Compositional Rule Of Inference Research Articles

Logical structure of fuzzy IF-THEN rules

This paper provides a logical basis for manipulation with fuzzy IF-THEN rules. Our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to Lotfi A. Zadeh but also other kinds of approximate reasoning methods, e.g., perception-based deduction, provided that there exists a possibility to characterize them within a formal logical system. In contrast with other approaches employing variants of multiple-valued first-order logic, the approach presented here employs fuzzy type theory of V. Novák which has sufficient expressive power to present the essential concepts and results in a compact, elegant and justifiable form. Within the effectively formalized representation developed here, based on a complete logical system, it is possible to reconstruct numerous well-known properties of CRI-related fuzzy inference methods, albeit not from the analytic point of view as usually presented, but as formal derivations of the logical system employed. The authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy IF-THEN rule bases will be represented, formalized and backed up by proof within the well-founded logical representation presented here. An immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy IF-THEN rules are ‘first class citizens´ of the representation: there are clear, logically founded definitions for fuzzy IF-THEN rule bases to be consistent, complete, or independent.

The logic of tied implications, part 1: Properties, applications and representation

A conjunction T ties an implication operator A if the identity A ( a , A ( b , z ) ) = A ( T ( a , b ) , z ) holds [A.A. Abdel-Hamid, N.N. Morsi, Associatively tied implications, Fuzzy Sets and Systems 136 (2003) 291–311]. We study the class of tied adjointness algebras (which are five-connective algebras on two partially ordered sets), in which the implications are tied by triangular norms. This class contains, besides residuated implications, several other implications employed in fuzzy logic. Nevertheless, we show that the algebraic inequalities of residuated algebras remain true for our tied implications, but in forms that distribute roles over the five connectives of the algebra. We apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules. We prove its equivalence to a scheme with one compound rule, when both schemata are interpreted by the compositional rule of inference, and all connectives are taken from one tied adjointness algebra. Then we quote another application of this rich theory, a notion of many-valued rough sets, which exhibit the basic mathematical behaviour of the rough sets of Pawlak. A comparator H is said to be prelinear if it satisfies H ( y , z ) ∨ H ( z , y ) = 1 for all y , z (Hájek). We introduce prelinear tied adjointness algebras, in which two comparators are prelinear. We provide a representation of those algebras, as subdirect products of tied adjointness chains, on the lines of Hájek's representation of BL-algebras. But our representations are more economical, because we employ minimal prime filters (on residuated lattices) only; rather than all prime filters.

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.

POP-Yager: A novel self-organizing fuzzy neural network based on the Yager inference

A Pseudo-Outer Product based Fuzzy neural network using the Yager Rule of Inference [Keller, J. M., Yager, R. R., & Tahani, H. (1992). Neural network implementation of fuzzy logic. Fuzzy Sets and Systems, 45(1), 1-12.] called the POP-Yager FNN is proposed in this paper. The proposed POP-Yager FNN training consists of two phases; namely: the fuzzy membership derivation phase using the Modified Learning Vector Quantization (MLVQ) method [Ang, K. K. (1998). POPFNN-CRI(S): A Fuzzy neural network based on the compositional rule of inference. M. Phil. Dissertation, Nanyang Technological University.]; and the rule identification phase using the novel one-pass LazyPOP learning algorithm [Quek, C. & Zhou, R. W. (1999). The POP learning algorithms: Reducing work in identifying fuzzy rules. Neural Networks, 14(10), 1431-1445]. The proposed two-phase learning process effectively constructs the membership functions and identifies the fuzzy rules. Inference process in POP-Yager is based on the well-established Yager fuzzy inference rule [Keller, J. M., Yager, R. R. & Tahani, H. (1992). Neural network implementation of fuzzy logic. Fuzzy Sets and Systems, 45(1), 1-12]. Operations in POP-Yager strictly perform the logical processes of the Yager inference rule. This gives the novel network a strong theoretical foundation. Extensive experimental results based on the classification performance of the POP-Yager FNN using the Anderson's Iris data [Duda, R. O. & Hart, P. E. (1973). Pattern classification and scene analysis. Wiley.] are presented for discussion. Results show that the POP-Yager FNN possesses excellent recall and generalization abilities. The performance of the POP-Yager FNN as a general approximation of traffic flow data [Tan, G. K. (1997). Feasibility of predicting congestion states with neural networks. Final Year Project Report, Nanyang Technological University, CSE.] is analysed.

