Showing papers in "Information & Computation in 1981"
TL;DR: The number of queries required to identify a regular set given an oracle for the set and some auxiliary information about the set is considered and the problem considered by Pao and Carr (1978) is shown to be solvable in a polynomial number of query.
Abstract: 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.
193 citations
TL;DR: An undirected graph model for t-diagnosable systems is introduced to avoid certain difficulties in diagnostic testing which may arise in ordinary (digraph model) t- diagnoseable systems.
Abstract: In this paper we compare the performances, in the presence of faults, of two basic classes of computing systems: the modularly redundant systems and the t-diagnosable systems. We, then, introduce an undirected graph model for t-diagnosable systems to avoid certain difficulties in diagnostic testing which may arise in ordinary (digraph model) t-diagnosable systems. An optimal diagnosis algorithm for this new model is also presented.
185 citations
TL;DR: This report definessimple surface points and simple closed surfaces, and shows that any connected collection of simple surface points form a simple closed surface, thus proving a three-dimensional analog of the two-dimensional Jordan curve theorem.
Abstract: This is one of a series of reports on the digital geometry of three-dimensional images, such as those produced by computed tomography. In this report we define simple surface points and simple closed surfaces, and show that any connected collection of simple surface points form a simple closed surface, thus proving a three-dimensional analog of the two-dimensional Jordan curve theorem. We also show that the converse is not a theorem (in contrast to the two-dimensional case), and discuss more complex surface types. Finally, we show that the two-dimensional analog of our definition of simple closed surface characterizes simple closed curves, but that several other characterizations of 2 D curves, when extended to 3 D , are not adequate to characterize surfaces.
134 citations
TL;DR: Three-dimensional arcs and curves are defined and characterized for subsets of three-dimensional arrays for connectedness, cavities, and holes.
Abstract: Basic concepts of connectedness, cavities, and holes are defined for subsets of three-dimensional arrays. Three-dimensional arcs and curves are also defined and characterized.
114 citations
TL;DR: A combinatorial lemma is proved and used here to derive new results on ω -automata and to give simpler proofs of known ones and new normal form theorems and also decidability results are proved for the first-order and the monadic second-order theory of certain structures over the ordering of natural numbers.
Abstract: A combinatorial lemma is proved and used here to derive new results on ω -automata and to give simpler proofs of known ones. In particular, we reprove McNaughton's fundamental theorem (characterizing the ω -regular sequence-sets), without having to construct a sophisticated ω -automaton. The theorem is obtained by coding the behaviour of automata in a second-order language and a simple application of the lemma. In close analogy (now referring to a first-order language) a theory of counter-free ω-automata is developed; it is shown that these automata are appropriate for characterizing the ω -star-free sequence-sets. Finally, the lemma is applied in mathematical logic: Here new normal form theorems and also decidability results are proved for the first-order and the monadic second-order theory of certain structures over the ordering of natural numbers.
80 citations
TL;DR: 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.
Abstract: 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.
75 citations
TL;DR: The algebraic properties of fuzzy sets under these nex operations of bounded-sum and bounded-difference are investigated and the properties in the case where these new operations are combined with the well-known operations of union, intersection, algebraic product and algebraic sum are investigated.
Abstract: Among the basic operations which can be performed on fuzzy sets are the operations of union, intersection, complement, algebraic product and algebraic sum. In addition to these operations, new operations called “bounded-sum” and “bounded-difference” were defined by L. A. Zadeh to investigate the fuzzy reasoning which provides a way of dealing with the reasoning problems which are too complex for precise solution. This paper investigates the algebraic properties of fuzzy sets under these nex operations of bounded-sum and bounded-difference and the properties of fuzzy sets in the case where these new operations are combined with the well-known operations of union, intersection, algebraic product and algebraic sum.
73 citations
71 citations
TL;DR: Methods are presented for obtaining parametric measures of information from the non-parametric ones and from information matrices and the properties of these measures are examined.
Abstract: In this paper methods are presented for obtaining parametric measures of information from the non-parametric ones and from information matrices. The properties of these measures are examined. The one-dimensional parametric measures which are derived from the non-parametric are superior to Fisher's information measure because they are free from regularity conditions. But if we impose the regularity conditions of the Fisherian theory of information, these measures become linear functions of Fisher's measure.
61 citations
TL;DR: The complexity of the general coloring problem, i.e., of deciding for a given graph G whether some graph H is an interpretation of G, is investigated and it is shown that for many very simple undirected graphs G this question is NP -complete.
