Lecture Notes in Computer Science 5725 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany Symeon Bozapalidis George Rahonis (Eds.) Algebraic Informatics Third International Conference, CAI 2009 Thessaloniki, Greece, May 19-22, 2009 Proceedings 13 Volume Editors Symeon Bozapalidis George Rahonis Aristotle University of Thessaloniki Department of Mathematics 54124 Thessaloniki, Greece E-mail:{bozapali, grahonis}@math.auth.gr Library of Congress Control Number: Applied for CR Subject Classification (1998): F.4, I.1.3, F.1.1, F.4.1, F.4.3, F.4.2 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN 0302-9743 ISBN-10 3-642-03563-9 Springer Berlin Heidelberg New York ISBN-13 978-3-642-03563-0 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2009 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper SPIN: 12735013 06/3180 543210 Preface CAI 2009 was the Third International Conference on Algebraic Informatics. It was intended to cover the topics of algebraic semantics on graphs and trees, formal power series, syntactic objects, algebraic picture processing, finite and in- finite computations, acceptors and transducers for strings, trees, graphs, arrays, etc., decision problems, algebraic characterization of logical theories, process algebra, algebraic algorithms, algebraic coding theory, algebraic aspects of cryp- tography. CAI 2009 was dedicated to Werner Kuich on the occasion of his retirement. It was held in Thessaloniki, Greece, during May 19-22, 2009 and organized under the auspices of the Department of Mathematics of the Aristotle University of Thessaloniki. The opening lecture was given by Werner Kuich, the tutorials by Alessandra Cherubini and Wan Fokkink, and the other four invited lectures by Bruno Courcelle, Dietrich Kuske, Detlef Plump, and Franz Winkler. This volume contains 2 papers from the tutorials, 5 papers of the invited lectures, and 16 contributed papers. We received 25 submissions, the contributors being from 14 and countries, and the Program Committee selected 16 papers. We are grateful to the members of the Program Committee for the evaluation of the submissions and the numerous referees who assisted in this work. We should like to thank all the contributors of CAI 2009 and especially the honorary guest Werner Kuich and the invited speakers who kindly accepted our invitation to present their important work. Special thanks are due to Alfred Hofmann the Editorial Director of LNCS, who gave us the opportunity to publish the proceedings of our conference in the LNCS series, as well as to Anna Kramer from Springer for the excellent cooperation. We are also grateful to the members of the Organizing Committee and a group of graduate students who helped us with several organizing jobs. Last but not least we want to express our gratitude to the members of the Steering Committee for their constant interest and especially to Arto Salomaa for his support at Springer. The sponsors of CAI 2009, OPAP, Aristotle University of Thessaloniki, Attiko Metro S.A., Research Academic Computer Technology Institute (Fronts), and Ziti Publications are gratefully acknowledged. July 2009 Symeon Bozapalidis George Rahonis Organization Steering Committee Jean Berstel, Marne-la-Vall´ee Zoltan ´ Esik, Szeged Werner Kuich, Vienna Arto Salomaa, Turku Program Committee J¨urgen Albert, W¨urzburg Jos Baeten, Eindhoven Symeon Bozapalidis, Thessaloniki (Chairman) Flavio Corradini, Camerino Erzs´ebet Csuhaj-Varj´u, Budapest Frank Drewes, Ume˚a Manfred Droste, Leipzig Ioannis Emiris, Athens Dora Giammarresi, Rome Masami Ito, Kyoto Friedrich Otto, Kassel Dimitrios Poulakis, Thessaloniki Robert Rolland, Marseille Kai Salomaa, Kingston Ontario Paul Spirakis, Patras Magnus Steinby, Turku Sophie Tison, Lille Heiko Vogler, Dresden Sheng Yu, London Ontario Referees J. Albert Y. Aubry J. Baeten J. Berstel S. Bloom S. Bozapalidis M. ´ Ciri´c F. Corradini B. Courcelle E. Csuhaj-Varj´u M. Domaratzki F. Drewes I. Emiris Z. F¨ul¨op D. Giammarresi A. Grammatikopoulou J. H¨ogberg S. Jenei VIII Organization H. Jonker