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It is well known that least mean square estimation can be employed to calculate conditional means, a procedure called the implicit conditioning method in this paper. It is possible to construct a priori probability densities of tractable form that, when conditioned on certain sets of variables, reduce to conditional probability densities which are identical to the canonical probability densities occurring in the statistical mechanics of certain classical systems. This yields a new variational principle for the calculation of canonical mean values in classical statistical mechanics. In this paper, two versions of this variational principle are applied to a simple lattice system to yield approximate expressions for the canonical mean values of certain properties of physical interest.

It is an interesting point of view to consider automata games as a model of human behaviors in a society. Automata games have been proposed by Tsetlin and Krylov. In their model, however, the final expected value of winnings was not equal to the value of Von Neumann in the game theory. In this paper, we consider the two automata zero sum games between automata proposed by Fu and Li with two strategies, and show that if the game matrix G = [ g ij ] satisfies the condition then the final expected value of winnings is equal to the value of Von Neumann. Satisfactory results are obtained by computer simulations, as shown at the end of this paper.

If information science is to be considered a “science” in the true sense of the word, there is a clear need for the development of a set of general concepts and analytical expressions regarding the flow of information in any situation for which information serves as a resource. This paper discusses quantitative aspects of a theory leading to the development of such a pragmatic information theory . At the heart of a theory of pragmatic information is the measure of information contained in a set of data. At the pragmatic level, information has value to the extent that it is useful as a resource for purposeful activity. The primary “purposeful activity” in life is decision-making. Hence, information and decision-making are inextricably tied together; in fact, information is data of value in decision-making . In this paper, we discuss quantitatively a very general decision model which provides a formal and comprehensive representation of uncertainty in decision-making. Other decision models presented in the literature assume away a considerable and important portion of the uncertainty that exists in decision-making. This model is then used as a framework for examining the role of information in decision-making in a way that is also formal and comprehensive. Hence, the decision model suggested facilitates a meaningful analysis of information; this analysis culminates in the development of a measure of the value of the pragmatic information contained in a set of data.

In a radically new approach to linear estimation, Lainiotis [33, 36–37, 52–53], using the “partition theorem”-an explicit Bayes theorem-obtained fundamentally new linear filtering and smoothing algorithms both for continuous as well as discrete data. The new algorithms are given in explicit, integral expressions of a “partitioned” form, and in terms of decoupled forward filters. The “partitioned” algorithms were shown to be especially advantageous from a computational as well as from an analysis standpoint. They are essentially based on the decomposition of the innovations into partial or conditional innovations and residuals. In this paper, the “partitioned” algorithms are shown to be the natural framework in which to study such important concepts as observability, controllability, unbiasedness, and the solution of Riccati equations. Specifically, in this paper, the “partitioned” algorithms are re-examined yielding further insight as well as several significant new results on: 1. (a) unbiased estimation and filter initialization procedures; 2. (b) stochastic observability and stochastic controllability; 3. (c) the interconnection between stochastic observability, Fisher information matrix, and the Cramer-Rao bound; 4. (d) estimation error-bounds; and most importantly 5. (e) computationally effective “partitioned” solutions of time-varying matrix Riccati equations. In fact, all of the above results have been obtained for general, time-varying, lumped, linear systems. In addition, it is shown that previously established smoothing algorithms, such as the Meditch differential algorithm and the Kailath-Frost total innovation algorithm, are readily obtained from the “partitioned” algorithms. The properties of the “partitioned” algorithms are obtained, thoroughly examined, and compared to those of other algorithms.

In a previous paper ^[^1^] the use of the concept of fuzzy sets in clustering was proposed. The convenience of fuzzy clustering over conventional representation was then stressed. Assigning each point a degree of belongingness to each cluster provides a way of characterizing bridges, strays, and undetermined points. This is especially useful when considering scattered data. The classificatory process may be considered as the breakdown of the probability density function of the original set into the weighted sum of the component fuzzy set densities. Such decomposition should be performed so that the components really represent clusters. This is done by optimization of some functional defined over all possible fuzzy classifications of the data set. Several functionals were suggested in ^[^1^]. The bulk of this paper is concerned with numerical techniques useful in the solution of such problems. The first two formulas treated do not provide an acceptable fuzzy classification but yield good starting points for the minimization of a third functional. This last method obtains very good dichotomies and is characterized by slower convergence than the previous processes. Using that functional, a modification is suggested to obtain partitions in more than two sets. Numerous computational experiments are presented.