Abstract: A graph H is called an interpretation of a graph G if a morphic image of H is (isomorphic to) a subgraph of G . A graph H can be seen to be n -colorable iff H is an interpretation of K n (the complete graph on n vertices): hence colorability is a special case of the concept of interpretation. In this paper the complexity of the general coloring problem, i.e., of deciding for a given graph G whether some graph H is an interpretation of G , is investigated. It is shown that for many very simple undirected graphs G this question is NP -complete (this was previously known for the graphs K n only). In fact, it seems to be NP -complete for all but trivial exceptions. In the directed case, non-trivial graphs G for which the problem is in P , and rather simple graphs G for which the problem is NP -complete are presented. A characterization of all graphs G for which the problem at issue is in P remains an evasive open problem.
59 citations
TL;DR: This paper completely characterize all codes which meet the Griesmer bound with equality and for which d ⩽ 2usuk−1}.
Abstract: For any binary linear code of length n, dimension k, and minimum distance d, the Griesmer bound says that uif128-1}. In this paper we completely characterize all codes which meet the Griesmer bound with equality and for which d ⩽ 2usuk−1}. In particular we prove Belov's conjecture.
TL;DR: This paper studies the general problem of the existence of output (dynamic) stabilizers for a control system in terms of new notions which generalize the usual ones of stabilizability and detectability.
Abstract: This paper studies the general problem of the existence of output (dynamic) stabilizers for a control system. The controller is not assumed to have any special structure. Necessary and sufficient conditions are given in terms of new notions which generalize the usual ones of stabilizability and detectability. In the particular cases of analytic and bilinear systems, these conditions are considerably simplified.
TL;DR: Single error-correcting codes for the asymmetric channel with input and output alphabets being {0, 1,…, a — 1⩽} are studied and explicit expressions for the size and weight distribution are given.
Abstract: We study single error-correcting codes for the asymmetric channel with input and output alphabets being {0, 1,…, a — 1⩽. From an abelian group G of order N with elements g0 = 0, g1,…, gN—1, Constantin and Rao (1979, Inform. Contr. 40, 20–36) define Vg = {(b1,b2,...,bn−1) ∈ {O, 1,..., a − 1 }N−1 | ∑N−1i−1 bigi 0 g} and show that Vg correct single errors. We give explicit expressions for the size and weight distribution of these codes. We further give a short discussion of some constant weight codes obtained by a similar construction.
TL;DR: It is shown that the length of the transient in a threshold automata with values on the set {0, 1~} can be exponential.
Abstract: We study the length of the transient in a threshold automata with values on the set {0, 1~. The original conjecture was that in the symmetric case, this length was a linear function of the number of cells in the automata (Goles, E. (1981), “Comportement oscillatoire d'une famille d'automates cellulaires non uniformes,” These, IMAG, Grenoble, France; and Olivos, J. (1979), “Contribution a l'algorithmique,” These, Orsay, France). We shall show that this length can be exponential.
TL;DR: The paper contains a study of certain block codes, with integer coordinate symbols, devised for the correction of some types of additive errors, including a decoding method for generalized Varshamov codes, and principally with enumeration problems.
Abstract: The paper contains a study of certain block codes, with integer coordinate symbols, devised for the correction of some types of additive errors. The codes are defined by a check equation over a finite Abelian group; thus they appear as an extension of the Varshamov and Constantin—Rao codes. The main results of the paper are concerned with the error-correcting capability, including a decoding method for generalized Varshamov codes, and principally with enumeration problems. A general formula for spectral enumerators is obtained and applied to the Varshamov and Constantin—Rao codes.
TL;DR: It is shown that to test whether two DOL sequences in the binary case coincide it is enough toTest whether four first words of these sequences are the same, and the result is optimal.
Abstract: It is shown that to test whether two DOL sequences in the binary case coincide it is enough to test whether four first words of these sequences are the same. The result is optimal.
TL;DR: It is shown that this class of cellular automata can accept various string languages in O(log n) time, where n is the length of the input string defining the initial states of the leaf vertices in the tree.
Abstract: This paper introduces a special class of cellular automata, called triangle cellular automata, which accept as input strings of length a power of two. In particular, we study the special case in which state information can only move upward through a complete binary tree of finite-state automata. It is shown that this class of cellular automata can accept various string languages in O(log n) time, where n is the length of the input string defining the initial states of the leaf vertices in the tree. Extensions to two dimensions, defining a pyramid cellular automaton, are also given.
TL;DR: It is proved that a rational language L is h -bounded iff it is thin if f it is semi-discrete, i.e., L contains at most k words of any given length for some fixed k ∈ ℕ.