A. Kalampakas D. Kuske A. Lopes A. Maletti E. Mandrali O. Matz I. Meinecke M. Mignotte K. Ogata F. Otto A. Papistas U. Prange M. Pohst D. Poulakis R. Rabinovich G. Rahonis R. Rolland Y. Roos K. Salomaa P. Spirakis M. Steinby S. Tison N. Tzanakis G. Vaszil H. Vogler S. Yu S S. Yu Organizing Committee Archontia Grammatikopoulou Antonios Kalampakas Eleni Mandrali Athanasios Papistas Dimitrios Poulakis (Co-chairman) George Rahonis (Chairman) Sponsors OPAP Aristotle University of Thessaloniki Attiko Metro S.A. Research Academic Computer Technology Institute (Fronts) Ziti Publications. Table of Contents Invited Paper of Werner Kuich Cycle-Free Finite Automata in Partial Iterative Semirings 1 Stephen L. Bloom, Zoltan ´ Esik, and Werner Kuich Tutorials Picture Languages: From Wang Tiles to 2D Grammars 13 Alessandra Cherubini and Matteo Pradella Process Algebra: An Algebraic Theory of Concurrency 47 Wan Fokkink Invited Papers On Several Proofs of the Recognizability Theorem 78 Bruno Courcelle Theories of Automatic Structures and Their Complexity 81 Dietrich Kuske The Graph Programming Language GP 99 Detlef Plump Canonical Reduction Systems in Symbolic Mathematics 123 Franz Winkler Contributed Papers Solving Norm Form Equations over Number Fields 136 Paraskevas Alvanos and Dimitrios Poulakis A Note on Unambiguity, Finite Ambiguity and Complementation in Recognizable Two-Dimensional Languages 147 Marcella Anselmo and Maria Madonia Context-Free Categorical Grammars 160 Michel Bauderon, Rui Chen, and Olivier Ly An Eilenberg Theorem for Pictures 172 Symeon Bozapalidis and Archontia Grammatikopoulou X Table of Contents On the Complexity of the Syntax of Tree Languages 189 Symeon Bozapalidis and Antonios Kalampakas On the Reversibility of Parallel Insertion, and Its Relation to Comma Codes 204 Bo Cui, Lila Kari, and Shinnosuke Seki Computation of Pell Numbers of the Form pX 2 220 Konstantinos A. Draziotis Iteration Grove Theories with Applications 227 Z. ´ Esik and T. Hajgat´o Combinatorics of Finite Words and Suffix Automata 250 Gabriele Fici Polynomial Operators on Classes of Regular Languages 260 Ondˇrej Kl´ıma and Libor Pol´ak Self-dual Codes over Small Prime Fields from Combinatorial Designs 278 Christos Koukouvinos and Dimitris E. Simos A Backward and a Forward Simulation for Weighted Tree Automata 288 Andreas Maletti Syntax-Directed Translations and Quasi-alphabetic Tree Bimorphisms—Revisited 305 Andreas Maletti and C˘at˘alin Ionut¸Tˆırn˘auc˘a Polynomial Interpolation of the k-th Root of the Discrete Logarithm 318 Gerasimos C. Meletiou Single-Path Restarting Tree Automata 324 Friedrich Otto and Heiko Stamer Parallel Communicating Grammar Systems with Regular Control 342 Dana Pardubsk´a, Martin Pl´atek, and Friedrich Otto Author Index 361 Cycle-Free Finite Automata in Partial Iterative Semirings Stephen L. Bloom 1 ,Zoltan ´ Esik 2, ,andWernerKuich 3, 1 Dept. of Computer Science Stevens Institute of Technology Hoboken, NJ. USA 2 Dept. of Computer Science University of Szeged Hungary 3 Institut f¨ur Diskrete Mathematik und Geometrie Technische Universit¨at Wien Austria Abstract. We consider partial Conway semirings and partial iteration semirings, both introduced by Bloom, ´ Esik, Kuich [2]. We develop a theory of cycle-free elements in partial iterative semirings that allows us to define cycle-free finite automata in partial iterative semirings and to prove a Kleene Theorem. We apply these results to power series over a graded monoid with discounting. 