This paper provides another formalization of the concept of an effectively calculable function motivated by the LISP language. In addition to the lambda functional abstraction operator the calculus described in this paper has a gamma decision operator of four arguments. There are two primitive relations of the calculus. The first is a denotation relation which in computer terms is the relationship holding between a name for a memory location and the contents of the location. The second is an identity relationship. An applied lambdagamma calculus with a successor function as the only primitive function is described in the paper as a fully formal theory with axioms and rules of deduction.

In a prior paper [5]we have given a lower bound on the time to multiply in a group using a circuit of unit delay elements with limited fan-in. Also in that paper was a circuit realization for group multiplication requiring time at most one unit greater than the lower bound. In the present work we generalize that circuit construction method so as to render it applicable to any function f: X"1 x X"2 -> Y, where X"1 and X"2 are finite sets. We then examine the optimality of the method when S is a finite semigroup and f"l: S x S -> S is semigroup multiplication.

In the paper entitled “Estimation of jumps, reliability and hazard rate” [V. K. Murthy, in Ann. Math. Stat . 36 , 1032–1040 (1968)], the author has treated the problem of estimating the jumps of the probability distribution function. In this paper, the author proposes the sum of the squares of the jumps as a criterion of departure from absolute continuity of the distribution function and then solves the problem of estimation based on a random sample. More specifically, let F(x) be a probability distribution function. Assuming the singular part to be identically zero, F(x) can be decomposed into F ( x ) = F 1 ( x ) + F 2 ( x ), where f 1 ( x ) is an everywhere continuous function and F 2 ( x ) is a pure step function with steps of magnitude, say, S ζ j at the points x = ζ j , j = 1, 2,…, ∞ and that finally both F 1 ( x ) and F 2 (x) are nondecreasing and are uniquely determined. Based on a random sample X 1 , X 2 ,…, X N of size N , a consistent, and asymptotically normal class of estimators are obtained for the distance Δ = S ζ j 2 . If the singular part is not identically zero, the results are good almost everywhere.

The purpose of this paper is to consider the behavior of a finite automaton in a nonstationary random environment. The behavior of finite deterministic automata in stationary random environments was considered by Tsetlin. In this paper, the probabilistic automaton is introduced as a random environment in order to generalize the stationary random environment. The interaction between the probabilistic automaton and the two-state deterministic automaton is considered in the case where the probabilistic automaton has two inputs and two states and, besides, is completely isolated by the 0th approximation. And the limiting state probability distribution of this finite automaton is also obtained. Moreover, it is shown that, if the probabilistic automaton is completely isolated by the (0, k )th approximation and satisfies some conditions, then the finite automaton can behave expediently against the probabilistic automaton.

In their fundamental paper Kalman and Bucy solved the problem of explicitly designing the optimal estimator of the state of a certain type of Markov process at the time T + h ( h ⩾ 0) given observations of a certain number of linear combinations of the states for a certain observation time (0, T ). The problem with h = 0 is currently known as the filtering problem and the one with h > 0 as the prediction problem. Below is given a solution of the smoothing or interpolation problem (0 A ⩽ T ), which is left open in their paper. This problem has been tackled by many authors, but it is believed that the method of the present paper has some points of interest.

This paper describes a fuzzy logic anti-slip system for Heavy Duty Off Road Vehicles. The anti-slippage system is based on distributed detection and local/global fuzzy control of slippage within an interconnected system of mechatronic wheel motors. Within this paper, the system layout, implementation, sensor data preprocessing, slip detection, four different local/global and test results under real working conditions are described. The system developed may serve as an example for successfully applied fuzzy control for a competitive (soon) commercially available product.

The present paper is closely related to an earlier publication wherein the present authors proposed a mathematical formulation for continuous nonlinear minimal-variance filters that led to stochastic differential equations satisfying known existence and uniqueness conditions for such equations. The necessity for the proposed reformulation of the problem arose from the use of the white-noise model to represent the physical system. The adoption of such a model allows certain formal simplicities in the derivation of filter equations, but these simplicities must be paid for by new complexities introduced by the requirement for mathematical rigor, as discussed in the previous paper. It is now shown that the modified formulation, while necessary for assured mathematical validity, can be dispensed with in the context of a computational mechanization.