Abstract: Since every hypercode is finite, one may ask for the significance of the property that a language L admits a uniform upper bound on the size of hypercodes included in L . Such a language L is called h-bounded . We prove that a rational language L is h -bounded iff it is thin iff it is semi-discrete , i.e., L contains at most k words of any given length for some fixed k ∈ ℕ. Moreover, a representation of these languages by regular expressions is established. Concerning the general case, some properties of the syntactic monoid Synt( L ) of an h -bounded (semi-discrete) language are derived. If L is not disjunctive, then Synt( L ) contains a zero element. Every subgroup of Synt( L ) is a finite cyclic group. The idempotents of Synt( L )\{0, 1~ form an antichain with respect to the usual partial order.
TL;DR: The Elias-bound in the Lee-metric is modified to improve the known bounds for the code parameters in some cases and the result is applied to perfect codes to obtain nonexistence theorems.
Abstract: The Elias-bound in the Lee-metric is modified to improve the known bounds for the code parameters in some cases. The result is applied to perfect codes to obtain nonexistence theorems. For large alphabets it is proved that n ⩾ (e + 2)2/2.1, if n denotes the block length of a perfect Lee-code correcting e errors and e ⩾ 285.
TL;DR: The codingcapacity of channels with non-Gaussian noise having finite entropy with respect to Gaussian noise of the same covariance is shown not to exceed the coding capacity of the Gaussian channel.
Abstract: We evaluate the information capacity of channels for which the noise process is a Gaussian measure on a quasi-complete locally convex space. The coding capacity is calculated in this setting and for time-continuous Gaussian channels using the information capacity result. The coding capacity of channels with non-Gaussian noise having finite entropy with respect to Gaussian noise of the same covariance is shown not to exceed the coding capacity of the Gaussian channel. The sensitivity of the information capacity to deviations from normality in the noise process is also investigated.
TL;DR: Sequential and parallel ways of rewriting are investigated and compared in the framework of selective substitution grammars, and several new characterizations of known classes of languages are obtained.
Abstract: Sequential and parallel ways of rewriting are investigated and compared in the framework of selective substitution grammars. New aspects of the notion of generative determinism of a grammar and of the notion of symmetric context are studied. Several new characterizations of known classes of languages are obtained.
TL;DR: The proofs are based on a characterization of pure multi-pass attribute grammars in terms of paths through their dependency graphs, and it is shown that the problem of determining whether an arbitrary attribute grammar is pure multipass, is of inherently exponential time complexity.
Abstract: An attribute grammar is pure (left-to-right) multi-pass if a bounded number of left-to-right passes over the derivation tree suffice to compute all its attributes. There is no requirement, as for the usual multi-pass attribute grammars, that all occurrences of the same attribute are computed in the same pass. It is shown that the problem of determining whether an arbitrary attribute grammar is pure multipass, is of inherently exponential time complexity. For fixed k > 0, it can be decided in polynomial time whether an attribute grammar is pure k-pass. The proofs are based on a characterization of pure multi-pass attribute grammars in terms of paths through their dependency graphs. A general result on dependency paths of attribute grammars relates them to (finite-copying) top-down tree transducers. The formal power of k-pass attribute grammars increases with increasing k. Formally, multi-pass attribute grammars are less powerful than arbitrary attribute grammars.
TL;DR: Given a reduced, cycle-free context-free grammar G = (Φ, Σ, P, y1), the following statements are equivalent.
Abstract: Given a reduced, cycle-free context-free grammar G = (Φ, Σ, P, y1), the following statements are equivalent. (i) G is nonexpansive; (ii) the structure generating functions of the grammars Gi = (Φ, Σ, P, yi), yi ∈ Φ, are rational. Furthermore a helpful theorem for proving certain context-free languages to be inherently ambiguous is given.
TL;DR: It is proved that there exists a structure satisfying the unwind property for all flow charts, but not for recursive procedures, and also that algebraically closed and real closed fields with infinite degrees of transcendence over their prime subfields satisfy the unwinding property.
Abstract: In the present paper we investigate abstract algebras having the so-called unwind property, i.e., algebras where every total computable function can be defined by open formulas in an adequate first-order language. A method of proving the unwind property is given for a special kind of algebra. Applying this method we prove that there exists a structure satisfying the unwind property for all flow charts, but not for recursive procedures. We prove also that algebraically closed and real closed fields with infinite degrees of transcendence over their prime subfields satisfy the unwind property for all schemas.
TL;DR: There is an algorithm for testing the (string by string) equality of two homomorphisms on an ω-regular set of infinite strings and it is shown that it is decidable whether two functional finite-state transducers define the same function on infinite strings.
