# Information and Control

Published by Elsevier

Online ISSN: 0019-9958

Published by Elsevier

Online ISSN: 0019-9958

Publications

Conference Paper

A real time computational method is presented for the identification of linear discrete dynamic systems with unknown parameters. It is shown that the method is globally convergent to the true parameters in a stochastic environment. Experimental simulation of second and fourth-order systems further affirms the practicality of the method.

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Article

In 0L-languages, words are produced from each other by the simultaneous transition of all letters according to a set of production rules; the context is ignored.(i)0L-languages are not closed under the operations usually considered.(ii)0L-languages over a one-letter alphabet are discussed separately; a characterization is given of a subclass.(iii)0L-languages are incomparable with regular sets, incomparable with context-free languages, and strictly included in context-sensitive languages.

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Article

We consider inductive inference of formal languages, as defined by Gold (1967), in the case of positive data, i.e., when the examples of a given formal language are successive elements of some arbitrary enumeration of the elements of the language. We prove a theorem characterizing when an indexed family of nonempty recursive formal languages is inferrable from positive data. From this theorem we obtain other useful conditions for inference from positive data, and give several examples of their application. We give counterexamples to two variants of the characterizing condition, and investigate conditions for inference from positive data that avoids “overgeneralization.”

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Article

In this paper explicit formulas for the weight enumerators for several classes of subcodes of the 2nd-order binary Reed-Muller codes are derived. A large set of the codes are shown to have the same weight enumerators. The classes of codes studied in this paper contain the (0, 2)th-order Euclidean Geometry codes and the codes studied by Berlekamp as subclasses.

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Article

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.

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Article

We consider the number of queries required to identify a regular set given an oracle for the set and some auxiliary information about the set. If the auxiliary information is n, the number of states of the canonical finite state acceptor for the language, then the upper and lower bounds on the number of queries are exponential in n. If the auxiliary information consists of a set of strings guaranteed to reach every live state of the canonical acceptor for the language, then the upper and lower bounds are polynomial in n and the size of the given set of strings. As a corollary, the problem considered by Pao and Carr (1978) is shown to be solvable in a polynomial number of queries.

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Article

This paper derives both necessary and sufficient conditions for the existence of the ε-property (Asymptotic Equipartition Property) for a finite alphabet source.It is shown that ergodicity (or even stationarity) are not necessary conditions for the existence of the ε-property. The class of sources which possess the ε-property is larger than the class of all ergodic sources and includes some nonstationary sources as well. The proof is elementary in the sense that neither the ergodic theorem nor martingales theorems are used.

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Article

In a foundational paper on the theory of automata A. W. Burks and H. Wang (1957) conjectured that a certain complexity measure involving the size of the strong components of a logical net formed a hierarchy for net behavior. This conjecture was established by Rhodes and Krohn. In this paper a strengthened version of the conjecture is proved by establishing that any logical net can be interpreted as a series-parallel composition of nets associated with its strong components. Some properties of the periodic behavior of machines, shown to be preserved under simulation and composition operations, are used to complete the proof. The relationship of this approach to algebraic proofs of series-parallel irreducibility is discussed.

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Article

This paper is devoted to the formulation and proof of an abstract alphabet version of the fundamental theorem of Shannon's rate distortion theory. The validity of the theorem is established for both discrete and continuous parameter information sources satisfying a certain regularity condition intermediate in restrictiveness between ergodicity and weak mixing. For ease of presentation, only single letter fidelity criteria are considered during the main development after which various generalizations are indicated.

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Article

A linear context-free language which is not acceptable by a finite probabilistic automaton is given, and it is shown that the family of stochastic languages is not closed under concatenation and homomorphism.

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Article

Multitape Turing machines which can use their storage tapes only as counters or as pushdown stores are investigated. The memory access restrictions are produced by regarding the machines as small computers (as in the formalism of Wang) and by restricting the instruction repertoires. Relationships are given linking machines which only accept or reject inputs and machines which emit output sequences as a function of their input. It is shown that without restrictions on computing time or amount of tape used that only six distinct classes of sets of strings (languages) are produced by the above memory access restrictions.