In this paper a non-gaussian discrete time stochastic pursuit problem is solved. Then this solution is applied to a class of linear stochastic feedback problems with arbitrarily generated disturbances and incomplete state information. As a by-product, we obtain a generalized version of the well-known “separation theorem.” In particular, the solution is applied to obtain explicit feedback laws for systems with imperfectly known disturbance parameters.

This paper presents a survey of some recent results in the fields of nonlinear filtering and stochastic control resulting from the use of Gaussian sum approximations to certain density functions. When trying to estimate and control the state of a nonlinear stochastic dynamic system from noisy measurement data, the a posteriori density of the state conditioned on the measurement data contains all of the available information about the system state. If these densities were available, it should be possible to develop “optimal” estimation and control policies. The well-known Bayesian recursion relations describe the evolution of this a posteriori density in terms of given a priori densities, the system dynamics and measurement equations and the available data. Unfortunately, it is seldom possible to solve these closed forms for the required a posteriori densities. Over the last few years it has been shown that when the a priori densities are approximated by a sum of Gaussian densities with positive weighting coefficients that linear non-Gaussian problems can be solved analytically and that new natural solutions to nonlinear problems are available. Recently this method has been extended to dual deterministic optimal control problems.

This paper describes procedures for the design of experiments to determine whether or not a given finite-state machine is operating correctly. The machines in question are assumed to be strongly connected and in a reduced form. The procedures presented are valid for any type of faults that do not increase the number of states of the original machine. More efficient procedures are presented in the last section, when certain restrictions are imposed on the class of possible faults.

In this paper, systems of natural deduction are described in which the usually cumbersome quantifier rules (∀-introduction and ∃-elimination) are replaced by somewhat more convenient ones. These systems have deducibility from a set of assumption formulas as a primitive notion which is shown to have the compactness property. They are equivalent to the system obtained from a logistic system of Kleene by excluding variation of variables, admit the usual subsystems, have some separation properties for logical operators and, together with Kleene's system under exclusion of variation of variables, are complete with respect to logical consequence.

This paper presents the innovations approach to space-time or distributed parameter filtering and smoothing. The innovations process is defined for systems with space-time coordinates and properties of this process are derived. Development of a special operator notation allows a very simple presentation of the algorithms for estimation. The filtering results are in agreement with those obtained by other derivations whereas it is believed that the fixed-interval and fixed-point smoothing algorithms for space-time systems are new.

In this paper we construct for each integer n , an lba which is universal for all lba 's whose set of tape symbols numbers no more than n . We also obtain the corresponding result for deterministic lba 's. These results are used to show that for each n there is a set accepted by a deterministic lba , but not by any deterministic lba with fewer than n symbols in its alphabet. The fundamental idea is to modify the construction of the universal Turing machine so that it becomes universal for an appropriate class of linear-bounded automata.

Hierarchical Control is an attractive solution when the optimization of a large scale system is requested. This paper introduces a consistent approach to the coordination problem in the dynamic case. Two basic transformations on the system afford real decomposition into subproblems by means of the generalized Lagrangian. Two specific hierarchical structures fulfilling efficiently the INPRE and INBAL principles are deduced. This method gives a convenient realization algorithm in both cases. Moreover, these principles are shown, in a sense, to be dual.

This paper deals with a new method for optimizing control of multimodal systems by using fuzzy automata. Two classes of fuzzy automata corresponding to the Mealy and Moore type of ordinary automata are formulated in which the transition may be executed on the basis of the membership function between these two states, and these automata are used as the optimizing controller for a global search of the optimum of a multimodal systems. The concepts of the higher order transition in the automata and the partition of the domain of objective function are introduced to the method of optimizing control, and the control systems can hold the true optimum at small hunting loss without keeping any local optimum. The operation of self-organization is performed to optimize the system in the automata.

The object of this paper is to characterize binary relations by their consistent composite formation properties. Systems composed of a binary relation between lattices are studied. A group of dual and inverse operators are defined on such systems. Twelve containment properties of a binary relation in a system are then defined and a thirteenth property is defined for the special case of Boolean algebras. All consistent combinations of the binary values of the properties are determined by the use of reduction theorems and by the construction of example systems. There are 147 consistent combinations of property values. Diagrams of a minimum set of forty representative example systems are displayed. Examples for all 147 cases can be generated from the representative systems by the group of system transformations.

