Showing papers in "Information & Computation in 1976"
TL;DR: This paper provides a methodology for solution of certain basic fuzzy relational equations, with fuzzy sets defined as mappings from sets into complete Brouwerian lattices, covering a large class of types of fuzzy sets.
Abstract: This paper provides a methodology for solution of certain basic fuzzy relational equations, with fuzzy sets defined as mappings from sets into complete Brouwerian lattices, covering a large class of types of fuzzy sets.
986 citations
TL;DR: This paper investigates the algebraic structures of fuzzy grades under the operations of join ⊔, meet ⊓, and negation ┐ which are defined by using the extension principle, and shows that convex fuzzy grades form a commutative semiring and normal convex fuzzies form a distributive lattice under ⊢ and ⊡.
Abstract: The concept of fuzzy sets of type 2 has been defined by L. A. Zadeh as an extension of ordinary fuzzy sets. The fuzzy set of type 2 can be characterized by a fuzzy membership function the grade (or fuzzy grade) of which is a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1]. This paper investigates the algebraic structures of fuzzy grades under the operations of join ⊔, meet ⊔, and negation ┐ which are defined by using the extension principle, and shows that convex fuzzy grades form a commutative semiring and normal convex fuzzy grades form a distributive lattice under ⊔ and ⊓. Moreover, the algebraic properties of fuzzy grades under the operations and which are slightly different from ⊔ and ⊓, respectively, are briefly discussed.
725 citations
TL;DR: By counting the number of polynomials which are affine equivalent to each representative polynomial in the set, this work derives explicit formulas for the enumerators of all weights less than 2.5d of Reed—Muller codes, where d is the minimum weight.
Abstract: Let Pr be the set of all polynomials of degree r in m variables over GF(2). Polynomial ƒ in Pr is said to be affine equivalent to polynomial g in Pr, if ƒ is transformable to g by an invertible affine transformation of the variables. Any polynomial of weight less than 2m−r+1 + 2m−r−1 in Pr is shown to have a simple structure. By using this fact, we find out a set of representative polynomials such that any polynomial of weight less than 2m−r+1 + 2m−r−1 in Pr is affine equivalent to one and only one polynomial of the set. By counting the number of polynomials which are affine equivalent to each representative polynomial in the set, we derive explicit formulas for the enumerators of all weights less than 2.5d of Reed—Muller codes, where d is the minimum weight.
86 citations
TL;DR: It is proved that two notions of being “balanced” and being "hard" are necessary and sufficient conditions for global maps C ; → C and C → C of tessellation automata to be one to one, respectively.
Abstract: In this paper, we introduce two notions of being “balanced” and being “hard.” And we prove that these notions are necessary and sufficient conditions for global maps C ; → C and C → C of tessellation automata to be one to one, respectively, where CC denotes the set of all the (finite) configurations of tessellation automata.
65 citations
TL;DR: An algorithm for finding an optimal Bayesian estimate of an unknown image from the projection data is proposed, it is shown that the algorithm is geometrically convergent, reports on its computer implementation, and its performance on a medical problem is demonstrated.
Abstract: We attack the problem of recovering an image (a function of two variables) from experimentally available integrals of its grayness over thin strips. This problem is of great importance in a large number of scientific areas. An important version of the problem in medicine is that of obtaining the exact density distribution within the human body from X-ray projections. This paper proposes an algorithm for finding an optimal Bayesian estimate of an unknown image from the projection data, shows that the algorithm is geometrically convergent, reports on its computer implementation, and demonstrates its performance on a medical problem.
51 citations
TL;DR: It is shown that the relative succinctness that may be achieved by describing deterministic context-free languages by general unambiguous grammars rather than by deterministic pushdown automata is not bounded by any recursive function.
Abstract: It is shown that the relative succinctness that may be achieved by describing deterministic context-free languages by general unambiguous grammars rather than by deterministic pushdown automata is not bounded by any recursive function.
51 citations
TL;DR: The main part of the paper is devoted to a proof of the properties of the “dual” families of nonlinear codes which were announced in a letter by Goethals (1974) .
