Fuzzy modus ponens: A new model suitable for applications in knowledge-based systems

Models for fuzzy deductive reasoning are becoming important in the field of knowledge-based systems. Zadeh suggested a compositional rule of inference, also called Generalized Modus Ponens. In order to study the fine structure of this model, we test seven rules of implication, usually found in the fuzzy literature, and two kinds of compositional rule with several properties which seems to be required by commonsense reasoning. But, as no rule is shown to satisfy these properties, we propose a new model for fuzzy Modus Ponens. the main idea of this model is to find a fuzzy extension of the meta-level rule of inference, in comparison with the object-level extension of the disjunctive syllogism such as the Zadeh's model. It is shown that our model satisfies all the intuitively required properties, and moreover is computationally very easy to apply.     

NOTE ON THE ARITHMETIC RULE BY ZADEH FOR FUZZY CONDITIONAL INFERENCE

Abstract

This paper shows that Zadeh's arithmetic rule for fuzzy conditional propositions “If x is A then y is B” and “If x is A then y is B else y is C” can infer quite reasonable consequences in a fuzzy conditional inference if new compositions of “max-[Odot] composition” and “max- composition” are used in the compositional rule of inference, though, as was pointed out before, this arithmetic rule cannot get suitable consequences in the compositional rule of inference which uses max-min composition. Moreover, it is shown that the arithmetic rule satisfies a syllogism under these two compositions.     

模糊推理的合成规则及其合成运算的选择研究

本文首先提出当知识库中含有多条取自于同论域上的模糊推理规则时，应用通常的六条合成推理规则所推得的结论将随着模糊推理规则数的增多而越来越偏离真实度这一问题。本文针对该问题对通常的六条合成推理规则及其三角模下的六条推广合成推理规则进行了理论比较研究，并对合成运算进行了比较研究。研究表明：合成推理规则（或Ｒ￣４）以合成运算ｍａｘ－Ｔ０是克服上述问题的最佳选择。这一重要结果对于模糊知识库的设计具有指导意义。     

Fuzzy reasoning based on the extension principle*

According to the operation of decomposition (also known as representation theorem) (Negoita CV, Ralescu, DA. Kybernetes 1975;4:169–174) in fuzzy set theory, the whole fuzziness of an object can be characterized by a sequence of local crisp properties of that object. Hence, any fuzzy reasoning could also be implemented by using a similar idea, i.e., a sequence of precise reasoning. More precisely, we could translate a fuzzy relation “If A then B” of the Generalized Modus Ponens Rule (the most common and widely used interpretation of a fuzzy rule, A, B, are fuzzy sets in a universe of discourse X, and of discourse Y, respectively) into a corresponding precise relation between a subset of P(X) and a subset of P(Y), and then extend this corresponding precise relation to two kinds of transformations between all L-type fuzzy subsets of X and those of Y by using Zadeh's extension principle, where L denotes a complete lattice. In this way, we provide an alternative approach to the existing compositional rule of inference, which performs fuzzy reasoning based on the extension principle. The approach does not depend on the choice of fuzzy implication operator nor on the choice of a t-norm. The detailed reasoning methods, applied in particular to the Generalized Modus Ponens and the Generalized Modus Tollens, are established and their properties are further investigated in this paper. © 2001 John Wiley & Sons, Inc.     

POPFNN-CRI(S): pseudo outer product based fuzzy neural network using the compositional rule of inference and singleton fuzzifier

A pseudo-outer product based fuzzy neural network using the compositional rule of inference and singleton fuzzifier [POPFNN-CRI(S)] is proposed in this paper. The correspondence of each layer in the proposed POPFNN-CRI(S) to the compositional rule of inference using standard T-norm and fuzzy relation gives it a strong theoretical foundation. The proposed POPFNN-CRI(S) training consists of two phases; namely: the fuzzy membership derivation phase using the novel fuzzy Kohonen partition (FKP) and pseudo Kohonen partition (PFKP) algorithms, and the rule identification phase using the novel one-pass POP learning algorithm. The proposed two-phase learning process effectively constructs the membership functions and identifies the fuzzy rules. Extensive experimental results based on the classification performance of the POPFNN-CRI(S) using the Anderson's Iris data are presented for discussion. Results show that the POPFNN-CRI(S) has taken only 15 training iterations and misclassify only three out of all the 150 patterns in the Anderson's Iris data.     