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Article

A study of a class of multiple-user channels including general multiple-access channels with many correlated sources and many simultaneous receivers is presented. The main result is summarized as Theorems 4.1 and 5.1 which establish a simple characterization of the capacity region on the basis of the polymatroidal structure of a set of (conditional) mutual informations. The results include, as special cases, the Slepian and Wolf's result (1973) as well as Ulrey's (1975). These may be regarded as further developments along the line shown by Ahlswede (1971) and Liao (1972). Furthermore, a finite upper bound for the cardinalities of the ranges of auxiliary variables is given in Theorems 4.2 and 5.2. Finally, the relation between Slepian—Wolf's formalism and ours is clarified.

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Article

The form of capacity achieving input distribution is specified for a class of finite state channels with side information available to the transmitter. It is shown that the best random codes consist of strategy function sequences whose elements are selected independently. It is a result of Theorem 4 (which is of independent interest) that the required distribution of the strategy functions is for the present class of channels specified by ah instead of the apparent ah parameters, so that a random selection of optimal codes is practically possible. As an application, an extremely simple capacity expression for finite state galois additive noise channels is obtained, and it is shown that the value of capacity will not increase if the transmitter is supplied with information about the channel states.

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Article

In this paper we study automata which work on directed ordered acyclic graphs, in particular those graphs, called derivation dags (d-dags), which model derivations of phrase-structure grammars. A rather complete characterization of the relative power of the following features of automata on d-dags is obtained: parallel versus sequential, deterministic versus nondeterministic and finite state versus a (restricted type of) pushdown store. New results concerning trees follows as special cases. Closure properties of classes of d-dag languages definable by various automata are studied for some basic operations. Characterization of general directed ordered acyclic graphs by these automata is also given.Contents.Abstract. 1. Introduction. 2. Definitions of the graphs. 3. Parallel dag automata. 4. Finite state relabeling. 5. Two-way dag-walking automata. 6. Comparison. 7. Closure properties of d-dag languages. 8. Recognition of doags. Acknowledgment. References.

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Article

In this paper we shall consider some design aspects of a computer which has two new modes of operation, which we call adaptive. One new mode of operation is the computer's ability to carry out useful computation even when component failures are present. This operation may be achieved at the expense of computational rate or accuracy. The ability to achieve this mode of operation is called graceful degradation, and its implementation differs from the redundancy techniques normally used to increase the reliability of a computer.The second mode of operation consists of the computer's ability to automatically increase the throughput at the expense of computation time and accuracy. Both hardware and software procedures for accomplishing these goals will be outlined.The results of this work are applicable to the design of those space-borne computers which need be operational for long periods of time, such as a year or two. Normally, the effective life of a piece of equipment is measured in terms of its mean time between failure (MTBF). However, for the computer system to be described, we are more interested in how long the system can carry out its overall functional goal, rather than the MTBF of its hardware.

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Article

The Bayes learning in adaptive control processes is defined by the learning structure in which the unknown probability distribution is re-estimated a posteriori by the use of Bayes' theorem after the random variable is observed at each stage of the process. Three kinds of measures which evaluate the effect of the Bayes' learning are presented. Using three examples in which the adaptive control problem is completely solved, it is shown that Bayes learning is sometimes unreasonable, in a certain sense, if the length of the programming period is not large.

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Article

This paper describes an adaptive controller for discrete-time stochastic environments. The controller receives the environment's current state and a reward signal which indicates the desirability of that state. In response, it selects an appropriate control action and notes its effect. The cycle repeats indefinitely. The control environments to be tackled include the well-known n-armed bandit problem, and the adaptive controller comprises an ensemble of n-armed bandit controllers, suitably interconnected. The design of these constituent elements is not discussed. It is shown that, under certain conditions, the controller's actions eventually become optimal for the particular control task with which it is faced, in the sense that they maximize the expected reward obtained in the future.