By extracting the basic properties common to the formal grammars appeared in existing literatures, we develop a general formulation of formal grammars. We define a pseudo grammar and derive from it the well-known probabilistic, fuzzy grammars and so on. Moreover, several interesting grammars such as ⊔∗ grammars, ⊔ ⊓ grammars, ⊔ ⊓ grammars, composite B -fuzzy grammars, and mixed fuzzy grammars, which have never appeared in any other papers before, are derived.

This paper presents a general and complete duality theory for optimal control of systems governed by linear ordinary differential equations with a convex cost criterion. Existence theorems for optimal control problems are obtained using the duality theory and a direct relationship between duality and Pontryagin's Maximum Principle is exhibited. Finally, applications to decomposition of optimal control problems are presented.

In the past, automatic procedures for the design of pattern classifiers have usually used a machine of fixed size. These procedures require that prior to learning we know, or can guess, the complexity of the decision surface used by the teacher. Techniques have been discovered for changing the complexity of a pattern classifier by adding or removing parts of it. These methods, respectively called “growing” and “pruning” use well established learning rules applied alternately with the addition or removal of comput,ing equipment. Growing is thus a process by which we can increase the complexity of a classifier until it fits the problem. Pruning removes computing equipment from a classifier whose performance is satisfactory, but inefficient. The paper describes our experimental evaluation of these procedures. These studies have shown that these techniques are capable of designing a classifier which closely models a complex teacher with a minimum of storage.

This paper is mainly concerned with two problems in the area of stochastic languages. For the λ-stochastic languages, a flexible stochastic experimental procedure based on Chebyshev's inequality is proposed. Then, the class of maximum-likelihood stochastic languages is defined on the basis of the maximum-likelihood final-state distribution. Properties of this class of languages and its relationship to λ-stochastic languages are investigated.

This paper develops theories for constructing filing schemes for formatted files with unequal-valued attributes when the query set contains all queries which specify two values from two attributes. These filing schemes provide a set of buckets for storing accession numbers of records. The retrieval rule is based on identifying a bucket from a query by solving algebraic equations over finite fields. The theories underlying these filing schemes are based on properties of deleted finite geometries. Expressions for retrieval time and storage redundancy are also given.

In this paper, properties of normalized stochastic languages are discussed and alternative procedures for constructing the Chomsky and Greibach normal forms for normalized stochastic context-free grammar (nscfg) are presented. A normalized stochastic context-free language (nscf l) is defined in terms of a nscfg. Furthermore, stochastic languages accepted by stochastic pushdown automata (spda) are defined, and relationships between stochastic context-free languages and spda are studied. The class of languages accepted by a spda with a 0 cutpoint is precisely the class of scf l.

This paper considers the problem of pattern classification in the frequency domain. The test statistic depends on the spectral density of each pattern class which may or may not be known. When the spectral density of the unknown input pattern to be classified is not known then the technique of time series analysis is applied to estimate its spectra, and asymptotic properties of the test statistic are investigated. A simple example is solved to illustrate the validity of this approach.

This paper proposes the concept of equivocation from information theory as a criterion for feature selection. An expression for equivocation is derived in terms of class conditional probabilities. By making the assumption that the patterns in each class are normally distributed, a transformation matrix that minimizes the equivocation in the reduced dimension is obtained. Furthermore, a relationship between the equivocation and the expected divergence between any pair of classes is presented.

In this paper, the algorithm to generate prime implicants presented in Ref. 5 is extended to a new algorithm. The new algorithm does not require the Boolean function to be in conjunctive normal form and is complete in the sense that it generates all the prime implicants. The author also shows that this new algorithm can be used to efficiently solve the selection problem ^[^4^]. By using this algorithm, the original selection problem can be solved by a branch-and-bound approach ^[^1^] which avoids an exhaustive search of the solution space and reduces the amount of necessary calculations.

One approach to abstract pattern recognition is to represent the discriminant functions such that they are characterized by linearly appearing parameters, e.g., a general multivariate polynomial. This paper discusses a class of methods for deriving values for these parameters given a finite set of samples of each pattern class; the methods discussed are those which use least-square or least-mean-square approximation to an unknown probability density or partially specified discriminant function. The algorithms which result are feasible for the solution of complex problems using a general-purpose digital computer. This viewpoint emphasizes the common denominator of many procedures in abstract pattern recognition.