Fuzzy conditional inference under max-⊙ composition

This paper shows that the majority of fuzzy inference methods for a fuzzy conditional proposition “If x is A then y is B,” with A and B fuzzy concepts, can infer very reasonable consequences which fit our intuition with respect to several criteria such as modus ponens and modus tollens, if a new composition called “max-⊙ composition” is used in the compositional rule of inference, though reasonable consequences cannot always be obtained when using the max-min composition, which is used usually in the compositional rule of inference. Furthermore, it is shown that a syllogism holds for the majority of the methods under the max-⊙ composition, though they do not always satisfy the syllogism under the max-min composition.     

Fuzzy logic-based formulation of the organizer of intelligent robotic systems

A fuzzy logic-based methodology is proposed to model the organization level of an intelligent robotic system. The user input commands to the system organizer are linguistic in nature and the primitive events-tasks from the task domain of the system are, in general, interpreted via fuzzy sets. Fuzzy relations are introduced to connect every event with a specific user input command. Approximate reasoning is accomplished via a modifier and the compositional rule of inference, whereas the application of the conjunction rule generates those fuzzy sets with elements all possible (crisp) plans. Themost possible plan among all those generated, that is optimal under an application dependent criterion, is chosen and communicated to the coordination level. Off-line feedback information from the lower levels is considered asa-priori known and is used to update all organization level information. An example demonstrates the applicability of the proposed algorithm to intelligent robotic systems.     

Approximation of fuzzy context-free grammars

Fuzzy context-free max- grammar (or FCFG, for short), as a straightforward extension of context-free grammar, has been introduced to express uncertainty, imprecision, and vagueness in natural language fragments. Li recently proposed the approximation of fuzzy finite automata, which may effectively deal with the practical problems of fuzziness, impreciseness and vagueness. In this paper, we further develop the approximation of fuzzy context-free grammars. In particular, we show that a fuzzy context-free grammar under max- compositional inference can be approximated by some fuzzy context-free grammar under max-min compositional inference with any given accuracy. In addition, some related properties of fuzzy context-free grammars and fuzzy languages generated by them are studied. Finally, the sensitivity of fuzzy context-free grammars is also discussed.     

On subsumption in distributed derivations

In this paper we study the subsumption inference rule in the context of distributed deduction. It is well known that the unrestricted application of subsumption may destroy the fairness and thus the completeness of a deduction strategy. Solutions to this problem in sequential theorem proving are known. We observe that in distributed automated deduction, subsumption may also thwartmonotonicity, a dual property of soundness, in addition to completeness. Not only do the solutions for the sequential case not apply, even proper subsumption may destroy monotonicity in the distributed case.We present these problems and propose a general solution that treats subsumption as a composition of a replacement inference rule,replacement subsumption, and a deletion inference rule,variant subsumption. (Proper subsumption, in this case, becomes a derived inference rule.) We define a newdistributed subsumption inference rule, which has all the desirable properties: it allows subsumption, including subsumption of variants, in a distributed derivation, while preserving fairness and monotonicity. It also works in both sequential and distributed environments.We conclude the paper with some discussion of the different behavior of subsumption in different architectures.     

Knowledge modeling based on interval-valued fuzzy rough set and similarity inference: prediction of welding distortion

Knowledge-based modeling is a trend in complex system modeling technology. To extract the process knowledge from an information system, an approach of knowledge modeling based on interval-valued fuzzy rough set is presented in this paper, in which attribute reduction is a key to obtain the simplified knowledge model. Through defining dependency and inclusion functions, algorithms for attribute reduction and rule extraction are obtained. The approximation inference plays an important role in the development of the fuzzy system. To improve the inference mechanism, we provide a method of similaritybased inference in an interval-valued fuzzy environment. larity based approximate reasoning, an inference result is Combining the conventional compositional rule of inference with simideduced via rule translation, similarity matching, relation modification, and projection operation. This approach is applied to the problem of predicting welding distortion in marine structures, and the experimental results validate the effectiveness of the proposed methods of knowledge modeling and similarity-based inference.     

