Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata

The purpose of Chapter 9 is to develop some classical results on formal languages and automata by an algebraic treatment using semirings, formal power series and matrices. The use of semirings, formal power series and matrices makes the proofs computational in nature and, consequently, more satisfactory from the mathematical point of view than the customary proofs.

[1]  J. Shepherdson,et al.  Computer programming and formal systems , 1965 .

[2]  Arto Salomaa.  Formal Languages and Power Series , 1990, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[3]  J. Berstel,et al.  Context-free languages , 1993, SIGA.

[4]  Werner Kuich.  Automata and Languages Generalized to omega-Continuous Semirings , 1991, Theor. Comput. Sci..

[5]  Christophe Reutenauer.  An Ogden-like iteration lemma for rational power series , 1980, Acta Informatica.

[6]  Arto Salomaa,et al.  Semirings, Automata, Languages , 1985, EATCS Monographs on Theoretical Computer Science.

[7]  Jonathan S. Golan,et al.  The theory of semirings with applications in mathematics and theoretical computer science , 1992, Pitman monographs and surveys in pure and applied mathematics.

[8]  Werner Kuich,et al.  On the Entropy of Context-Free Languages , 1970, Inf. Control..

[9]  Juha Honkala.  On Images of Algebraic Series , 1996, J. Univers. Comput. Sci..

[10]  Richard Edwin Stearns,et al.  A Regularity Test for Pushdown Machines , 1967, Inf. Control..

[11]  Rohit Parikh,et al.  On Context-Free Languages , 1966, JACM.

[12]  R. K. Shyamasundar,et al.  The Structure Generating Function of Some Families of Languages , 1976, Inf. Control..

[13]  Gérard Jacob.  La finitude des représentations linéaires des semi-groupes est décidable , 1978 .

[14]  Bruce E. Litow,et al.  A Context-Free Language Decision Problem , 1994, Theor. Comput. Sci..

[15]  I. N. Sneddon,et al.  Theory Of Automata , 1969 .

[16]  A. Bertoni,et al.  Counting Problems and Algebraic Formal Power Series in Noncommuting Variables , 1990, Inf. Process. Lett..

[17]  Friedrich J. Urbanek.  On Greibach Normal Form Construction , 1986, Theor. Comput. Sci..

[18]  Juha Honkala,et al.  On Lindenmayerian Algebraic Power Series , 1997, Theor. Comput. Sci..

[19]  Werner Kuich,et al.  Hauptkegel algebraischer Potenzreihen , 1987, J. Inf. Process. Cybern..

[20]  Kosaburo Hashiguchi,et al.  Improved Limitedness Theorems on Finite Automata with Distance Functions , 1990, Theor. Comput. Sci..

[21]  Philippe Flajolet.  Elements of a general theory of combinatorial structures , 1985, FCT.

[22]  Maurice Gross,et al.  Introduction to Formal Grammars , 1970, Springer Berlin Heidelberg.

[23]  Andreas Weber.  Finite-Valued Distance Automata , 1994, Theor. Comput. Sci..

[24]  Henning Fernau,et al.  Valuations of Languages, with Applications to Fractal Geometry , 1995, Theor. Comput. Sci..

[25]  Helmut Prodinger,et al.  On the Height of Derivation Trees , 1979, ICALP.

[26]  Monoides et semi-anneaux continus , 1988 .

[27]  Gérard Jacob,et al.  Sur un Théorème de Shamir , 1975, Inf. Control..

[28]  Juha Honkala,et al.  A Necessary Condition for the Rationality of the Zeta Function of a Regular Language , 1989, Theor. Comput. Sci..

[29]  Andreas Weber,et al.  Exponential Upper and Lower Bounds for the Order of a Regular Language , 1994, Theor. Comput. Sci..

[30]  Marcel Paul Schützenberger,et al.  On the Definition of a Family of Automata , 1961, Inf. Control..

[31]  Philippe Flajolet,et al.  Analyse d'algorithmes de manipulation d'arbres et de fichiers , 1981 .

[32]  H. Weinert.  Generalized semialgebras over semirings , 1988 .

[33]  Jean Berstel,et al.  Recognizable Formal Power Series on Trees , 1982, Theor. Comput. Sci..

[34]  Michael A. Arbib,et al.  An Introduction to Formal Language Theory , 1988, Texts and Monographs in Computer Science.

[35]  Werner Kuich,et al.  Languages and the enumeration of planted plane trees , 1970 .

[36]  Grzegorz Rozenberg,et al.  The mathematical theory of L systems , 1980 .

[37]  Eliahu Shamir,et al.  A Representation Theorem for Algebraic and Context-Free Power Series in Noncommuting Variables , 1967, Inf. Control..

[38]  Philippe Flajolet,et al.  Birthday Paradox, Coupon Collectors, Caching Algorithms and Self-Organizing Search , 1992, Discret. Appl. Math..

[39]  G. Karner.  On limits in complete semirings , 1992 .

