devised by W. Marek, G. Schwarz and M. Truszczynski. In particular, the normal fragments of the two important classical ranges of modal non-monotonic logics remain intact: many-valued autoepistemic logic is captured by any non-monotonic logic in K5 - KD45 and many-valued reflexive autoepistemic logic corresponds to KTw5 - Sw5.

Reasoning with truth values on compacted fuzzy chained rules

In this paper, we consider the problem of executing a fuzzy knowledge base (FKB) with rule chaining. The inference process used as starting point is the one based on forward reasoning functions which, obtained from the compositional rule of inference, permits performing the execution of rules in the truth space. This way the process is totally independent from the universes of discourse in which the different variables are defined, allowing a homogeneous treatment for all the variables in the FKB. The execution of the rules is interpreted as the "propagation" of linguistic truth values of the linguistic truth variable that reflect the linguistic degree of fulfillment of each of the propositions in the rules. This execution process is analyzed in two fields of application: control systems, where it is customary to assume t-norm operators as implication functions, and the aggregation process is implemented through the maximum operator and expert systems applications, where other implication functions may be needed and t-norm operators are generally used as aggregation operators. For both of these situations, we present a compaction mechanism which allows a noticeable part of the operations to be performed a priori, thus achieving an important computation time saving.

A new type of fuzzy neural network based on a truth space approach for automatic acquisition of fuzzy rules with linguistic hedges

Fuzzy reasoning methods are generally classified into two approaches: the direct approach and the truth space approach. Several researches on the relationships between these approaches have been reported. There has been, however, no research which discusses their utility. The authors have previously proposed four types of fuzzy neural networks (FNNs) called Type I, II, III, and IV. The FNNs can identify the fuzzy rules and tune the membership functions of fuzzy reasoning automatically, utilizing the learning capability of neural networks. Types III and IV, which are based on the truth space approach, can acquire linguistic fuzzy rules with the fuzzy variables in the consequences labeled according to their linguistic truth values (LTVs). However, the expressions available for the linguistic labeling are limited, since the LTVs are singletons. This paper presents a new type of FNN based on the truth space approach for automatic acquisition of the fuzzy rules with linguistic hedges. The new FNN, called Type V, has the LTVs defined by fuzzy sets for fuzzy rules and can express the identified fuzzy rules linguistically using the fuzzy variables in the consequences with linguistic hedges. Two simulations are done for demonstrating the feasibility of the new method. The results show that the truth space approach makes the fuzzy rules easy to understand.

An efficient technique for obtaining unate implementation of functions through input encoding

Unate circuits have the desirable property that a single stuck-at fault manifests as multiple unidirectional error at the output. If the valid outputs belong to a unidirectional error detecting code, we can add circuitry to determine if the outputs have suffered any error and implement a self-checking system. The key contribution here is that we can start with an arbitrary circuit with arbitrary number of inputs and outputs and convert that to a unate circuit with coded input outputs. Earlier techniques were based on enumeration of truth space and therefore suffered from a combinatorial explosion for large number of inputs. The technique presented here proceeds with local transformations, producing a sub-optimal solution as a consequence, as opposed to no solution. The results are shown to be within acceptable bounds.

ファジィ推論における直接法と間接法に関する考察

ファジィ推論における直接法と間接法に関する考察

Approximations and truth spaces

Approximations form an essential part of scientific activity and they come in different forms: conceptual approximations (simplifications in models), mathematical approximations of various types (e.g. linear equations instead of non-linear ones, computational approximations), experimental approximations due to limitations of the instruments and so on and so forth. In this paper, we will consider one type of approximation, namely numerical approximations involved in the comparison of two results, be they experimental or theoretical. Our goal is to lay down the conceptual and formal foundations of a local theory of partial truth. This is done by introducing and exploring the concept of truth space.

A new approach to inference in approximate reasoning

Abstract We introduce a new approach to inference in approximate reasoning based on truth value restriction. The degree to which the actual given value A of a variable X agrees with the antecedent value B in a production “If X is B then Y is C ” is represented as a fuzzy subset of a truth space. We introduce a new form of implication based on the exponential operation. A new compatibility relation based on the input value A and the antecedent value B is defined. A fuzzy truth value true derived from the antecedent value B is also defined. It is shown that by mapping this compatibility relation into the truth space, a fuzzy truth value is generated which produces an exponential family of the fuzzy truth value true . We show that the inferred consequent can be generated from this truth value. Theoretical results showing the validity of this method to (1) chain rules consisting of several single fuzzy conditional propositions, and (2) conjunctive fuzzy conditional propositions, are also presented. Theoretical and simulation results demonstrate that our method is superior to the existing methods when intuitively correct responses are required.

A new approach to approximate reasoning using a fuzzy logic

An approach to approximate reasoning based upon fuzzy logic is given. The method has the advantage of being computationally very simple and efficient even for multiple compound implication statements. This efficiency is the result of always processing ‘implication linguistic statements’ in a simple fuzzy truth space, so allowing for easy pencil and paper calculation. New concepts introduced in this paper are: 1. (i) the definition of a new set of fuzzy truth values 2. (ii) inverse truth functional modification for calculating fuzzy truth value restrictions 3. (iii) approximate reasoning using fuzzy truth value restrictions and an implication composition in truth space.

Fuzzy truth definition of possibility measure for decision classification

A new definition of possibility measure is presented which is calculated on truth space and is shown to be equivalent to Zadeh's original definition. This alternative formulation is shown to be the more natural in the context of decision classification because it clearly demonstrates the need for determining both the possibility of a category and not that category in a selection criterion. A number of useful possibility theorems are presented and their application to decision classification is demonstrated in a simplistic medical diagnosis problem, which also employs entropy measure as an additional parameter. The truth space formulation of possibility measure is shown to be of further value in problems of high dimensional state and an important possibility theorem relating to such problems is presented.