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Showing papers in "Information & Computation in 1973"


Journal ArticleDOI
TL;DR: A method is suggested to calculate the weight distributions of cosets of a code and a “dual concept” of that of perfect codes is presented and examined in detail.
Abstract: Starting from the distance distribution of an unrestricted code and its Mac Williams transform, one defines four parameters that, in the linear case, reduce to the minimum weight and the number of distinct weights of the given code and of its dual. In the general case, one exhibits the combinatorial meaning of these parameters and, using them, one obtains various results on the distance properties of the code. In particular, a method is suggested to calculate the weight distributions of cosets of a code. A “dual concept” of that of perfect codes is also presented and examined in detail.

310 citations


Journal ArticleDOI
TL;DR: The general form that the weight distribution of a self-dual code over GF (2) and GF (3) can have is described and an explicit formula for this weight distribution when the minimum distance between codewords is made as large as possible is given.
Abstract: Gleason has described the general form that the weight distribution of a self-dual code over GF (2) and GF (3) can have. We give an explicit formula for this weight distribution when the minimum distance d between codewords is made as large as possible. It follows that for self-dual codes of length n over GF (2) with all weights divisible by 4, d ⩽ 4[ n /24] + 4; and for self-dual codes over GF (3), d ⩽ 3[ n /12] + 3; where the square brackets denote the integer part. These results improve on the Elias bound. A table of this extremal weight distribution is given in the binary case for n ⩽ 200 and n = 256.

279 citations


Journal ArticleDOI
TL;DR: Two sufficient conditions of global observability of nonlinear systems are presented: the ratio condition which is the generalization of Fujisawa and Kuh's (1971) ratio condition of circuit theory and the strongly positive semidefinite condition.
Abstract: The purpose of this paper is to investigate the problem of observability of nonlinear systems. Two sufficient conditions of global observability of nonlinear systems are presented: (1) the ratio condition which is the generalization of Fujisawa and Kuh's (1971) ratio condition of circuit theory, (2) the strongly positive semidefinite condition. The relationships between these two conditions as well as the condition of positive definiteness of Fitts (1970) are given.

140 citations


Journal ArticleDOI
TL;DR: Generative models of picture languages with array rewriting rules are presented and each letter of the alphabet of different sizes can be generated by a context-free array grammar.
Abstract: Generative models of picture languages with array rewriting rules are presented. The rewriting rules are regular, context-free or context-sensitive with arrays of terminals in the place of strings of terminals. Derivations are restricted by the condition for row and column catenation. The grammars describe a wide variety of pictures and are more powerful than the matrix grammars for digital pictures introduced in our earlier paper. A distinct hierarchy is shown to exist between the different classes introduced. The models are closed under reflection (about base and rightmost vertical), halfturn, quarter-turn, transpose, and conjugation. Further closure properties such as union, product, star and homomorphism are examined. The models can be applied to generate several interesting patterns of kolam and to describe the repetitive patterns of two-dimensional crystallography. Each letter of the alphabet of different sizes can be generated by a context-free array grammar.

139 citations


Journal ArticleDOI
TL;DR: The problem of the existence of a unifying substitution between two terms is shown to be undecidable, even if the objects of the language are restricted to third order.
Abstract: The problem of the existence of a unifying substitution between two terms is considered in type theory. This problem is shown to be undecidable, even if we restrict the objects of the language to third order. This means that we are not able to recognize whether two terms have a common instance or not. This result has important implications for the mechanization of higher order logic.

128 citations


Journal ArticleDOI
TL;DR: TOL languages form an infinite hierarchy with respect to “natural” complexity measures introduced in this paper, and are contained in the family of context-free programmed languages.
Abstract: We discuss a family of systems and languages (called TOL) which have originally arisen from the study of mathematical models for the development of some biological organisms. From a formal language theory point of view, a TOL system is a rewriting system where at each step of a derivation every symbol in a string is rewritten in a context-free way, but different rewriting steps may use different sets of production rules and the language consists of all strings derivable from the single fixed string (the axiom). The family of TOL languages (as well as its different subfamilies considered here) is not closed with respect to usually considered operations; it is “incomparable” with context-free languages, but it is contained in the family of context-free programmed languages. TOL languages form an infinite hierarchy with respect to “natural” complexity measures introduced in this paper.