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Article

Adaptive signal detection and pattern recognition can be viewed as a problem in statistical classification wherein the partitioning of an n-dimensional sample space into category (signal) regions is determined through estimation from a set of samples from the categories. When the correct associations of the samples are known, the problem is the commonly treated supervised one. This paper, examining the nonsupervised case wherein the correct associations of the samples are unknown, demonstrates that it is possible under extremely general conditions to achieve effective adaptation without supervision. With particular emphasis on a two-category (binary detection) model, general conditions are described under which nonsupervised adaptation is possible, and specific simple yet rapidly convergent techniques are presented under varying degrees of prior knowledge of the statistical properties of the data.Most of the paper is concerned with a two-category case where the corresponding (equiprobable) distributions differ only in location. The paper proceeds by examining the over-all probability distribution comprised of the two component category distributions, and the adaptation treated is directed toward determining the decision boundary, or the distribution parameters necessary for defining it. For univariate normal distributions various estimators (and their convergence properties) of the over-all mean are examined. For multivariate monotone (including normal) distributions the over-all sample covariance matrix is used to obtain the component covariance matrices when these are general (including the colored noise case), or simply to obtain the principal eigenvector (of the overall matrix) when the component distributions are spherically symmetric (white noise). A hill-climbing algorithm is included. These results for the important model of binary signal detection in gaussian noise demonstrate that no prior knowledge of the signal or noise parameters is required for nonsupervised adaptation to the optimum detector. It is shown in one dimension that for equiprobable component distributions of almost any functional form and differing only by translation we can obtain category distribution estimators which converge uniformly over the real line with probability 1. Considered also are the case of different a priori probabilities, the problem of tracking, and some aspects of the multiple-category problem.

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Article

We give upper and lower bounds of the capacity C(X) of a channel with additive noise X under a constraint on input signals in terms of the second order moments. It is proved that C(X0)⩽C(X)⩽C(X0)+HX0(X), where C(X0) is the capacity of the channel with additive Gaussian noise X0 with same covariance as that of X and HX0(X) is the entropy of the measure induced by X in the functional space with respect to that induced by X0.

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Article

We consider the use of an (n, k) lineastock code over a channel from the viewpoint of a mean-square error criterion. The code is over a finite field GF(q) and the data to be transmitted have a natural q-adic association with the first zk integers. It is assumed that the transition probabilities of the channel satisfy an additive property. Optimum performance is judged by the mean-square of the overall error between input and output. On the basis of this criterion, we determine the optimum encoding rule and standard array decoding rule. Our analysis and synthesis of the coding rules employ the Fourier transforms of the channel transition probabilities. The optimum parameters of these rules are determined in the transform domain. The optimum decoder configuration for a rate k/n code resembles a bank of k generalized bandpass filters and is similar to a digital filter. It uses complex-valued arithmetic operations as opposed to finite field operations. The decoding system can efficiently employ fast transform techniques. Decoder storage requirements are also considered.

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Article

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.

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Article

First, some general properties of codes which are invariant under the permutation groups are given. For these codes an interesting relation is given between the minimum weights of dual codes. Secondly, results on the weight distributions of a class of binary cyclic codes are presented. Finally, some results on cross-correlation functions between two maximum-length sequences are introduced.

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Article

We investigate derivation-controlled K-iteration grammars, called (Γ, K)-iteration grammars, where Γ can be any family of control languages. We prove that already under very weak restrictions on Γ and K the following hold: (i) Regular control does not increase the generating power of K-iteration grammars, (ii) for each (Γ, K)-iteration grammar there exists an equivalent propagating (Γ, K)-iteration grammar, (iii) the family of (Γ, K)-iteration languages is a full hyper-AFL, (iv) for each (Γ, K)-iteration grammar there exists an equivalent (Γ, K)-iteration grammar with exactly two substitutions. We also discuss some additional properties and applications of (uncontrolled) K-iteration grammars and controlled (deterministic) ETOL systems and their languages in a wider context.

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Article

F-bounded erasing operator in abstract family of language for mapping, applying to families defined by tape-bounded Turing acceptors

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Article

Byzantine Agreement involves a system of n processes, of which some t may be faulty. The problem is for the correct processes to agree on a binary value sent by a transmitter that may itself be one of the n processes. If the transmitter sends the same value to each process, then all correct processes must agree on that value, but in any case, they must agree on some value. An explicit solution not using authentication for n = 3t + 1 processes is given, using 2t + 3 rounds and O(t3 log t) message bits. This solution is easily extended to the general case of n ⩾ 3t + 1 to give a solution using 2t + 3 rounds and O(nt + t3 log t) message bits.

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Article

Algebraic power series in noncommuting variables are obtained as solutions of polynomial systems of equations. These systems can be regarded as context-free grammars with weights. Some properties of algebraic power series are derived, in particular their representation as transductions of certain canonical Dyck sets.