Abstract: This paper is concerned with sets of infinite strings (ω-languages) and mappings between them. The main result is that there is an algorithm for testing the (string by string) equality of two homomorphisms on an ω-regular set of infinite strings. As a corollary we show that it is decidable whether two functional finite-state transducers define the same function on infinite strings (are ω-equivalent).
TL;DR: The Shannon sampling theorem is proved for weak sense stationary stochastic processes using the theory of integral functions of exponential type and certain generalized sampling series leads to a better behaviour of approximation.
Abstract: The Shannon sampling theorem is proved for weak sense stationary stochastic processes using the theory of integral functions of exponential type. Estimates together with rates of convergence are given for the error which arises when a non-bandlimited random signal function is approximated by its sampling series expansion. The construction of certain generalized sampling series leads to a better behaviour of approximation.
TL;DR: It is shown how to produce from these polynomials and their periods and indices all irreducible trinomials T(N, K) = x N + x K + 1 with N ≡ ±3 (mod 8).
Abstract: We list the 19 values of N for which the trinomial T(N ,2) = x N + x} 2 + 1 is irreducible over GF (2) with N ⩽ 60,000. We then show how to produce from these polynomials and their periods and indices all irreducible trinomials T(N, K) = x N + x K + 1 with N ≡ ±3 (mod 8). These are members of infinite classes of irreducible trinomials T(NR, 2R) , where R has all of its prime factors divisors of the period of T(N , 2) and none of its prime factors divisors of the index of T(N , 2).
TL;DR: This paper discusses the concept of a reconfigurable cellular computer, in which each p/sub i/ can receive information from a set s/ Sub i/ of the other ps, and the s/ sub i/s are all of bounded size, but they need not remain fixed throughout a computation.
Abstract: When a collection of processors c=(p/sub 1/, ..., p/sub n/) operates in parallel, it is desirable that at any given stage of the computation, each p/sub i/ should have a task of about the same size to perform, and each p/sub i/ should require about the same amount of information from the other ps in order to perform its task. To the extent that these conditions are violated, parallelism is impaired, in the sense that the ps are not all used with equal efficiency. In cellular computers, e.g. as they might be used for parallel image processing, these conditions are maintained by having the ps all perform similar computations on different parts of the input data, and by allowing each p/sub i/ to receive information from a fixed set of the others (its neighbours), where these sets are all of bounded size. This paper discusses, on an abstract level, the concept of a reconfigurable cellular computer, in which each p/sub i/ can receive information from a set s/sub i/ of the other ps, and the s/sub i/s are all of bounded size, but they need not remain fixed throughout a computation. Requiring the s/sub i/s to have bounded size impliesmore » that most ps cannot communicate directly; the expected time required for two arbitrary ps to communicate depends on the graph structure defined by the sets s/sub i/. The question of how to change the s/sub i/s in parallel during the course of a computation is also discussed. 15 references.« less
TL;DR: It is shown that for a class of languages uif110-5}, if uifuscL}1 is closed under homomorphicism and inverse homomorphism, then for each alphabet σ, the following two statements are equivalent.
Abstract: A context-free grammar G with terminal vocabulary σ is left universal for a class of languages uif110-1} with respect to a class of languages uif110-2} if for each language L ⊂ σ* in uif110-3}, a control language C in uif110-4} can be found such that G controlled by C generates L by leftmost derivations. We show that for a class of languages uif110-5} and a class of languages uif110-6}, if uifuscL}1 is closed under homomorphism and inverse homomorphism, then for each alphabet σ, the following two statements are equivalent. (1) There exists a context-free grammar with terminal vocabulary σ which is left universal for uif110-7} with respect to uif110-8}. (2) There exist a context-free language L1 and a homomorphism h such that each language L ⊂ σ* in uif110-9} equals uith(L1 ∩ L′)} for some L′ in uif110-10}. We also give some applications of this result.
TL;DR: It is proved that expected cell-distribution and expected growth equivalence are decidable, that the question of whether or not a Markov DT0L system generates a dead word is equivalent to an open question for Z-rational series, and that it is decidable whether orNot a given word survives in a given propagating MarkovDT0L systems.
Abstract: We consider Markov DT0L systems, that is, DT0L systems with a Markov chain control. Our main concern is life and death in such systems. We prove that expected cell-distribution and expected growth equivalence are decidable, that the question of whether or not a Markov DT0L system generates a dead word is equivalent to an open question for Z-rational series, and that it is decidable whether or not a given word survives in a given propagating Markov DT0L system.