Semilinear Sets and Counter Machines: a Brief Survey

Semilinear sets are one of the most important concepts in theoretical computer science, as illustrated by the fact that the set of nonnegative integer solutions to any system of Diophantine equations is semilinear. Parikh's theorem enables us to represent any semilinear set as a pushdown automaton PDA. We summarize recent results on the descriptional complexity of conversions among different representations of a semilinear set: as a vector set conventional, a finite automaton FA, a PDA, etc.. We also discuss semilinearity-preserving operations like union, intersection, and complement. We use Parikh's theorem to enlarge the class of finite-state machines that can represent semilinear sets. In particular, we give a simpler proof of a known result that characterizes semilinear sets in terms of machines with reversal-bounded counters. We then investigate the power of such a machine with only one counter in the context of a long-standing conjecture about repetition on words.

[1]  Shinnosuke Seki,et al.  Operational State Complexity under Parikh Equivalence ? (Extended Abstract) , 2014 .

[2]  Eitan M. Gurari,et al.  Two-Way Counter Machines and Diophantine Equations , 1982, JACM.

[3]  Oscar H. Ibarra,et al.  Characterizations of Bounded semilinear Languages by One-Way and Two-Way Deterministic Machines , 2012, Int. J. Found. Comput. Sci..

[4]  Tao Jiang,et al.  New Decidability Results Concerning Two-Way Counter Machines , 1995, SIAM J. Comput..

[5]  Luca Aceto,et al.  A Fully Equational Proof of Parikh's Theorem , 2001, RAIRO Theor. Informatics Appl..

[6]  Jeffrey Shallit,et al.  Unary Context-Free Grammars and Pushdown Automata, Descriptional Complexity and Auxiliary Space Lower Bounds , 2002, J. Comput. Syst. Sci..

[7]  Sheila A. Greibach,et al.  An Infinite Hierarchy of Context-Free Languages , 1969, JACM.

[8]  Eitan M. Gurari,et al.  The Complexity of Decision Problems for Finite-Turn Multicounter Machines , 1981, J. Comput. Syst. Sci..

[9]  Oscar H. Ibarra,et al.  One-reversal counter machines and multihead automata: Revisited , 2011, Theor. Comput. Sci..

[10]  Oscar H. Ibarra,et al.  Reversal-Bounded Multicounter Machines and Their Decision Problems , 1978, JACM.

[11]  Joost Engelfriet,et al.  Finitary Compositions of Two-way Finite-State Transductions , 2007, Fundam. Informaticae.

[12]  Marvin Minsky,et al.  Computation : finite and infinite machines , 2016 .

[13]  Masami Ito,et al.  Formal Languages Consisting of Primitive Words , 1993, FCT.

[14]  Thiet-Dung Huynh.  The Complexity of Semilinear Sets , 1982, J. Inf. Process. Cybern..

[15]  Anthony Widjaja Lin,et al.  Model Checking Recursive Programs with Numeric Data Types , 2011, CAV.

[16]  Javier Esparza.  Petri Nets, Commutative Context-Free Grammars, and Basic Parallel Processes , 1995, FCT.

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

[18]  H. Wilf,et al.  Uniqueness theorems for periodic functions , 1965 .

[19]  Pierre Ganty,et al.  Parikhʼs theorem: A simple and direct automaton construction , 2010, Inf. Process. Lett..

[20]  Oscar H. Ibarra,et al.  On the Parikh Membership Problem for FAs, PDAs, and CMs , 2014, LATA.

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

[22]  William F. Ogden,et al.  A helpful result for proving inherent ambiguity , 1968, Mathematical systems theory.

[23]  Shinnosuke Seki,et al.  Converting nondeterministic automata and context-free grammars into Parikh equivalent one-way and two-way deterministic automata , 2012, Inf. Comput..

[24]  Rich Schroeppel,et al.  A Two Counter Machine Cannot Calculate 2N , 1972 .

[25]  Anthony Widjaja Lin,et al.  Parikh Images of Grammars: Complexity and Applications , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[26]  Oscar H. Ibarra,et al.  A Technique for Proving Decidability of Containment and Equivalence of Linear Constraint Queries , 1999, J. Comput. Syst. Sci..

[27]  Tamás Gergely,et al.  First-Order Logics , 2010 .

[28]  Javier Esparza.  Petri Nets, Commutative Context-Free Grammars, and Basic Parallel Processes , 1997, Fundam. Informaticae.

[29]  Andreas Malcher,et al.  First-order logics: some characterizations and closure properties , 2012, Acta Informatica.

[30]  Oscar H. Ibarra,et al.  On the solvability of a class of diophantine equations and applications , 2006, Theor. Comput. Sci..

[31]  Jean-Marc Talbot,et al.  Properties of Visibly Pushdown Transducers , 2010, MFCS.

[32]  Anthony Widjaja To,et al.  Parikh Images of Regular Languages: Complexity and Applications , 2010, 1002.1464.

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

[34]  Oscar H. Ibarra,et al.  On two-way nondeterministic finite automata with one reversal-bounded counter , 2005, Theor. Comput. Sci..

[35]  Oscar H. Ibarra,et al.  On the containment and equivalence problems for two-way transducers , 2012, Theor. Comput. Sci..

[36]  Grzegorz Rozenberg,et al.  Handbook of formal languages, vol. 1: word, language, grammar , 1997 .

[37]  B. S. Baker,et al.  Reversal-bounded multipushdown machines. [Turing acceptors for context free languages] , 1974 .

[38]  S. Ginsburg,et al.  BOUNDED ALGOL-LIKE LANGUAGES^) , 1964 .

[39]  Oscar H. Ibarra,et al.  On two-way FA with monotonic counters and quadratic Diophantine equations , 2004, Theor. Comput. Sci..

[40]  Stanislav Böhm,et al.  Equivalence of deterministic one-counter automata is NL-complete , 2013, STOC '13.

[41]  Brenda S. Baker,et al.  Reversal-Bounded Multipushdown Machines , 1974, J. Comput. Syst. Sci..

[42]  Oscar H. Ibarra,et al.  On Counter Machines, Reachability Problems, and Diophantine Equations , 2008, Int. J. Found. Comput. Sci..