Decomposition of a fuzzy controller based on the inference break-up method

A concept called the decomposition of multivariable control rules is presented. Fuzzy control is the application of the compositional rule of inference and it is shown how the inference of the rule base with complex rules can be reduced to the inference of a number of rule bases with simple rules. A fuzzy logic based controller is applied to a simple magnetic suspension system. The controller has proportional, integral and derivative separate parts which are tuned independently. This means that all parts have their own rule bases. By testing it was found that the fuzzy PID controller gives better performance over a typical operational range then a traditional linear PID controller. The magnetic suspension system and the contact-less optical position measurement system have been developed and applied for the comparative analysis of the real-time conventional PID control and the fuzzy control.     

基于可拓规则的故障诊断专家系统推理机的研究

针对传统产生式规则在知识表示、匹配冲突等方面存在的局限,提出了一种将可拓规则用于故障诊断专家系统推理机的方法;该方法重点研究了可拓规则的匹配原理和可拓推理机算法思想,提出了匹配度计算方法并用来计算故障条件与规则前件的匹配度;根据研究表明,利用可拓规则进行推理,不仅在知识表示上比传统产生式规则推理有所提高,而且还解决了传统专家系统容易出现匹配冲突等问题;最后以AMU故障推理为例,说明可拓推理机具有推理速度快、效率高等优点,取得了较好的推理效果.     

A new belief rule base knowledge representation scheme and inference methodology using the evidential reasoning rule for evidence combination

In this paper, we extend the original belief rule-base inference methodology using the evidential reasoning approach by i) introducing generalised belief rules as knowledge representation scheme, and ii) using the evidential reasoning rule for evidence combination in the rule-base inference methodology instead of the evidential reasoning approach. The result is a new rule-base inference methodology which is able to handle a combination of various types of uncertainty.Generalised belief rules are an extension of traditional rules where each consequent of a generalised belief rule is a belief distribution defined on the power set of propositions, or possible outcomes, that are assumed to be collectively exhaustive and mutually exclusive. This novel extension allows any combination of certain, uncertain, interval, partial or incomplete judgements to be represented as rule-based knowledge. It is shown that traditional IF-THEN rules, probabilistic IF-THEN rules, and interval rules are all special cases of the new generalised belief rules.The rule-base inference methodology has been updated to enable inference within generalised belief rule bases. The evidential reasoning rule for evidence combination is used for the aggregation of belief distributions of rule consequents.     

基于证据推理的确定因子规则库推理方法

不确定性推理方法是人工智能领域的一个主要研究内容,If-then规则是人工智能领域最常见的知识表示方法. 文章针对实际问题往往具有不确定性的特点,提出基于证据推理的确定因子规则库推理方法.首先在If-then规则的基础上给出确定因子结构和确定因子规则库知识表示方法,该方法可以有效利用各种类型的不确定性信息,充分考虑了前提、结论以及规则本身的多种不确定性. 然后,提出了基于证据推理的确定因子规则库推理方法. 该方法通过将已知事实与规则前提进行匹配,推断结论并得到已知事实条件下的前提确定因子;进一步,根据证据推理算法得到结论的确定因子. 文章最后,通过基于证据推理的确定因子规则库推理方法在UCI数据集分类问题的应用算例,说明该方法的可行性和高效性.     