84 citations


Journal ArticleDOI
R.M. Capocelli1, A. De Luca1
TL;DR: A measure of the total amount of uncertainty that arises in making decisions has been proposed in the general case and the Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions are derived.
Abstract: The problem of making decisions to classify the objects of a certain universe into two or more suitable classes has been considered in the setting of fuzzy sets theory. A measure of the total amount of uncertainty that arises in making decisions has been proposed in the general case. This quantity reduces to the “entropy” of a fuzzy set in the case of two classes. Other quantities which play a relevant role in this theory are the “energy” and the “effective power” of a fuzzy set, defined as ∑ i = 1 N w i f i a n d φ ∑ i = 1 N f i , respectively, where w is a nonnegative weight function and φ a nonnegative constant. If w = constant and φ ≠ 0, the energy is proportional to the effective power and, therefore, to the “power” of the fuzzy set. The maximum of the uncertainty has been calculated in some cases of interest, keeping constant the total energy and effective power. In particular the Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions are derived. Some applications to decision theory are considered in the case of both deterministic and probabilistic decisions. Finally, the analogies that exist between the previous concepts and the thermodynamic ones are discussed.

80 citations


Journal ArticleDOI
TL;DR: Growth functions of informationless Lindenmayer systems are investigated from the point of view of integral sequential word functions and some of the inclusion relations between language families do not remain valid for the corresponding families of growth functions.
Abstract: Growth functions of informationless Lindenmayer systems are investigated from the point of view of integral sequential word functions. Algorithms are obtained for the solution of equivalence, minimization and construction problems. It is found out that some of the inclusion relations between language families do not remain valid for the corresponding families of growth functions. Some results concerning context-dependent Lindenmayer systems, as well as growth relations of OL-systems are also obtained.

63 citations


Journal ArticleDOI
TL;DR: A criterion to measure derivational complexity of formal grammars and languages is proposed and discussed and it is shown that for each nonnegative integer k, there exists a context\3-free language whose rank is k.
Abstract: A criterion to measure derivational complexity of formal grammars and languages is proposed and discussed. That is, the associate language and the L-associate language are defined for a grammar such that the former represents all the valid derivations and the latter represents all the valid leftmost derivations. It is shown that for any phrase\3-structure grammar, the associate language is a contex\3-sensitive language and the L\3-associate language is a context\3-free language. Necessary and sufficient conditions for an associate language to be a regular set and to be a context\3-free language are found. The idea in the above necessary and sufficient conditions is extended to the notion of “rank≓ for a measure of derivational complexity of context\3-free grammars and languages. It is shown that for each nonnegative integer k, there exists a context\3-free language whose rank is k. The paper also includes a few solvable decision problems concerning derivational complexity of grammars.

53 citations


Journal ArticleDOI
TL;DR: Two classes of restricted top-down parsing algorithms modeling “recursive descent” are considered and it is shown that the smaller class recognizes all deterministic context free languages, and that both classes can be simulated in linear time on a random access machine.
Abstract: Two classes of restricted top-down parsing algorithms modeling “recursive descent” are considered. We show that the smaller class recognizes all deterministic context free languages, and that both classes can be simulated in linear time on a random access machine. Certain generalizations of these parsing algorithms are shown equivalent to the larger class. Finally, it is shown that the larger class has the property that loops and other “failures” can always be eliminated.

43 citations


Journal ArticleDOI
TL;DR: A probabilistic algorithm is presented to calculate the Boolean product of two n × n Boolean matrices using an expected number of elementary operations of 0(n2).
Abstract: A probabilistic algorithm is presented to calculate the Boolean product of two n × n Boolean matrices using an expected number of elementary operations of 0(n2). Asymptotically in n, almost all pairs of matrices may be multiplied using this algorithm in 0(n2+e) elementary operations for any e > 0.

Journal ArticleDOI
TL;DR: The theory of constructing observer–estimators for linear, continuous-time systems is described and the case that some observations are noise-free while others are noisy is considered.
Abstract: This paper describes the theory of constructing observer–estimators for linear, continuous-time systems. Both deterministic and stochastic cases are considered; in particular, the case that some observations are noise-free while others are noisy is considered. Asymptotic properties for both time-varying and time-invariant systems are analyzed and the influence of observability and detectability assumptions is considered.

Journal ArticleDOI
TL;DR: The asymptotic stability of solutions of a class of third and fourth order delaydifferential equations is presented and the results are an extension to Krasovskii's second order system.
Abstract: The asymptotic stability of solutions of a class of third and fourth order delaydifferential equations is presented. The results are an extension to Krasovskii's second order system.

