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The Expression Of Graph Properties And Graph Transformations In Monadic SecondOrder Logic
, 1997
"... By considering graphs as logical structures, one... ..."
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By considering graphs as logical structures, one...
ContextFree Languages and PushDown Automata
 Handbook of Formal Languages
, 1997
"... Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.1 Grammars : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : ..."
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Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.1 Grammars : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2. Systems of equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.1 Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 Resolution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.3 Linear systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2.4 Parikh's theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generaliza ..."
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Cited by 5 (3 self)
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We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language  the largest set commuting with a language , of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago  asking whether or not the centralizer of any rational language is rational  in the case of periodic, binary, and ternary sets of words, as well as for rational ccodes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an ccode X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.
Convergence of Newton’s Method over Commutative Semirings ⋆
"... Abstract. We give a lower bound on the speed at which Newton’s method (as defined in [5, 6]) converges over arbitrary ωcontinuous commutative semirings. From this result, we deduce that Newton’s method converges within a finite number of iterations over any semiring which is “collapsed at some k ∈ ..."
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Abstract. We give a lower bound on the speed at which Newton’s method (as defined in [5, 6]) converges over arbitrary ωcontinuous commutative semirings. From this result, we deduce that Newton’s method converges within a finite number of iterations over any semiring which is “collapsed at some k ∈ N ” (i.e. k = k + 1 holds) in the sense of [1]. We apply these results to (1) obtain a generalization of Parikh’s theorem, (2) to compute the provenance of Datalog queries, and (3) to analyze weighted pushdown systems. We further show how to compute Newton’s method over any ωcontinuous semiring. 1
The Equational Theory of Fixed Points with Applications to Generalized Language Theory
, 2001
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Semilinear Parikh Images of Regular Expressions via Reduction
"... Abstract. A reduction system for regular expressions is presented. For a regular expression t, the reduction system is proved to terminate in a state where the mostreduced expression readily yields a semilinear representation for the Parikh image of the language of t. 1 ..."
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Abstract. A reduction system for regular expressions is presented. For a regular expression t, the reduction system is proved to terminate in a state where the mostreduced expression readily yields a semilinear representation for the Parikh image of the language of t. 1
Path algorithms on regular graphs
"... Abstract. We consider standard algorithms of finite graph theory, like for instance shortest path algorithms. We present two general methods to polynomially extend these algorithms to infinite graphs generated by deterministic graph grammars. 1 ..."
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Abstract. We consider standard algorithms of finite graph theory, like for instance shortest path algorithms. We present two general methods to polynomially extend these algorithms to infinite graphs generated by deterministic graph grammars. 1
From Parikh’s Theorem to ManySorted Spectra
"... Abstract. We discuss a generalization of Parikh’s Theorem for contextfree languages to classes of manysorted relational structures which are both definable in Monadic Second Order Logic and which are of bounded patchwidth. Patchwidth is a generalization of both treewidth and cliquewidth. This g ..."
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Abstract. We discuss a generalization of Parikh’s Theorem for contextfree languages to classes of manysorted relational structures which are both definable in Monadic Second Order Logic and which are of bounded patchwidth. Patchwidth is a generalization of both treewidth and cliquewidth. This gives a powerful unifying tool to prove that certain classes of graphs are of unbounded width. For R. Parikh, at the occasion of his 70th birthday 1 Generalizing Parikh’s Theorem R. Parikh’s celebrated theorem, first proved in [Par66], counts the number of occurrences of letters in words of a contextfree languages L over an alphabet of k letters. For a given word w, the numbers of these occurrences is denoted by a vector n(w) ∈ N k, and the theorem states Theorem 1 (Parikh 1966). For a contextfree language L, the set Par(L) = {n(w) ∈ N k: w ∈ L} is semilinear. A set X ⊆ Ns is linear in Ns iff there is vector ā ∈ Ns and a matrix M ∈ Ns×r such that X = Aā, M ¯ = { ¯b ∈ Ns: there is ū ∈ Nr with ¯b = ā + M · ū}. Singletons are linear sets with M = 0. If M ̸ = 0 the series is nontrivial. X ⊆ Ns is semilinear in Ns iff X is a finite union of linear sets Ai ⊆ Ns. For s = 1 the semilinear sets are exactly the ultimately periodic sets. The terminology is from [Par66], and has since become standard terminology in formal language theory. Several alternative proofs of Parikh’s Theorem have appeared since. D. Pilling [Pil73] put it into a more algebraic form, and more recently, L. Aceto, Z. Esik and A. Ingolfsdottir [AÉI02] showed that it depends only on a few equational properties of least prefixed points. B. Courcelle [Cou95]. has generalized Theorem 1 further to certain graph grammars, and relational structures which are