Abstract: An investigation is made of the family of codes which are supercodes of the first-order and subcodes of the second-order Reed—Muller codes. These codes are in a one-to-one correspondence with subsets of alternating bilinear forms and it is shown how their distance enumerators can be obtained. A nice duality relation is defined on the set of linear codes in this family which relates their weight enumerators. The best codes of this family are defined and constructed, most of which are new nonlinear codes. The main part of the paper is devoted to a proof of the properties of the “dual” families of nonlinear codes which were announced in a letter by Goethals (1974) .
41 citations
TL;DR: An in depth look at a complete statistical description of linear—quadratic—Gaussian (LQG) performance measures and exhibits the essential quadratic nature of the LQG class of systems.
Abstract: An in depth look at a complete statistical description of linear—quadratic—Gaussian (LQG) performance measures is presented in this paper. The development leads to computationally tractable formulas that form the basis for design performance analysis. The complete statistical description of performance developed here exhibits the essential quadratic nature of the LQG class of systems.
38 citations
TL;DR: In this paper, a new family of ω-languages (sets of infinite sequences) associated with context-free languages and pushdown automata are introduced, and their basic properties, such as inclusion relations, closure under the Boolean operations and periodicity, are studied and compared with the corresponding properties of the families accepted by finite automata.
Abstract: New families of ω-languages (sets of infinite sequences) associated with context-free languages and pushdown automata are introduced. Their basic properties, such as inclusion relations, closure under the Boolean operations and periodicity, are studied and compared with the corresponding properties of the families of ω-languages accepted by finite automata. Moreover, a number of solvability and unsolvability results are proved. The results obtained imply that there is a definite difference between the family of ω-languages accepted by pushdown automata and the family associated with context-free languages.
36 citations
TL;DR: A sharpened sampling theorem is presented and results are used to derive similar theorems for bandlimited random processes.
Abstract: Several classes of bandlimited functions are defined and characterized in a variety of ways and the relationships between these classes are explored. A sharpened sampling theorem is presented. These results are used to derive similar theorems for bandlimited random processes.
34 citations
TL;DR: A solution to the closely related problem of developing a program which can be taught a language and is capable of handling problems of the order of typical programming languages is discussed.
Abstract: The classical definition of the grammatical inference problem is to find a grammar for a language, given a finite sample from that language. We discuss a solution to the closely related problem of developing a program which can be taught a language. The method is highly interactive and attempts to make maximum use of negative information. It has proved to be efficacious on context-free languages such as the one of expressions over the alphabet of letters, digits, parentheses, and arithmetic operators. We believe it is capable of handling problems of the order of typical programming languages.
TL;DR: It is shown that there are languages recognized by deterministic three-head one-way nonwriting finite automata that cannot be recognized by any nondeterministic two-headOne-way writing finite automaton.
Abstract: The family of languages recognized by one-way multihead writing finite automata is investigated. It is shown that there are languages recognized by deterministic three-head one-way nonwriting finite automata that cannot be recognized by any nondeterministic two-head one-way writing finite automaton.
TL;DR: It is shown that many automata-theoretic concepts, e.g., separation, connectivity, retrievability, strong connectivity, etc., have standard topological analogs under this topology and many results concerning these concepts are direct consequences of this observation.
Abstract: Let A = ( Q, X, δ ) be an X -automaton, with Q its state set. For a subset B ⊆ Q , the source of B is defined as σB = { q Ȩ Q | δ ( q, x ) Ȩ B for some x Ȩ X }. The source turns out to be a closure operator for Q and defines a topology on Q , vis. B ⊆ Q is closed iff B = σB . It is shown that many automata-theoretic concepts, e.g., separation, connectivity, retrievability, strong connectivity, etc., have standard topological analogs under this topology and many results concerning these concepts are direct consequences of this observation.
TL;DR: A proof of the duality in a special case is presented, and the interpretation of the logical calculus for sequential machine interconnection is given, which specializes to known open loop minimal realization theorems.