The extension of standard concepts of set theory (like union, intersection, etc.) to the theory of fuzzy sets is not obvious. Several policies for answering questions of this type have been discussed, but it appears that the operations of max and min play a central role in the arithmetic of fuzzy sets. Letting μA(x) denote our willingness to accept x as a member in the fuzzy set A, Zadeh defines intersection A ∩ B and union A ∪ B of the two fuzzy sets A and B by A ∩ B = {(x; min {μA(x), μB(x)})} and A ∪ B = {(x; max {μA(x), μb(x)})}. The object of this paper is to show that these definitions are not only natural, but under quite reasonable assumptions the only ones possible.

This paper uses optimization theory and continuous invariant imbedding in order to derive computationally efficient sequential algorithms for the maximum likelihood estimation of bias errors in recursive filtering with noise corrupted input observations and correlated plant and measurement noise. Error analysis algorithms are derived for adaptive and nonadaptive systems with bias and modeling errors. Examples demonstrate the efficacy of the adaptive estimation algorithms and the error analysis algorithms for estimation with bias uncertainty.

The purpose of this paper is to investigate various properties of some learning automata operating in random environments. Based on the strategies of ''many-armed bandit problems,'' two sequential learning models, A"r","k and B"r","k","b, having deterministic and stochastic transition rules, respectively, are formulated. Automata using these strategies have shown the desired learning behavior, which is similar to the performance of linear-strategy automata and certain type of stochastic automata. Furthermore, the learning behavior of fuzzy automata in random environments is also considered. Comparison among various models has been made. Computer-simulated examples for the application of proposed models include games against nature and games between automata. The results from simulation have been considered quite satisfactory.

This paper presents the problem of unsupervised learning structure and parameter adaptive pattern recognition. Different near optimal solutions that alleviate the infinite or exploding memory requirements of the optimal solution are suggested. The learning capabilities of these methods are compared with that of the optimal supervised scheme by presenting an example.

In this work the embedding of a graph X onto a graph Y is discussed, given a set of constraints on correspondences or noncorrespondences of nodes and arcs of the two graphs. This problem is relevant in different areas, such as operation research, computer aided design, etc. The embedding problem is approached by a covering technique on a Node Correspondence Table. The properties of such a table are investigated, and in particular, emphasis is given to the study of coverability. The paper is mainly divided into two parts (Sections II and III). The effect of the constraints on node correspondence only is studied first, and the basic properties of the Node Correspondence Tables are derived. Then, the addition of constraints on arc correspondence is discussed. In both such cases, a partially enumerative procedure is presented for determining all the possible solutions. Finally, two examples are thoroughly discussed and solved, one of which belongs to the category of subgraph isomorphism problems.

A novel approach to obtaining series representations in the stochastic mean for weakly, and therefore mean square, continuous stochastic processes is presented. Two distinct orthogonal series representations are derived for the entire class of weakly continuous stochastic processes over a broad class of Lebesgue sets of the real line, and a constructive procedure to obtain them explicitly is given. They include as particular cases all earlier representations. Also they are shown to converge almost surely in the norm of an L"2 space. Two general results, on which the development of this paper is based, are also presented.

This paper deals with a new clustering technique using the concept of fuzzy set introduced by Zadeh. A cluster is defined as a subset characterized by a membership function which associates with each element a real number in the unit interval. A membership function is proposed and a method to select the cluster elements is derived using the separation theorem of the fuzzy sets. The clusters obtained by this way are controlled overlapping groupings.

This paper deals with a nonlinear learning algorithm characterized by a threshold effect. The algorithm is derived in relation to a learning process in pattern recognition where steps must be taken to minimize the effect of spurious samples carrying unreliable information. With the assumption of a multivariate normal density for each pattern population, the nonlinear algorithm reduces to discarding the spurious samples with a threshold element and in the meantime performing an ordinary linear algorithm for the remaining samples. Optimal properties of this algorithm are studied and a criterion for selecting a proper threshold is discussed. A computer-simulated experiment in character recognition is presented to illustrate a possible application for the nonlinear algorithm. Results of this experiment have shown significant improvement over an ordinary linear algorithm in handling character samples with various imperfections due to the poor quality in writing or printing encountered in practice.

This paper presents an algorithm for pattern classification using the concept of characteristic vector of a Boolean function first proposed by Dertouzos for single-threshold element realization. It is shown that the algorithm is convergent for a finite number of iterations. Depending on the choice of a parameter, three correction rules which are the fixed increment rule, the absolute correction rule and the fractional correction rule can be obtained from the algorithm. The quantity (b − bj) used in the algorithm is shown to be equivalent to the augmented pattern used in pattern classification in the literature.