[40]  Seymour Ginsburg,et al.  Two Families of Languages Related to ALGOL , 1962, JACM.

[41]  Kosaburo Hashiguchi,et al.  Algorithms for Determining Relative Star height and Star Height , 1988, IFIP Congress.

[42]  Juha Honkala,et al.  On Parikh Slender Languages and Power Series , 1996, J. Comput. Syst. Sci..

[43]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[44]  I. Anderson,et al.  Graphs and Networks , 1981, The Mathematical Gazette.

[45]  Philippe Dumas,et al.  Additive Cellular Automata and Algebraic Series , 1993, Theor. Comput. Sci..

[46]  M. Nivat.  Laboratoire d'informatique théorique et programmation , 1981 .

[47]  Georg Karner.  Nivat's Theorem for Pushdown Transducers , 1992, Theor. Comput. Sci..

[48]  Robert W. Floyd,et al.  Syntactic Analysis and Operator Precedence , 1963, JACM.

[49]  W. Wechler.  The concept of fuzziness in automata and language theory , 1978 .

[50]  Jacques Sakarovitch.  Kleene's theorem revisited , 1986, IMYCS.

[51]  Manfred Kudlek,et al.  On Distributed Catenation , 1997, Theor. Comput. Sci..

[52]  Friedrich J. Urbanek,et al.  Infinite Linear Systems and one Counter Languages , 1983, Theor. Comput. Sci..

[53]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[54]  Daniel J. Lehmann,et al.  Algebraic Structures for Transitive Closure , 1976, Theor. Comput. Sci..

[55]  Seymour Ginsburg,et al.  Algebraic and Automata Theoretic Properties of Formal Languages , 1975 .

[56]  Christophe Reutenauer.  Sur les Series Associees a Certains Systemes de Lindenmayer , 1979, Theor. Comput. Sci..

[57]  Werner Kuich.  Lindenmayer Systems Generalized to Formal Power Series and Their Growth Functions , 1993, Developments in Language Theory.

[58]  Alexandru Mateescu.  On (Left) Partial Shuffle , 1994, Results and Trends in Theoretical Computer Science.

[59]  Noam Chomsky,et al.  The Algebraic Theory of Context-Free Languages* , 1963 .

[60]  Imre Simon,et al.  Recognizable Sets with Multiplicities in the Tropical Semiring , 1988, MFCS.

[61]  Christophe Reutenauer.  On a Question of S. Eilenberg , 1977, Theor. Comput. Sci..

[62]  Philippe Flajolet,et al.  Ambiguity and Transcendence , 1985, ICALP.

[63]  Arto Salomaa,et al.  Formal languages , 1973, Computer science classics.

[64]  Aristid Lindenmayer,et al.  Mathematical Models for Cellular Interactions in Development , 1968 .

[65]  Noam Chomsky,et al.  On Certain Formal Properties of Grammars , 1959, Inf. Control..

[66]  Daniel J. Rosenkrantz,et al.  Matrix Equations and Normal Forms for Context-Free Grammars , 1967, JACM.

[67]  M. Schützenberger,et al.  Rational sets in commutative monoids , 1969 .

[68]  Juha Honkala.  On D0L Systems with Immigration , 1993, Theor. Comput. Sci..

[69]  J. Conway.  Regular algebra and finite machines , 1971 .

[70]  Arto Salomaa,et al.  Automata-Theoretic Aspects of Formal Power Series , 1978, Texts and Monographs in Computer Science.

[71]  S C Kleene,et al.  Representation of Events in Nerve Nets and Finite Automata , 1951 .

[72]  Wolfgang Wechler,et al.  Universal Algebra for Computer Scientists , 1992, EATCS Monographs on Theoretical Computer Science.

[73]  D. L. Pilling,et al.  Commutative Regular Equations and Parikh's Theorem , 1973 .

[74]  Oscar H. Ibarra,et al.  On Sparseness, Ambiguity and other Decision Problems for Acceptors and Transducers , 1986, STACS.

[75]  R. Kemp,et al.  On the number of words in the language {w in sigma* | W = wR}2 , 1982, Discret. Math..

[76]  Juha Honkala,et al.  On Morphycally Generated Formal Power Series , 1995, RAIRO Theor. Informatics Appl..

[77]  Juha Honkala,et al.  On Lindenmayerian Series in Complete Semirings , 1993, Developments in Language Theory.

[78]  Dominique Perrin,et al.  Finite Automata , 1958, Philosophy.

[79]  Juha Honkala.  On Algebraic Generalized Zeta Functions of Formal Power Series , 1991, Theor. Comput. Sci..

[80]  Karel Culik,et al.  A Note on the Equivalence Problem of Rational Formal Power Series , 1986, Inf. Process. Lett..

[81]  F. L. Bauer,et al.  Automaten und formale Sprachen , 1976 .

[82]  Holger Petersen,et al.  The Ambiguity of Primitive Words , 1994, STACS.