Abstract: Although the distinction between software and hardware is a posteriori, there is an a priori distinction that masquerades as the software—hardware distinction. This is the distinction between procedure interconnection, the semantics of flow chart diagrams, which is known to be described by the regular expression calculus; and system interconnection, the semantics of network diagrams, which is described by a certain logical calculus, dual to a calculus of regular expressions. This paper presents a proof of the duality in a special case, and gives the interpretation of the logical calculus for sequential machine interconnection. A minimal realization theorem for feedback systems is proved, which specializes to known open loop minimal realization theorems.
TL;DR: It is proved that the necessary and sufficient condition for maximizing the mutual information between a state and the estimate is to minimize the entropy of the estimation error.
Abstract: The purpose of this paper is to study estimation problems from the viewpoint of the information theory. It is proved that the necessary and sufficient condition for maximizing the mutual information between a state and the estimate is to minimize the entropy of the estimation error. Based on this relation between the mutual information and the error entropy, the Kalman filter for a discrete-time linear system is derived by an application of the information theory. Furthermore, the optimal filter for a continuous-time linear system is also constructed by an analogous approach.
TL;DR: It is shown that time-invariant algorithms can use knowledge of the sample size to obtain lower error rates than in the infinite sample problem and the minimal error rate achievable after N samples goes to zero.
Abstract: This paper explores the structure and performance of optimal finite state machines used to test between two simple hypotheses. It is shown that time-invariant algorithms can use knowledge of the sample size to obtain lower error rates than in the infinite sample problem. The existence of an optimal rule is established and its structure is found for optimal time-varying algorithms. The structure of optimal time-invariant rules is partially established. The particular problem of testing between two Gaussian distributions differing only by a shift is then examined. It is shown that the minimal error rate achievable after N samples goes to zero like exp[−(ln N)1/2].
TL;DR: The notion of an iterated Γ map, that is, a nondeterministic generalized sequential machine with accepting states, is introduced, which leads to the notions of Γ systems and languages.
Abstract: The notion of an iterated Γ map, that is, a nondeterministic generalized sequential machine with accepting states, is introduced, which leads to the notions of Γ systems and languages. It is shown that the family of Γ languages is equal to the family of recursively enumerable languages. A detailed study is presented of the properties of subfamilies of the family of L languages and their relationship with the various families of L languages is investigated. An example result is that the family of e -free context—sensitive languages is equal to the family of languages generated by extensions of propagating deterministic Γ systems with at most three Γ maps.
TL;DR: This note discusses some of these inequivalence results, and points out a relationship between local picture processing operations and a special class of finite-state languages, the strictly locally testable languages.
Abstract: Finite-state two-dimensional languages are much less well-behaved than their one-dimensional counterparts. Various definitions of language classes that are equivalent for strings are inequivalent for arrays. This note discusses some of these inequivalence results, and also points out a relationship between local picture processing operations and a special class of finite-state languages, the strictly locally testable languages.
TL;DR: It is shown that there is always a linear encoding rule which is one part of the optimum pairs of permissible encoding—decoding rules and an exact expression for the performance of this type of suboptimal decoder is examined.
Abstract: We are concerned with the use of linear codes in the channel coding part of a numerical data transmission system in which the performance criterion is the mean-square error. The optimum decoding rule is developed for a fixed code over the finite field GF ( q ). It is a mapping from the channel alphabet into the real numbers and as such combines both the error-correcting and digital-to-analog converter functions found in most systems. We employ abstract Fourier analysis on groups to determine an optimum 1—1 encoding rule to be used with the corresponding optimum decoder. A procedure for the simultaneous optimization over both rules is presented. We show that there is always a linear encoding rule which is one part of the optimum pairs of permissible encoding—decoding rules. A system which implements the optimum decoding rule is detailed. In this realization the outputs from a bank of generalized bandpass filters are arithmetically combined with the output of a normalizing lowpass filter to produce the optimum estimate in the decoder. These filters are mechanized in the Fourier domain by weighting the spectrum of the indicator function of the received word by selected values of the transforms of the channel transition probabilities. This approach uses complex-valued arithmetic operations as opposed to finite field operations usually found in decoders. We examine a method for reducing the complexity of the decoder and determine an exact expression for the performance of this type of suboptimal decoder.
TL;DR: It is proved that deterministic array automata which can travel across the unmarked portions of the input array, but not write on these portions, are no more powerful than deterministic arrays which are restricted to the marked portion of their input arrays.
