Linear automaton transformations
01 Apr 1958-Vol. 9, Iss: 4, pp 541-544
TL;DR: For a finite commutative ring with unit the authors determine which linear transformations M: RN—+RN can be realized by finite automata.
Abstract: Let R be a nonempty set, let N consist of all non-negative rational integers, and denote by RN the set of all functions on N to R. If R is a ring, a map M: R"—>P^ is linear if M(rxfx+r2f2)=rx(Mfx) +r2(Mf2) for rx, r2 in R, fx, f2 in RN. For a finite commutative ring with unit we determine which linear transformations M: RN—+RN can be realized by finite automata. More precisely, let A, B he finite nonempty sets. A map M: AN—>BN is an automaton transformation if there exists a finite set Q, maps Mq: A X£>—><2, Mb: A XQ-*B, elements h in B, q in Q such that corresponding to each/ in AN there exists an h in QN satisfying
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Book•
25 Apr 2008
TL;DR: Principles of Model Checking offers a comprehensive introduction to model checking that is not only a text suitable for classroom use but also a valuable reference for researchers and practitioners in the field.
Abstract: Our growing dependence on increasingly complex computer and software systems necessitates the development of formalisms, techniques, and tools for assessing functional properties of these systems. One such technique that has emerged in the last twenty years is model checking, which systematically (and automatically) checks whether a model of a given system satisfies a desired property such as deadlock freedom, invariants, and request-response properties. This automated technique for verification and debugging has developed into a mature and widely used approach with many applications. Principles of Model Checking offers a comprehensive introduction to model checking that is not only a text suitable for classroom use but also a valuable reference for researchers and practitioners in the field. The book begins with the basic principles for modeling concurrent and communicating systems, introduces different classes of properties (including safety and liveness), presents the notion of fairness, and provides automata-based algorithms for these properties. It introduces the temporal logics LTL and CTL, compares them, and covers algorithms for verifying these logics, discussing real-time systems as well as systems subject to random phenomena. Separate chapters treat such efficiency-improving techniques as abstraction and symbolic manipulation. The book includes an extensive set of examples (most of which run through several chapters) and a complete set of basic results accompanied by detailed proofs. Each chapter concludes with a summary, bibliographic notes, and an extensive list of exercises of both practical and theoretical nature.
4,905 citations
Cites result from "Linear automaton transformations"
...The existence of minimal DFAs relies on results stated by Myhill [309] and Nerode [313]....
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TL;DR: Finite automata are considered as instruments for classifying finite tapes as well as generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined.
Abstract: Finite automata are considered in this paper as instruments for classifying finite tapes. Each one-tape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms.
1,930 citations
Journal Article•
TL;DR: This work recalls classical theorems and gives new ones characterizing sequential string-to-string transducers, including algorithms for determinizing and minizizing these transducers very efficiently, and characterizations of the transducers admitting determinization and the corresponding algorithms.
Abstract: Finite-machines have been used in various domains of natural language processing. We consider here the use of a type of transducer that supports very efficient programs: sequential transducers. We recall classical theorems and give new ones characterizing sequential string-to-string transducers. Transducers that outpur weights also play an important role in language and speech processing. We give a specific study of string-to-weight transducers, including algorithms for determinizing and minizizing these transducers very efficiently, and characterizations of the transducers admitting determinization and the corresponding algorithms. Some applications of these algorithms in speech recognition are described and illustrated.
1,052 citations
TL;DR: The question of whether there is an automaton with n states which agrees with a finite set D of data is shown to be NP-complete, although identification-in-the-limit of finite automata is possible in polynomial time as a function of the size of D.
Abstract: The question of whether there is an automaton with n states which agrees with a finite set D of data is shown to be NP-complete, although identification-in-the-limit of finite automata is possible in polynomial time as a function of the size of D. Necessary and sufficient conditions are given for D to be realizable by an automaton whose states are reachable from the initial state by a given set T of input strings. Although this question is also NP-complete, these conditions suggest heuristic approaches. Even if a solution to this problem were available, it is shown that finding a minimal set T does not necessarily give the smallest possible T.
819 citations
TL;DR: The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages and it is suggested that such undecidability is common in these and other dynamical systems.
Abstract: Self-organizing behaviour in cellular automata is discussed as a computational process. Formal language theory is used to extend dynamical systems theory descriptions of cellular automata. The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages. Many examples are given. The sizes of the minimal grammars for these languages provide measures of the complexities of the sets. This complexity is usually found to be non-decreasing with time. The limit sets generated by some classes of cellular automata correspond to regular languages. For other classes of cellular automata they appear to correspond to more complicated languages. Many properties of these sets are then formally non-computable. It is suggested that such undecidability is common in these and other dynamical systems.
579 citations
References
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15 Dec 1951
TL;DR: This memorandum is devoted to an elementary exposition of the problems and of results obtained on the McCulloch-Pitts nerve net during investigations in August 1951.
Abstract: An elementary exposition of the problems and results obtained during investigations in August, 1951, of the kinds of events any finite automation can respond to by assuming one of certain states.
1,799 citations
"Linear automaton transformations" refers background in this paper
...Then M is certainly causal due to (1)....
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...Further, the least number of states required in order to induce M as in (1) is the number of intrinsic states....
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...If X = (x0> • • • , xf) is in a(A), define qx to be the h(j) determined from (1) by letting f(n) =x„ for all n<j....
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