基于作用模糊子集推理的单片机模糊控制实现原理及其应用

针对DCRI模糊推理方法的复杂性,首先提出了作用模糊子集推理方法;然后将该推理方法与单片机数字运算少年叮结合,提出了基于作用模糊子集推理的单片机模糊控制实现原理,研制了开发了80C552型单片机模糊控制器;最后以建筑热工系统为被控对象,试验研究了测试室温度模糊控制过程。     

Comparison of two inference methods for P-type fuzzy logic control through experimental investigation using a hydraulic manipulator

This paper presents a comparison of the two important inference schemes: “individual-rule-based inference” and “compositional rule of inference” as applied to fuzzy logic control, through experimental investigation. The techniques are implemented on a hydraulic manipulator of an industrial machine with P-type fuzzy control. The fuzzy logic controller is designed for automatic positioning of the cutter blade of an automated fish-cutting machine. The features of the machine, which uses hydraulic servo control for cutter positioning, are outlined. The performance of the machine under the two inference schemes is examined and contrasted. Some practical implementations of the results are indicated.     

Unsupervised grammar induction using history based approach

Grammar induction, also known as grammar inference, is one of the most important research areas in the domain of natural language processing. Availability of large corpora has encouraged many researchers to use statistical methods for grammar induction. This problem can be divided into three different categories of supervised, semi-supervised, and unsupervised, based on type of the required data set for the training phase. Most current inductive methods are supervised, which need a bracketed data set for their training phase; but the lack of this kind of data set in many languages, encouraged us to focus on unsupervised approaches. Here, we introduce a novel approach, which we call history-based inside-outside (HIO), for unsupervised grammar inference, by using part-of-speech tag sequences as the only source of lexical information. HIO is an extension of the inside-outside algorithm enriched by using some notions of history based approaches. Our experiments on English and Persian languages show that by adding some conditions to the rule assumptions of the induced grammar, one can achieve acceptable improvement in the quality of the output grammar.     

Issues on adjointness in multiple-valued logics

We contribute to the theory of implications and conjunctions related by adjointness, in multiple-valued logics. We suggest their use in Zadeh’s compositional rule of inference, to interpret generalized modus ponens inference schemata. We provide new complete characterizations of implications that distinguish left arguments, implications that satisfy the exchange principle, divisible conjunctions, commutative conjunctions, associative conjunctions and triangular norms. We also introduce and characterize pseudo-strict and pseudo-continuous implications and conjunctions, and we explore the close relationship between these two notions.     

A neuro-fuzzy system for inferencing

We justify the need for a connectionist implementation of compositional rule of inference (COI) and propose a network architecture for the same. We call it COIN—the compositional rule of inferencing. Given a relational representation of a set of rules, the proposed architecture can realize the COI. The outcome of COI depends on the choice of the implication function and also on choice of inferencing scheme. The problem of choosing an appropriate implication function is avoided through neural learning. The system automatically finds an “optimal” relation to represent a set of fuzzy rules. We suggest a suitable modeling of connection weights so as to ensure learned weights lie in [0, 1]. We demonstrate through numerical examples that the proposed neural realization can find a much better representation of the rules than that by usual implication and hence results in much better conclusions than the usual COI. Numerical examples exhibit that COIN outperforms not only usual COI but also some of the previous neural implementations of fuzzy logic. ©1999 John Wiley & Sons, Inc.     

Extension of the temporal synchrony approach to dynamic variable bindingin a connectionist inference system

The relationship between symbolism and connectionism has been one of the major issues in recent artificial intelligence research. An increasing number of researchers from each side have tried to adopt the desirable characteristics of the approach. A major open question in this field is the extent to which a connectionist architecture can accommodate basic concepts of symbolic inference, such as a dynamic variable binding mechanism and a rule and fact encoding mechanism involving nary predicates. One of the current leaders in this area is the connectionist rule-based system proposed by Shastri and Ajjanagadde. The paper demonstrates that the mechanism for variable binding which they advocate is fundamentally limited, and it shows how a reinterpretation of the primitive components and corresponding modifications of their system can extend the range of inference which can be supported. Our extension hinges on the basic structural modification of the network components and further modifications of the rule and fact encoding mechanism. These modifications allow the extended model to have more expressive power in dealing with symbolic knowledge as in the unification of terms across many groups of unifying arguments.