Journal ArticleDOI
TL;DR: It is shown that any array language has an array grammar in which all terminal arrays are in fact connected; and that the set of terminal, finite connected components of non #'s that occur in the sentential forms of any array grammar is an array language.
Abstract: The language of an array grammar has been defined as the set of finite, connected terminal arrays, surrounded by #'s, that can be derived from an initial S surrounded by #'s. In the first section of this paper, it is shown that any array language has an array grammar in which all terminal arrays are in fact connected; and that the set of terminal, finite connected components of non #'s that occur in the sentential forms of any array grammar is an array language. Thus several possible ways of defining an array language are in fact all equivalent. In the second section of the paper, it is shown that array grammars need not use #'s as context. It is also shown that array grammars which start with arbitrary initial arrays of S's, and neither create nor destroy #'s, have exactly the same power as monotonic array grammars. In the third section, parallel array grammars are defined, and it is shown that any (monotonic) sequential array language is a (monotonic) parallel array language and vice versa.

Journal ArticleDOI
TL;DR: A general theory of prior probability models is presented, valid for both discrete and continuous random variables, even when the prior information about them has been obtained with errors.
Abstract: A general theory of prior probability models is presented, valid for both discrete and continuous random variables, even when the prior information about them has been obtained with errors. An example is included as an illustration.

Journal ArticleDOI
TL;DR: Several new types of grammars are introduced; some of them are proved to possess the same generative power as ordinary matrix Grammars; for some other types inclusion properties are obtained.
Abstract: The matrix grammar is a well-known concept of a grammar with restricted use of productions. By weakening the matrix restrictions imposed on context-free grammars several new types of grammars are introduced; some of them are proved to possess the same generative power as ordinary matrix grammars; for some other types inclusion properties are obtained.

Journal ArticleDOI
TL;DR: A characterization theorem for the measure directed divergence is given by assuming intuitively reasonable postulates and with the help of functional equations.
Abstract: Shannon's entropy was characterized by many authors by assuming different sets of postulates. One other measure associated with Shannon's entropy is directed divergence or information gain. In this paper, a characterization theorem for the measure directed divergence is given by assuming intuitively reasonable postulates and with the help of functional equations.

Journal ArticleDOI
TL;DR: It is demonstrated that the class of self-dual codes over GF( q), for q an arbitrary prime power, contains members that satisfy a modified Varshamov—Gilbert bound that is asymptotically the same as the usual bound.
Abstract: We demonstrate that the class of self-dual codes over GF( q ), for q an arbitrary prime power, contains members that satisfy a modified Varshamov—Gilbert bound that is asymptotically the same as the usual bound. This is an extension of the results in MacWilliams, Sloane, and Thompson (1972) for a subclass of self-dual codes over GF(2).

Journal ArticleDOI
TL;DR: An equation is derived for the posterior statistics of a Markov process that modulates the transition rates of an observed jump process that forms the basis of a nonlinear filtering theory for observed jump processes that is the counterpart of the Stratonovich\3-Kushner filtering theories for nonlinear observations in additive white Gaussian noise.
Abstract: An equation is derived for the posterior statistics of a Markov process that modulates the transition rates of an observed jump process. This equation forms the basis of a nonlinear filtering theory for observed jump processes that is the counterpart of the Stratonovich\3-Kushner filtering theory for nonlinear observations in additive white Gaussian noise.

Journal ArticleDOI
TL;DR: Whittaker's sampling theorem is extended to periodic and not necessarily simultaneous sampling of band-limited functions g ( t ) and their derivatives g l ( t ); l = 1,…, n − 1.
Abstract: Whittaker's sampling theorem is extended to periodic and not necessarily simultaneous sampling of band-limited functions g ( t ) and their derivatives g l ( t ); l = 1,…, n − 1. Uniqueness is established and explicit results for function reconstruction are provided. Further generalizations include arbitrary shifts of a finite number of sample points and sampling in the complex domain.


Journal ArticleDOI
TL;DR: A study is made of conditions on a language L which ensure that the smallest intersection-closed full AFL containing L (written ℱ ^ ∩ ( L ) ) does or does not contain all recursively enumerable languages.
Abstract: A study is made of conditions on a language L which ensure that the smallest intersection-closed full AFL containing L (written ℱ ^ ∩ ( L ) ) does or does not contain all recursively enumerable languages. For example, it is shown that if L = {ani/j⩾0} and limi→∞ inf(ni+1/ni) > 1, then ℱ ^ ∩ ( L ) contains all recursively enumerable languages. On the other hand, it is shown that if L ⊆ a* and the ratio of the number of words in L of length less than n to n goes to 1 as n → ∞, then ℱ ^ ∩ ( L ) does not contain all recursively enumerable languages.