RSPOP: Rough Set–Based Pseudo Outer-Product Fuzzy Rule Identification Algorithm

System modeling with neuro-fuzzy systems involves two contradictory requirements: interpretability verses accuracy. The pseudo outer-product (POP) rule identification algorithm used in the family of pseudo outer-product-based fuzzy neural networks (POPFNN) suffered from an exponential increase in the number of identified fuzzy rules and computational complexity arising from high-dimensional data. This decreases the interpretability of the POPFNN in linguistic fuzzy modeling. This article proposes a novel rough set-based pseudo outer-product (RSPOP) algorithm that integrates the sound concept of knowledge reduction from rough set theory with the POP algorithm. The proposed algorithm not only performs feature selection through the reduction of attributes but also extends the reduction to rules without redundant attributes. As many possible reducts exist in a given rule set, an objective measure is developed for POPFNN to correctly identify the reducts that improve the inferred consequence. Experimental results are presented using published data sets and real-world application involving highway traffic flow prediction to evaluate the effectiveness of using the proposed algorithm to identify fuzzy rules in the POPFNN using compositional rule of inference and singleton fuzzifier (POPFNN-CRI(S)) architecture. Results showed that the proposed rough set-based pseudo outer-product algorithm reduces computational complexity, improves the interpretability of neuro-fuzzy systems by identifying significantly fewer fuzzy rules, and improves the accuracy of the POPFNN.

A Compositional Framework for Formally Verifying Modular Systems

We present a tool-supported framework for proving that the composition of the behaviors of the separate parts of a complex system ensures a desired global property of the overall system. A compositional inference rule is formally introduced and encoded in the logic of the PVS theorem prover. Methodological considerations on the usage of the inference rule are presented, and the framework is then used to prove a meaningful property of a simple, but significant, control system.

Development of a New Max-Min Compositional Rule of Inference in Control Systems

Generally, Max-Min CRI (Compositional Rule of Inference ) method by Zadeh and Mamdani is used in the conventional fuzzy inference. However, owing to the problems of Max-Min CRI method, the inference often results in significant error regions specifying the difference between the desired outputs and the inferred outputs. In this paper, I propose a New Max-Min CRI method which can solve some problems of the conventional Max-Min CRI method. And then this method is simulated in a D.C.series motor, which is a bench marking system in control systems, and showed that the new method performs better than the other fuzzy inference methods.

DECISION MAKING BASED ON STATISTICAL DATA, SIGNED DISTANCE AND COMPOSITIONAL RULE OF INFERENCE

An assessment of a set of alternatives under certain evaluation criteria has difficulty in dealing with the priority of these alternatives, especially with a lack of precise information in an uncertain environment. Fuzzy numbers are usually applied to represent the imprecise numerical measurements of different alternatives. In this study statistical data are used to derive level (1-α,1-β) interval-valued fuzzy numbers to represent unknown alternative effectiveness scores, after which, by using the compositional rule of inference and signed distance to transform the fuzzy decision making problem into crisp one, one can conveniently obtain the order of these different alternatives and subsequently obtain the best alternative. The approach presented is computationally efficiency, and its underlying concepts are simple and comprehensible. By using this extended generalized method, two cases of an organizational type of rapid-transit-system selection problem are presented as examples to illustrate the applicability of the interval-valued fuzzy numbers and ranking system for decision making. The key contribution of the method is the seamless integration of the statistical data, interval-valued fuzzy number and signed distance to analyze multicriteria decision making problem. The innovation introduced in the model concerns interval-valued fuzzy number which is recognized as a determinant of the effectiveness score in fuzzy relation matrix.