Abstract: In this communication, it is proved that deterministic array automata which can travel across the unmarked portions of the input array, but not write on these portions, are no more powerful than deterministic array automata which are restricted to the marked portion of their input arrays. The similar question for nondeterministic automata is still open. The one-dimensional case is also examined.
TL;DR: In this work, an algorithm is presented for deciding whether an arbitrary homomorphism preserves star height or not and it is shown that all homomorphisms which preserve star height are injective.
Abstract: A homomorphism ƒ from the set of words over a finite alphabet Σ 1 to the set of words over another finite alphabet Σ 2 is said to preserve star height if for all regular events R ⊆ Σ 1 * , the star height of R is equal to that of ƒ ( R ). An algorithm is presented for deciding whether an arbitrary homomorphism preserves star height or not: in our result, the injectivity of homomorphisms plays an important role, and it is shown that all homomorphisms which preserve star height are injective.
TL;DR: The obtained results, as compared with the present status of the regulator theory for linear dynamical systems, allow us to point out strong similarities as well as definite differences, the latter being basically due to the essentially nonlinear nature of the systems considered herein.
Abstract: This paper deals with the problem of controlling the state of a finite automaton to some preassigned value; namely, the regulator problem. First, the simplest situation where the state of the system is currently known is briefly considered and the corresponding solution pointed out. Then, the much more difficult problem arising in the case of nontrivial output transformation is analyzed. This obviously calls for the preliminary solution of the state reconstruction problem in a partially or totally uncertain environment. The obtained results, as compared with the present status of the regulator theory for linear dynamical systems, allow us to point out strong similarities as well as definite differences, the latter being basically due to the essentially nonlinear nature of the systems considered herein.
TL;DR: When answers to questions are not necessarily truthful, a charge per question of the channel capacity of the associated channel and an appropriate terminal charge are shown to imply that the average cost of a questionnaire is bounded below by the Shannon entropy of a given a priori density, thus further exploring the relationship between optimal diagnostic questioning and information theory.
Abstract: This paper examines the problem of determining a sequence of questions, or a questionnaire, in order to optimally identify the unknown state of a system for the case where the answers to the questions asked are not necessarily truthful. The problem is modeled by a slight generalization of the partially observed Markov optimization problem for the special case where the underlying process is not allowed to change state during the period of questioning. In this formulation, each question is considered equivalent to a discrete memoryless channel and its associated channel matrix. Recent results in questionnaire theory and an application of restriction set theory to diagnosis are shown to be essentially equivalent to the case where all available channel matrices are deterministic, i.e., all questions receive truthful replies, and further results for this particular case are determined. When answers to questions are not necessarily truthful, a charge per question of the channel capacity of the associated channel and an appropriate terminal charge are shown to imply that the average cost of a questionnaire is bounded below by the Shannon entropy of a given a priori density, thus further exploring the relationship between optimal diagnostic questioning and information theory. Conditions sufficient for an optimal questionnaire to attain this lower limit are then presented.
TL;DR: The main properties of the generalized t -designs introduced by Delsarte (1973) are studied and used in a majority decoding method which slightly differs from Massey's threshold decoding.
Abstract: The main properties of the generalized t -designs introduced by Delsarte (1973) are studied and used in a majority decoding method which slightly differs from Massey's threshold decoding. The paper also contains a number of results concerning the existence of such designs in codes and a list of some codes which can be decoded by our method.
TL;DR: A balanced multiple valued filing scheme of order 2 is constructed analogous to that of Yamamoto, Teramoto, and Futagami (1972) by deleting a spread in a t -dimensional projective geometry.
Abstract: A balanced multiple valued filing scheme of order 2 is constructed analogous to that of Yamamoto, Teramoto, and Futagami (1972) by deleting a spread in a t -dimensional projective geometry. PG( t, s ), where t + 1 is not prime. Although the bucket structure is the same, the storage and retrieval algorithms are based on a difference set representation. The system is characterized by a set of parameters whose cardinality is a linear function of the number of attribute values. The buckets are identified by two parameters from this set which can easily be computed directly from the attribute values. The storage algorithm identifies the buckets corresponding to a record without scanning the list of buckets. The ideas are illustrated with examples based on PG(5, 2).