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Chapter

In this paper we have shown how algebraic semantics can be defined for recursive -flowchart schemes using the freeness results for reducible -flowcharts. This considerably extends the algebraic characterization of flowcharts as begun by Elgot and Shepherdson [ES1][ES2].
Our results contrast with those of Gallier [G1][G3] in that they are derived independent of the special choice of .
In [S2] an example is given where -flowcharts represent nondeterministic programs on a stack machine thus providing an extension of the target language used by Thatcher, Wagner, and Wright in [ADJ6]. The results of this paper might lead to an extension of their compiler correctness results to programming languages incorporating recursive structures.

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Article

This article gives new fast methods for decoding certain error-correcting codes by solving certain algebraic equations. As described by Peterson (1961), the locations of a Bose-Chaudhuri Hocquenghem code over a field of characteristic p are associated with the elements of an extension field, GF(pk). The code is designed in such a way that the weighted power-sum symmetric functions of the error locations can be obtained directly by computing appropriately chosen parity checks on the received word. Good methods for computing the elementary symmetric functions from the weighted power-sum symmetric functions have been presented by Berlekamp (1967). The elementary symmetric functions, σ1, σ2, …, σt are the coefficients of an algebraic equation whose roots are the error locations xt+σ1xt−1+σ2xt−1+⋯+σt=0Previous methods for finding the roots of this equation have searched all of the elements in GF(pk) (Chien, 1964) or looked up the answer in a large table (Polkinghorn, 1966). We present here improved procedures for extracting the roots of algebraic equations of small degrees.

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Article

We consider a class of nonlinear estimation problems possessing certain algebraic properties, and we exploit these properties in order to study the computational complexity of nonlinear estimation algorithms. Specifically, we define a class of finite-state Markov processes evolving on finite groups and consider noisy observations of these processes. By introducing some concepts from the theory of representations of finite groups, we are able to define a pair of “dual≓ filtering algorithms. We then study several specific classes of groups in detail, and, by developing a generalization of the fast Fourier transform algorithm, we derive an efficient nonlinear filtering algorithm. A continuous-time version of these ideas is developed for cyclic groups.

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Article

We present a new class of optimal (n, k) group codes over the general finito field GF(q), q, a prime power, which are obtained by systematically deleting or puncturing certain coordinates of the maximal length shift register (qk − 1, k) code. The algorithm for puncturing is algebraic in that the coordinates deleted form subgroups of the additive group of the (qk − 1) roots of unity, or cosets of the multiplicative group of the (qk − 1) roots of unity, modulo the multiplicative group of GF(q). The specific algebraic nature of this puncturing procedure for any particular k yields codes of length n greater than qk−1. Optimality is proven by generalizing the Griesmer Bound on group codes. Encoding and decoding procedures are presented for this class of codes.

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Article

Two different formulations of fuzzy algorithms are developed in the present paper. The first corresponds to Turing algorithms and the second to Markov normal algorithms. It is shown that the two formulations are equivalent in a certain sense.

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Article

Let WRAM [PRAM]be a parallel computer with p processors (RAMs) which share a common memory and are allowed simultaneous reads and writes [only simultaneous reads]. The only type of simultaneous writes allowed is a simultaneous AND: a subset of the processors may write 0 simultaneously into the same memory cell. Let t be the time bound of the computer. We design below families of parallel algorithms that solve the string matching problem with inputs of size n (n is the sum of lengths of the pattern and the text) and have the following performance in terms of p, t and n: (1) For WRAM: pt = O(n) for p ⩽ n/log n (i.e., t ⩾ log n).† (2) for PRAM: pt = O(n) for p ⩽ n/log2n (i.e., t ⩾ log2n). (3) For WRAM: t = constant for p = n1 + ε and any ε > 0. (4) For WRAM: t = O(log n/log log n) for p = n. Similar families are also obtained for the problem of finding all initial palindromes of a given string.

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Article

The packing of rectangles with both dimensions i.i.d. ∼U(0, 1) onto a semi-infinite fixed-width strip is considered. The expected efficiency, expressed in terms of unused area on the strip is calculated for three simple procedures, all of which are level-algorithms: Next Fit, Rotatable Next Fit (where pieces are possibly orientated before packing, so that their width always exceeds their height), and Next Fit Decreasing (pieces are presorted by their height). It becomes evident that the single most important determinant of this efficiency is the variance of level heights, and the procedures can be ranked by their success in keeping this variance down. It is indirectly demonstrated that worstcase behavior of simple packing algorithms is a poor predictor of their expected performance. Results were obtained via analysis, computer (symbolic) integration, and simulation.