Fault Detection in Systems-A Fuzzy Approach

The task of fault detection is important when dealing with failures of crucial nature. After detection of faults in a system, it is advisable to suggest maintenance action before occurrence of a failure. Fault detection may be done by observing various symptoms of the system during its operational stage. Sometimes, symptoms cannot be quantified easily but can be expressed in linguistic terms. Since linguistic terms are fuzzy quantifiers, these can be represented by fuzzy numbers. In this paper, two cases have been discussed, where a fault likely to affect a particular systemlsystems, is detected. In the first case, this is done by means of a compositional rule of inference. The second case is based on modified similarity measure. For both these cases, linguistic terms have been expressed as trapezoidal fuzzy numbers

GenSo-EWS: a novel neural-fuzzy based early warning system for predicting bank failures

Bank failure prediction is an important issue for the regulators of the banking industries. The collapse and failure of a bank could trigger an adverse financial repercussion and generate negative impacts such as a massive bail out cost for the failing bank and loss of confidence from the investors and depositors. Very often, bank failures are due to financial distress. Hence, it is desirable to have an early warning system (EWS) that identifies potential bank failure or high-risk banks through the traits of financial distress. Various traditional statistical models have been employed to study bank failures [J Finance 1 (1975) 21; J Banking Finance 1 (1977) 249; J Banking Finance 10 (1986) 511; J Banking Finance 19 (1995) 1073]. However, these models do not have the capability to identify the characteristics of financial distress and thus function as black boxes. This paper proposes the use of a new neural fuzzy system [Foundations of neuro-fuzzy systems, 1997], namely the Generic Self-organising Fuzzy Neural Network (GenSoFNN) [IEEE Trans Neural Networks 13 (2002c) 1075] based on the compositional rule of inference (CRI) [Commun ACM 37 (1975) 77], as an alternative to predict banking failure. The CRI based GenSoFNN neural fuzzy network, henceforth denoted as GenSoFNN-CRI(S), functions as an EWS and is able to identify the inherent traits of financial distress based on financial covariates (features) derived from publicly available financial statements. The interaction between the selected features is captured in the form of highly intuitive IF-THEN fuzzy rules. Such easily comprehensible rules provide insights into the possible characteristics of financial distress and form the knowledge base for a highly desired EWS that aids bank regulation. The performance of the GenSoFNN-CRI(S) network is subsequently benchmarked against that of the Cox's proportional hazards model [J Banking Finance 10 (1986) 511; J Banking Finance 19 (1995) 1073], the multi-layered perceptron (MLP) and the modified cerebellar model articulation controller (MCMAC) [IEEE Trans Syst Man Cybern: Part B 30 (2000) 491] in predicting bank failures based on a population of 3635 US banks observed over a 21 years period. Three sets of experiments are performed-bank failure classification based on the last available financial record and prediction using financial records one and two years prior to the last available financial statements. The performance of the GenSoFNN-CRI(S) network as a bank failure classification and EWS is encouraging.

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.

Evolutive Learning Algorithms for Fuzzy Modeling

This paper presents the evolutive learning for the newly proposed Generalized Fuzzy Model (GFM), which combines the inherent features of the existing Compositional Rule of Inference (CRI) and Takagi-Sugeno (TS) models. The evolutive learning is necessitated by the fact that local learning involving the hybridization of LSE and Gradient Descent (GD) techniques fails to yield the targeted performance for certain dynamic systems. The LSE is used to estimate the consequent part, and GD is used to estimate the premise part of the IF-THEN fuzzy rule. Further learning by the hybrids of Genetic and Simulated Annealing techniques, known as GA hybrid and SA hybrid, provides improved performance. The results are demonstrated on stock market data.