TL;DR: This article proposes to define and to study the useful information and the useful routing length of a questionnaire in the general case, and to exhibit some properties relative to a product of questionnaires, corresponding to dependent or independent processes.
Abstract: A questionnaire is an inquiry process, using a probabilistic latticoid. We suppose that a positive valuation, called an utility, characterizes every terminal vertex. The useful information and the useful routing length of a questionnaire have been introduced in a particular case. We propose, in this article, to define and to study these quantities in the general case, and to exhibit some properties relative to a product of questionnaires, corresponding to dependent or independent processes.
TL;DR: The general theory of structural equivalence of phrase-structure languages is applied to a particular subclass of context-free languages, the class of tree language counterparts of regular languages.
Abstract: It is proposed that structural equivalence of phrase-structure languages be defined by means of introducing, for each such language, a class of topological structures on the language. More specifically, given a phrase-structure language (either as a set of trees or as a set of strings), we introduce a class of topological spaces associated with finite sets of “phrases.” A function from one language to another, where both are equipped with such classes of topological spaces, is said to be structurally continuous, if for any topological space belonging to the first, there is a space belonging to the second such that the function is continuous with respect o these spaces. Then phrase-structure languages, or grammars that generate such languages, may be classified into structurally homeomorphic types in the obvious way. Two different methods of topologizing phrase-structure languages (one dependent on the other) are considered, and it is shown that for the class of context-free languages, one method provides a finer classification of languages (or grammars) than the other. In Part 2 we apply the general theory to a particular subclass of context-free languages, the class of tree language counterparts of regular languages.
TL;DR: It is shown that regular control on regular and linear kolam array grammars does not increase theirGenerative capacity, while CF and CS controls increase their generative capacity.
Abstract: It is shown that regular control on regular and linear kolam array grammars does not increase their generative capacity, while CF and CS controls increase their generative capacity.
TL;DR: The behavior of an information-lossless finite-state automaton (herein called a complete automaton) when a complete sequence is the input is examined and the results obtained are used to study the partial ordering defined by Rayna on degrees of finite- state transformability.
Abstract: Let ( Q, Σ, δ, ω ) be a finite-state automaton with input-output alphabet Σ . Let Σ + be the free semigroup generated by Σ and Σ N be the set of all infinite sequences of elements of Σ . Rayna, in “Degrees of Finite-State Transform-ability,” Information and Control (1974), calls a sequence x ∈ Σ N complete if and only if every A ∈ Σ + is contained in x . We call a degree of finite-state transformability complete if and only if it contains a complete sequence. Huffman, in “Canonical Forms for Information-Lossless Finite-State Logical Machines,” IRE Transactions on Circuit Theory (1959), calls a finite-state automaton information-lossless iff there exist no two states q i and q f and no two different equal-length input blocks A, B ∈ ∑ + and an output block C ∈ Σ + such that δ ( q i , A ) = q f = δ ( q i , B ) and ω ( q i , A ) = C = ω ( q i , B ). In this paper we examine the behavior of an information-lossless finite-state automaton (herein called a complete automaton) when a complete sequence is the input and use the results obtained to study the partial ordering [ x ] ⩾ [ y ] defined by Rayna on degrees of finite-state transformability. In particular we show that (1) if [ x ] is a complete degree and [ y ] is an incomplete degree with [ x ] > [ y ], then there exists a complete degree [ x ′] and an incomplete degree [ y ′] such that [ x ] > [ x ′] > [ y ′] > [ y ], and (2) a complete degree [ x ] has a cover, i.e., a complete degree [ x ′] > [ x ] with no complete degree strictly in between, if and only if every complete degree has a cover.
TL;DR: A locally linear language is one whose words can be pumped up almost anywhere and there are striking similarities between the properties of families of locally linear languages and slip families.
Abstract: A locally linear language is one whose words can be pumped up almost anywhere. There are striking similarities between the properties of families of locally linear languages and slip families. In particular, there exists a largest AFL of locally linear languages.