[83]  Armando B. Matos.  Periodic Sets of Integers , 1994, Theor. Comput. Sci..

[84]  Volker Diekert.  Transitive Orientations, Möbius Functions, and Complete Semi-Thue Systems for Free Partially Commutative Monoids , 1988, ICALP.

[85]  Tero Harju,et al.  The Equivalence Problem of Multitape Finite Automata , 1991, Theor. Comput. Sci..

[86]  S. Bozapalidis.  Psendo-Boolean Programming with Special Restraints. Applications to Graph Theory. , 1965 .

[87]  Philippe Flajolet,et al.  Analytic Analysis of Algorithms , 1992, ICALP.

[88]  Imre Simon,et al.  On Semigroups of Matrices over the Tropical Semiring , 1994, RAIRO Theor. Informatics Appl..

[89]  Werner Kuich,et al.  The Characterization of Nonexpansive Grammars by Rational Power Series , 1981, Inf. Control..

[90]  Werner Kuich,et al.  The Kleene and the Parikh Theorem in Complete Semirings , 1987, ICALP.

[91]  Michael A. Harrison,et al.  Introduction to formal language theory , 1978 .

[92]  Maurice Nivat,et al.  Transduction des langages de Chomsky , 1968 .

[93]  Georg Karner.  A Topology for Complete Semirings , 1994, STACS.

[94]  Grzegorz Rozenberg,et al.  TOL Schemes and Control Sets , 1975, Inf. Control..

[95]  A. Lindenmayer.  Mathematical models for cellular interactions in development. II. Simple and branching filaments with two-sided inputs. , 1968, Journal of theoretical biology.

[96]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[97]  Alberto Bertoni,et al.  The Complexity of Computing the Number of Strings of Given Length in Context-Free Languages , 1991, Theor. Comput. Sci..

[98]  Werner Kuich.  Omega-Continuous Semirings, Algebraich Systems and Pushdown Automata , 1990, ICALP.

[99]  Werner Kuich.  The Algebraic Equivalent of AFL Theory , 1995, ICALP.

[100]  Jean Berstel,et al.  Properties of Infinite Words: Recent Result , 1989, STACS.

[101]  Robert Cori,et al.  Enumeration des graphes planaires a l'aide des series formelles en variables non commutatives , 1972, Discret. Math..

[102]  Sheila A. Greibach,et al.  A New Normal-Form Theorem for Context-Free Phrase Structure Grammars , 1965, JACM.

[103]  Hing Leung.  A Note on Finitely Ambiguous Distance Automata , 1992, Inf. Process. Lett..

[104]  Jay R. Goldman,et al.  Formal Languages and Enumeration , 1978, J. Comb. Theory, Ser. A.

[105]  Hing Leung.  Limitedness Theorem on Finite Automata with Distance Functions: An Algebraic Proof , 1991, Theor. Comput. Sci..

[106]  Hermann A. Maurer,et al.  The Structure Generating Function and Entropy of Tuple Languages , 1971, Inf. Control..

[107]  Jean Berstel,et al.  Zeta functions of formal languages , 1990 .

[108]  Marcel P. Schützenberger.  Un problème de la théorie des automates , 1960 .

[109]  Kosaburo Hashiguchi,et al.  Limitedness Theorem on Finite Automata with Distance Functions , 1982, J. Comput. Syst. Sci..

[110]  Juha Honkala,et al.  On a Power Series Generalization of ETOL Languages , 1996, Fundam. Informaticae.

[111]  Jean Berstel,et al.  Transductions and context-free languages , 1979, Teubner Studienbücher : Informatik.

[112]  Philippe Flajolet,et al.  Analytic Models and Ambiguity of Context-free Languages* , 2022 .

[113]  Joseph S. Ullian,et al.  Partial Algorithm Problems for Context Free Languages , 1967, Inf. Control..

[114]  Zoltán Ésik,et al.  Some varieties of iteration theories , 1984, Bull. EATCS.

[115]  Georg Karner.  On Transductions of Formal Power Series over Complete Semirings , 1992, Theor. Comput. Sci..

[116]  Dexter Kozen.  A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events , 1994, Inf. Comput..

[117]  Juha Honkala.  On Lindenmayerian Algebraic Sequences , 1997, Theor. Comput. Sci..

[118]  Daniel Krob,et al.  The Equality Problem for Rational Series with Multiplicities in the tropical Semiring is Undecidable , 1992, Int. J. Algebra Comput..

[119]  A. Lindenmayer.  Mathematical models for cellular interactions in development. I. Filaments with one-sided inputs. , 1968, Journal of theoretical biology.

[120]  Philippe Flajolet,et al.  Prefixes of Infinite Words and Ambiguous Context-Free Languages , 1987, Inf. Process. Lett..

[121]  H. Lausch,et al.  Algebra of Polynomials , 1974 .

[122]  Bernd Mahr,et al.  Iteration and summability in semirings , 1984 .

[123]  Azaria Paz,et al.  Introduction to Probabilistic Automata , 1971 .