A generalized time quantifier approach to approximate reasoning

In this paper, a rather expressive fuzzy temporal logic for linear time is introduced. First, this logic is a multivalued generalization (Lukasiewicz style) of a two-valued linear-time temporal logic based on, e.g., the “until” operator. Second, it is obtained by introducing a generalized time quantifier (a generalization of the partition operator investigated by Shen) applied to fuzzy time sets. In this fuzzy temporal logic, generalized compositional rules of inference, suitable for approximate reasoning in a temporal setting, are presented as valid formulas. Some medical examples illustrate our approach.

A new perspective on reasoning with fuzzy rules

This article expresses the idea that information encoded on a computer may have a negative or positive emphasis. Negative information corresponds to the statement that some situations are impossible. Often, it is the case for pieces of background knowledge expressed in a logical format. Positive information corresponds to observed cases. It is encountered often in data-driven mathematical models, learning, etc. The notion of an “if …, then …” rule is examined in the context of positive and negative information. It is shown that it leads to the three-valued representation of a rule, after De Finetti, according to which a given state of the world is an example of the rule, a counterexample to the rule, or is irrelevant for the rule. This view also sheds light on the typology of fuzzy rules. It explains the difference between a fuzzy rule modeled by a many-valued implication and expressing negative information and a fuzzy rule modeled by a conjunction (a la Mamdani) and expressing positive information. A new compositional rule of inference adapted to conjunctive rules, specific to positive information, is proposed. Consequences of this framework on interpolation between sparse rules are also presented. © 2003 Wiley Periodicals, Inc.

Compositional rule of inference as an analogical scheme

We discuss fuzzy deductive reasoning achieved through the compositional rule of inference from the analogy point of view, merging it into the analogical scheme. Necessary and sufficient conditions on operators involved are investigated to fit the compositional rule of inference into the analogical scheme. Several examples including well known models are included.

Design of a Lukasiewicz rule-driven fuzzy processor

Rule-driven processing is a proven way of achieving high-speed in fuzzy processing. Up to now, rule- driven architectures where designed to work with mini- mum or product as T-norm. Nevertheless, a Lukasiewicz T-norm is typically used with the compositional rule of inference in expert systems applications that are based on a fuzzy inference engine. This paper presents a rule-driven processing architecture for systems with a Lukasiewicz T-norm and partition of unity membership functions with an overlap factor of two.

A triangular norm-based fuzzy predicate logic

Fuzzy predicate calculi are powerful tools in analyzing topics of Zadeh's agenda, such as fuzzy modus ponens, compositional rule of inference, fuzzy functions and fuzzy control, etc. So far, however, predicate logics which are complete with respect to the semantics over [0,1] are rather scarce. In this paper, a new type of fuzzy predicate logic K L ∗ , which has a recursive axiomatization that is complete with respect to the semantics over [0,1], is introduced. Furthermore, its important applications are also discussed.

On generalized modus ponens with multiple rules and a residuated implication

We consider a multiple-rule, generalized modus ponens inference scheme, with an interpretation based on compositional rule of inference (CRI) and a residuated implication. We show that such a system is equivalent, as far as CRI is concerned, to a system that satisfies the “basic requirement for fuzzy reasoning”, proposed by Turksen and Tian (Fuzzy Sets and Systems 58 (1993) 3–40). We establish an analogous conclusion for an alternative interpretation (due to Magrez and Smets) of generalized modus ponens, which we call consequent dilation rule (CDR). This method is computationally faster than CRI, but it produces less specific inference results.

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.

A neuro-fuzzy framework for inferencing

Earlier we proposed a connectionist implementation of compositional rule of inference (COI) for rules with antecedents having a single clause. We first review this net, then generalize it so that it can deal with rules with antecedent having multiple clauses. We call it COIN, the compositional rule of inferencing network. 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 both the implication function and the inferencing scheme. The problem of choosing an appropriate implication function is avoided through neural learning. COIN can automatically find a ‘good’ relation to represent a set of fuzzy rules. We model the connection weights so as to ensure learned weights lie in [0,1]. We demonstrate through extensive 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.