The Nearest Neighbour Classifier accepts patterns in the form of points in a descriptor space and classifies them by reference to a number of stored labelled points called locates. Training procedures for such classifiers have in the past consisted of a set of rules for selecting various members of the training set for use as locates. This paper describes an alternative approach in which the locates do not necessarily represent real patterns, and the training procedure consists of adjusting the positions of the locates so as to optimise the decision surface.

A set S of clauses is said to be substitutably unsatisfiable if and only if there exist a set M of variants (copies) of clauses in S and a substitution @q such that [email protected] is truth-functionally unsatisfiable. In this paper, we have introduced ''variable-constrained resolution,'' or V-resolution for short. Using V-resolution, it is guaranteed that the empty clause @? can always be derived from a substitutably unsatisfiable set of clauses. Comparisons between (ordinary) resolution and V-resolution are given. V-resolution has been implemented in a LISP program. Experimental results show that V-resolution is much more efficient than resolution.

This paper is concerned with the analysis of a class of problems involving mixture variates (functions of several random variables). The explicit inclusion of a mixture as a separate category in pattern recognition problems is studied and plausible forms for the mixing densities in such problems are derived.

The point of departure in this paper is the definition of a language, L, as a fuzzy relation from a set of terms, T = x, to a universe of discourse, U = y. As a fuzzy relation, L is characterized by its membership function @m"L:T x U -> [0,1], which associates with each ordered pair (x,y) its grade of membership, @m"L(x,y), in L. Given a particular x in T, the membership function @m"L(x,y) defines a fuzzy set, M(x), in U whose membership function is given by @m"M"("x")(y) = @m"L(x,y). The fuzzy set M(x) is defined to be the meaning of the term x, with x playing the role of a name for M(x). If a term x in T is a concatenation of other terms in T, that is, x = x"1 ... x"n, x"i @e T, i = 1,...,n, then the meaning of x can be expressed in terms of the meanings of x"1,...,x"n through the use of a lambda-expression or by solving a system of equations in the membership functions of the x"i which are deduced from the syntax tree of x. The use of this approach is illustrated by examples.

This paper establishes two concepts of identifiability for the unknown parameters of linear, constant, stable, discrete-time dynamic systems. Necessary and sufficient conditions are obtained for the system parameters to be identifiable in the deterministic sense (Theorems 1-2) and stochastically identifiable (Theorem 3). These conditions are shown to be significant in studying the asymptotic properties of maximum likelihood estimators (Corollary 3.1). Generalizations of these concepts to include the identification of the system's initial state are also discussed (Theorems 2a and 3a).

This paper describes the proceedings of the workshop on Biomedical Statistics and Computation of the Fourth Annual Symposium on the Interface. This was a two day symposium held at the University of California at Irvine on September 17 & 18, 1970. It was sponsored jointly by the ACM, ASA, and the California Mathematics Council. Dr. Tarter chaired this workshop. The thirty participants in this workshop represented all of the disciplines which combine to become Biocomputer Statistics. There were physicians both interested in and active in computer applications to medicine, directors of biomathematics and biostatistics programs, computer specialists, and mathematical and applied statisticians, as well as medical students. It should be noted that the three other workshops at the Interface Symposium were more concerned with computer hardware and software and thus attracted most of the pure computer specialists. This tended to weight this workshop with a large proportion of statisticians.

The paper points out that under certain conditions the Nearest Neighbour and Potential Function Classifiers produce similar decision surfaces. A new classifier is defined which is closely related to the Nearest Neighbour Classifier, and which implements a piecewise hyperspherical decision surface. This decision surface is similar to that produced by a Potential Function Classifier under a wider variety of conditions.

This paper extends the concept of fuzzy set, introduced by L. A. Zadeh, to a time dependent setting. Time dependence enters into a number of areas including planning and pattern recognition. Results are given for topologies of fuzzy sets and applied to shadows. Time dependent fuzzy relations are defined and discussed.

In many realistic estimation problems, the presence of the state vector to be estimated is uncertain. In this instance, use of the linear sequential conditional mean filter algorithms (the Kalman filter) results in suboptimum performance. This paper derives nonlinear sequential filter algorithms for conditional mean estimation of a Gauss Markov process when there is uncertainty as to the presence of the Gauss Markov process (signal process) in the observation. Associated estimation error variances are determined and a simple example is considered.