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Article

The edit distance between strings a1 … am and b1 … bn is the minimum cost s of a sequence of editing steps (insertions, deletions, changes) that convert one string into the other. A well-known tabulating method computes s as well as the corresponding editing sequence in time and in space O(mn) (in space O(min(m, n)) if the editing sequence is not required). Starting from this method, we develop an improved algorithm that works in time and in space O(s · min(m, n)). Another improvement with time O(s · min(m, n)) and space O(s · min(s, m, n)) is given for the special case where all editing steps have the same cost independently of the characters involved. If the editing sequence that gives cost s is not required, our algorithms can be implemented in space O(min(s, m, n)). Since s = O(max(m, n)), the new methods are always asymptotically as good as the original tabulating method. As a by-product, algorithms are obtained that, given a threshold value t, test in time O(t · min(m, n)) and in space O(min(t, m, n)) whether s ⩽ t. Finally, different generalized edit distances are analyzed and conditions are given under which our algorithms can be used in conjunction with extended edit operation sets, including, for example, transposition of adjacent characters.

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Article

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].

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Article

The class NC consists of problems solvable very fast (in time polynomial in log n) in parallel with a feasible (polynomial) number of processors. Many natural problems in NC are known; in this paper an attempt is made to identify important subclasses of NC and give interesting examples in each subclass. The notion of NC1-reducibility is introduced and used throughout (problem R is NC1-reducible to problem S if R can be solved with uniform log-depth circuits using oracles for S). Problems complete with respect to this reducibility are given for many of the subclasses of NC. A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in NC2 (solvable by uniform Boolean circuits of depth O(log2n) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, in “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York).

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Article

A distributed algorithm is presented, for allocating a large number of identical resources (such as airline tickets) to requests which can arrive anywhere in a distributed network. Resources, once allocated, are never returned. The algorithm searches sequentially, exhausting certain neighborhoods of the request origin before proceeding to search at greater distances. Choice of search direction is made non-deterministically. Analysis of expected response time is simplified by assuming that the search direction is chosen probabilistically, that messages require constant time, that the network is a tree with all leaves at the same distance from the root, and that requests and resources occur only at leaves. It is shown that the response time is approximated by the number of messages of one type that are sent during the execution of the algorithm, and that this number of messages is a nondecreasing function of the interarrival time for requests. Therefore, the worst case occurs when requests come in so far apart that they are processed sequentially. The expected time for the sequential case of the algorithm is analyzed by standard techniques. This time is shown to be bounded by a constant, independent of the size of the network. It follows that the expected response time for the algorithm is bounded in the same way.

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Article

Perfect codes in the Lee metric are proved to be impossible for (3⩽n⩽5;e⩾n−1;q⩾2e+1) and .

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Article

In this note we investigate the properties of two classes of binary primitive alternant codes that are generalizations of the primitive BCH codes. For these codes we establish certain equivalence and invariance relations and obtain values of d and d*, the minimum distances of the prime and dual codes.

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Article

Languages accepted by alternating auxiliary pushdown automata using simultaneously a(n) alternations and s(n) space are shown to be members of the class of languages accepted by nondeterministic Turing machines using a(n) 2es(n) space for some c > 0. This result is used to show that the hierarchy of classes of languages accepted by pushdown automata based on the number of alternations collapses at the second level of the hierarchy. The power of alternation bounded pushdown automata without auxiliary storage is also investigated.

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Article

The ambiguity problem for general context-free phrase structure grammars has been shown undecidable. This paper proves the undecidability of a new form of Post's correspondence problem. Using this result, the undecidability of the ambiguity problem for minimal linear grammars is obtained.

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Article

Optimum weighting functions are derived for least-mean-square reconstruction of N-dimensional stochastic fields from discrete sample measurements of amplitude and gradient. Exact interpolation can be achieved when the intensity spectrum of the field is wave-number-limited and the sampling lattice is periodic, provided that the spectral images induced by sampling overlap nowhere more than (N + 1) times. Compared with conventional (amplitude-only) sampling, a network (N + 1) times less dense is thus required.Examples include a rederivation of the one-dimensional bandlimited case, and the calculation of weighting functions and reconstruction errors for a second-order “Butterworth” process under various postulated sampling schemes. Weighting functions are derived for a two-dimensional isotropic wave-number-limited process, sampled to yield a 3:1 spectral overlap.