Journal ArticleDOI
TL;DR: It is shown that by allowing finite rather than empty sets as the starting points in the recursive process for obtaining a language defined by the ALGOL-like equations, the authors get exactly the family of E0L-languages.
Abstract: E0L-languages arose in the study of developmental systems in biology. In this paper we characterize E0L-languages in terms of ALGOL-like equations. We show that by allowing finite rather than empty sets as the starting points in the recursive process for obtaining a language defined by the ALGOL-like equations, we get exactly the family of E0L-languages. Corollaries obtained include a normal form theorem, the elimination of e -rules for E0L-systems and a comparison with the extended definable languages.

Journal ArticleDOI
TL;DR: It is shown that Church random sequences must have very high minimal-program complexity when the computation resources are effectively restricted; whereas, it is known that when there are no restrictions on resources, there are Church random sequence with extremely low minimal- program complexity.
Abstract: In this paper we investigate the minimal-program complexity of several types of pseudo-recursive and pseudo-random sequences when the resources used for their computation are restricted. For several types of these sequences their minimal-program complexity increases enormously whenever the resources for their computation is restricted in any effective manner at all. In particular, it is shown that Church random sequences must have very high minimal-program complexity when the computation resources are effectively restricted; whereas, it is known that when there are no restrictions on resources, there are Church random sequences with extremely low minimal-program complexity.

Journal ArticleDOI
TL;DR: This paper provides a complete solution of the problem of finding the cycle representatives of a cyclic code for all cyclic codes having a parity check polynomial which satisfies either of the following conditions: (i) h ( x ) consists entirely of factors having exponent n, or (ii) the exponents of the factors of h (x ) are relatively prime to each other.
Abstract: The problem of finding the cycle representatives of a cyclic code is considered. In a recent paper, Tavares, Allard and Shiva, provided a solution to this problem for all cyclic codes having a parity check polynomial which satisfies either of the following conditions: (i) h ( x ) consists entirely of factors having exponent n , (ii) the exponents of the factors of h ( x ) are relatively prime to each other. For other cyclic codes, the solution remained incomplete. In this paper we provide a complete solution of the problem.

Journal ArticleDOI
TL;DR: It is shown that every type 1 language can be generated by an e-free simple matrix grammar (of order 2), if a certain leftmost restriction is imposed on the application of matrices.
Abstract: It is shown that every type 1 language can be generated by an e-free simple matrix grammar (of order 2), if a certain leftmost restriction is imposed on the application of matrices. Dropping the e-free restriction, every type 0 language can be obtained. This strengthens a result previously established for matrix grammars.

Journal ArticleDOI
TL;DR: A method of automatic document classification is optimized to give best agreement with manually assigned classification of a set of test documents to measure the indexing worth of additional keywords.
Abstract: A method of automatic document classification is optimized to give best agreement with manually assigned classification of a set of test documents. The effectiveness of the automatic classification depends on the correlation between the relevance and the mutual keyword content of pairs of documents. An expression is obtained to measure the indexing worth of additional keywords. The resolution of the automatic classification is dependent on the correlation between mutual keyword content and the uniqueness of the association of categories with documents.


Journal ArticleDOI
TL;DR: It is shown that each context-free matrix language is a homomorphic image of the intersection of a state language with unconditional transfer and a regular set.
Abstract: The context-free matrix grammar and the state grammar without any restriction in applying productions are considered. It turned out that these grammars are equivalent in the generative power. Another type of state grammar called the state grammar with unconditional transfer is introduced, and it is shown that each context-free matrix language is a homomorphic image of the intersection of a state language with unconditional transfer and a regular set.

Journal ArticleDOI
TL;DR: One of the most interesting and difficult (mathematically) open problems was whether or not null symbols can be dispensed without sacrificing the weak generative capacity, and the answer is negative.
Abstract: Several open problems concerning local adjunct languages are considered and solved. One of the most interesting (from a linguistic point of view) and difficult (mathematically) open problems was whether or not null symbols can be dispensed without sacrificing the weak generative capacity. This problem is solved and the answer is negative. Also considered are some problems concerning one-sided grammars, homomorphisms of languages (it is shown that local adjunct languages are not closed under homomorphism), β-linear languages and mixed adjunct grammars.