1 Introduction Cycle-free power series r ∈ S Σ ∗  ,whereS is a semiring and Σ is an alphabet, are defined by the condition that (r, ε), the coefficient of r at the empty word ε,is nilpotent. Transferring this notion via its transition matrix to a finite automaton assures that the behavior of a cycle-free finite automaton is well defined. This fact makes it possible to generalize classical finite automata with ε-moves to weighted cycle-free finite automata (see Kuich, Salomaa [11], ´ Esik, Kuich [9]). In this paper, we take an additional step of generalization. We consider cycle- free elements in a partial iterative semiring and consider cycle-free finite au- tomata. This generalization preserves all the nice results of weighted cycle-free finite automata and allows us to prove the usual Kleene Theorem stating the coincidence of the sets of recognizable and rational elements. This paper consists of this and three more sections. In Section 2 we consider partial iterative semirings and partial Conway semirings, both introduced by Bloom, ´ Esik, Kuich [9]. Moreover, we define cycle-free elements in partial itera- tive semirings and prove several identities involving these cycle-free elements. In Section 3 we introduce cycle-free finite automata in partial iterative semirings,  Partially supported by grant no. K 75249 from the National Foundation of Scientific Research of Hungary, and by Stiftung Aktion ¨ Osterreich-Ungarn.  Partially supported by Stiftung Aktion ¨ Osterreich-Ungarn. S. Bozapalidis and G. Rahonis (Eds.): CAI 2009, LNCS 5725, pp. 1–12, 2009. c  Springer-Verlag Berlin Heidelberg 2009 2 S.L. Bloom, Z. ´ Esik, and W. Kuich define recognizable and rational elements and prove a Kleene Theorem: an ele- ment is recognizable iff it is rational. In Section 4 we apply the results to power series over a finitely generated graded monoid with discounting. 2 Cycle-Free Elements in Partial Iterative Semirings Suppose that S is a semiring and I is an ideal of S,sothat0∈ I, I + I ⊆ I and IS ∪ SI ⊆ I.AccordingtoBloom, ´ Esik, Kuich [2], S is a partial iterative semiring over I if for all a ∈ I and b ∈ S the equation x = ax + b has a unique solution in S. We denote this unique solution by a ∗ b. Example. This is a running example for the whole paper. Let S be a semiring and Σ an alphabet, and consider the power series semiring S Σ ∗  .Apower series r ∈ S Σ ∗  is called proper if (r, ε) = 0. Clearly, the collection of proper power series forms an ideal I = {r ∈ S Σ ∗  | (r, ε)=0}. By Theorem 5.1 of Droste, Kuich [4], S Σ ∗  is a partial iterative semiring over the ideal I,where the ∗ of a proper power series r is defined by r ∗ =  j≥0 r j . ✷ In the rest of this section we suppose that S is a partial iterative semiring over I.Moreover,weletJ denote the set of all a ∈ S such that a k ∈ I for some k ≥ 1. Note that if a k ∈ I then a m ∈ I for all m ≥ k.Whena k is in I,wesaythata is cycle free with index k. We clearly have I ⊆ J. Proposition 1. If a ∈ I and b ∈ J then a + b ∈ J. Moreover, if a, b ∈ S with ab ∈ J then ba ∈ J. Proof. If a ∈ I and b ∈ J with b k ∈ I,then(a + b) k is a sum of terms which are k-fold products over {a, b}. Each such product is in I since it is either b k or contains a as a factor. Since I is closed under sum, it follows that (a + b) k is in I and thus a + b is in J. Suppose now that a, b ∈ S with (ab) k ∈ I for some k ≥ 1. Then (ba) k+1 = b(ab) k b ∈ I,provingthatba ∈ J. ✷ The following fact was shown in Bloom, ´ Esik, Kuich [2]. Proposition 2. Suppose that a ∈ J and b ∈ S. Then the equation x = ax+b has a unique solution. Moreover, its unique solution is a ∗ b,wherea ∗ is the unique solution of the equation x = ax +1. Thus,wehaveapartial ∗ -operation S → S defined on the set J of cycle-free elements. Proposition 3. Suppose that a, b ∈ J and c ∈ S.Ifac = cb,thena ∗ c = cb ∗ .In particular, a(a m ) ∗ =(a m ) ∗ a, for all a ∈ J and m ≥ 1. Proof. We have acb ∗ + c = cbb ∗ + c = c(bb ∗ +1) = cb ∗ ,sothata ∗ c = cb ∗ by uniqueness. ✷ Proposition 4. Suppose that a, b ∈ S such that ab ∈ J.Thenba ∈ J, moreover, (ab) ∗ a = a(ba) ∗ and a(ba) ∗ b +1=(ab) ∗ . [...]