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Article

A specific isomorphism is constructed via the transform domains between the analog signal space L2 (−∞, ∞) and the digital signal space l2. It is then shown that the class of linear time-invariant realizable filters is invariant under this isomorphism, thus demonstrating that the theories of processing signals with such filters are identical in the digital and analog cases. This means that optimization problems involving linear time-invariant realizable filters and quadratic cost functions are equivalent in the discrete-time and the continuous-time cases, for both deterministic and random signals. Finally, applications to the approximation problem for digital filters are discussed.

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Article

The frameworks of unconditional and conditional Term Rewriting and Applicative systems are explored with the objective of using them for defining functions. In particular, a new operational semantics, Tue-Reduction, is elaborated for conditional term rewriting systems. For each framework, the concept of evaluation of terms invoking defined functions is formalized. We then discuss how it may be ensured that a function definition in each of these frameworks is meaningful, by defining restrictions that may be imposed to guarantee termination, unambiguity, and completeness of definition. The three frameworks are then compared, studying when a definition may be translated from one formalism to another.

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Article

In this paper we investigate inductive inference identification criteria which permit infinitely many errors in explanations, but which require that the “density” of these errors be no more than a certain, prespectified amount. We introduce three hierarchies of such criteria, each of which has the same order type as the real unit interval. These three hierarchies are progressively more strict in the way they measure density of errors of explanations. The strictest of the three turns out to have all of its members, save one, incomparable to the identification criterion which permits finitely many errors in explanations.

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Article

The problem of constructing minimum-redundancy prefix codes for the general discrete noiseless channel without constraints is solved for unequal code letter costs, provided that the symbols encoded are assumed to be equally probable. A graphical technique is developed for solving the problem for which the code words are equally probable and are constructed from r symbols where r is greater than or equal to two. A method is given for constructing an optimal exhaustive prefix code. This method is then generalized to the extent that the exhaustive constraint is deleted, thereby resulting in an algorithm, designated ACE for arbitrary symbol cost and equal code word probability, which solves the stated problem.

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Article

The computation of the minimum distance for q-nary cyclic AN-codes is treated in a systematic manner. By means of a new method expressions are obtained for the minimum distance in many of different cases. This includes several known results as well as some new ones. The weight distribution problem may be solved the same way.

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Article

Given N distinct memory modules, the elements of an (infinite) array in storage are distributed such that any set of N elements arranged according to a given data template T can be accessed rapidly in parallel. Array embeddings that allow for this are called skewing schemes and have been studied in connection with vector processing and SIMD machines. In 1975 Shapiro (IEEE Trans. Comput.C-27 (1978), 421–428) proved that there exists a valid skewing scheme for a template T if and only if T tessellates the plane. A conjecture of Shapiro is settled and it is proved that for polyominos P a valid skewing scheme exists if and only if there exists a valid periodic skewing scheme. (Periodicity implies a rapid technique to locate data elements.) The proof shows that when a polyomino P tessellates the plane without rotations or reflections, then it can tessellate the plane periodically, i.e., with the instances of P arranged in a lattice. It is also proved that there is a polynomial time algorithm to decide whether a polyomino tessellates the plane, assuming the polyominos in the tessellation should all have an equal orientation.

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Article

AN encoding in residue number systems allows construction of a class of nonlinear arithmetic error-correcting codes. The properties of these codes are further investigated and the redundancy necessary and sufficient to ensure multiple error correction is determined. An error-correcting procedure holding for codes using the minimal redundancy is presented. An error-correcting approach based upon the Euclidean algorithm is also reconsidered and it is shown that its redundancy requirements are lower than previously known.

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Article

Robbins has proposed a finite memory constraint on the two-armed bandit problem in which the coin to be tossed at each stage may depend on the history of the previous tosses only through the outcomes of the last r tosses. Letting the choice of coin depend on the time at which the toss is made, we exhibit a deterministic rule with memory r = 2, the description of which is independent of the coin biases p1 and p2, which achieves, with probability one, a limiting proportion of heads equal to max \{{p1, p2\}}. Thus this rule is asymptotically uniformly best among the class of time-varying finite memory rules.

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