... scanned during a scanning process and together with property 4 forbids the existence of holes in the picture during the scanning process In [4] several examples of continuous normalized scanning strategies are given, showing the richness of possibilities in 2D case, and they produce, for suitable data structures, different definitions of tiling automata Here we introduce a formal definition of tiling automata... tiles admitting a valid tiling admits a quasiperiodic valid tiling The quasi-periodicity function for a quasi periodic tiling τ is the function that associate to each integer x the minimal size n of the squares in which one can find all the patterns of size x appearing in the tiling This function enables to characterize quasi periodic tilings that are periodic Proposition 2 A quasi periodic tiling is periodic... A pattern appears in a tiling τ : Z2 → T if the tiling is the extension of the image of the pattern by a shift A valid tiling τ : Z2 → T is called quasi-periodic if for each pattern M appearing in τ there is an integer n such that M appears in all n × n squares in τ A valid quasi periodic tiling that is not periodic is called strictly quasi-periodic In [24] Durand proved the following Picture Languages:... there are finite sets of tiles which admit only non-periodic tilings of the plane A finite set of Wang tiles which admits only non-periodic valid tiling is said aperiodic In 1966 Berger [8], proved the following Theorem 1 The plane tiling problem is undecidable His proof is based on encoding the halting problem of Turing Machine in the valid tiling of an arbitrary large square portion of the plane Moreover,... Then, using a counting argument on trees suitably associated to valid tilings, Durand obtains the following Theorem 3 If a tile set can be used to form a strictly quasi-periodic tiling of the plane, then it can form an uncountable number of different tilings It is important to note that valid tilings could be defined in several different ways For instance one could arrange all edge colors in complementary... of local rules L ⊆ T 4 A tiling τ satisfies the local rules L if and only if all 2 × 2 patterns appearing in the tiling are in L In [26] the authors give via this approach a new short proof of the existence of aperiodic tilings Besides the strong connections with first order and description logics [25] yet arising from its original motivation, tiling problems have appeared in many branches of physics... series Theoretical Computer Science 366, 199–227 (2006) 6 Droste, M., Sakarovitch, J., Vogler, H.: Weighted automata with discounting Information Processing Letters 108, 23–28 (2008) 7 Esik, Z., Kuich, W.: Equational axioms for a theory of automata In: Martin-Vide, C., Mitrana, V., Paun, G (eds.) Formal Languages and Applications Studies in Fuzziness and Soft Computing, 148, pp 183–196 Springer, Heidelberg... corresponding non-deterministic automata In the deterministic case the family of languages recognized by tiling automata depends on the chosen scanning strategy, so L(DTA) denotes the set of all languages recognized by a deterministic d-tiling automata for each scanning strategy in any direction d ∈ c2c and DREC = L(DTA) Moreover the family L(4DFA) of languages recognized by a deterministic 4-way automaton... pictures with at least two equal columns is in REC, but not in UREC Hence Theorem 5 ([5]) UREC is strictly included in REC The notion of determinism for tiling systems has to be referred to a direction, like in 1D case The considered direction is one of the four main directions from a corner to another (c2c) Definition 5 A tiling system (Σ, Γ, Θ, π) is tl2br-deterministic 4 if for any γ1 , γ2 , γ3 ∈ Γ ∪... semirings Fundamenta Informaticae 86, 19–40 (2008) 3 Conway, J.H.: Regular Algebra and Finite Machines Chapman and Hall, Boca Raton (1971) 4 Droste, M., Kuich, W.: Semirings and formal power series In: Droste, M., Kuich, W., Vogler, H (eds.) Handbook of Weighted Automata EATCS Monographs on Theoretical Computer Science, ch 1 Springer, Heidelberg (to appear, 2009) 5 Droste, M., Kuske, D.: Skew and in nitary . Lecture Notes in Computer Science 5725 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis,. Berlin Heidelberg 2009 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper SPIN: 12735013 06/3180 543210 Preface CAI. semirings, both introduced by Bloom, ´ Esik, Kuich [9]. Moreover, we define cycle-free elements in partial itera- tive semirings and prove several identities involving these cycle-free elements. In Section