On the rational subset problem for groups

We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for G-automaton subsets of M. It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership.

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

[2]  Ronald L. Rivest.  On the Notion of Pseudo-Free Groups , 2004, TCC.

[3]  Robert Gilman Michael Shapiro.  On groups whose word problem is solved by a nested stack automaton , 1998 .

[4]  Ilya Kapovich,et al.  Foldings, Graphs of Groups and the Membership Problem , 2005, Int. J. Algebra Comput..

[5]  Brigitte Servatius,et al.  Groups assembled from free and direct products , 1992, Discret. Math..

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

[7]  Vladimir Shpilrain.  Assessing security of some group based cryptosystems , 2003, IACR Cryptol. ePrint Arch..

[8]  M. Benois.  Descendants of Regular Language in a Class of Rewriting Systems: Algorithm and Complexity of an Automata Construction , 1987, RTA.

[9]  Ronald V. Book,et al.  Monadic Thue Systems , 1982, Theor. Comput. Sci..

[10]  Stuart W. Margolis,et al.  On one-relator inverse monoids and one-relator groups , 2001 .

[11]  Alfred V. Aho,et al.  Nested Stack Automata , 1969, Journal of the ACM.

[12]  Robert H. Gilman,et al.  Formal languages and infinite groups , 1995, Geometric and Computational Perspectives on Infinite Groups.

[13]  John R. Stallings,et al.  Algorithms in geometric group theory , 1999 .

[14]  Nellie Clarke Brown.  Trees , 1896, Savage Dreams.

[15]  David E. Muller,et al.  Groups, the Theory of Ends, and Context-Free Languages , 1983, J. Comput. Syst. Sci..

[16]  Mark Kambites,et al.  On commuting elements and embeddings of graph groups and monoids , 2006, Proceedings of the Edinburgh Mathematical Society.

[17]  Robert H. Gilman,et al.  Geometric and Computational Perspectives on Infinite Groups , 1995 .

[18]  M. J. Dunwoody.  The accessibility of finitely presented groups , 1985 .

[19]  Benjamin Steinberg.  Inverse automata and profinite topologies on a free group , 2002 .

[20]  M. Dehn.  Über unendliche diskontinuierliche Gruppen , 1911 .

[21]  Jeffrey D. Ullman,et al.  Formal languages and their relation to automata , 1969, Addison-Wesley series in computer science and information processing.

[22]  A. Karrass,et al.  Finite and infinite cyclic extensions of free groups , 1973, Journal of the Australian Mathematical Society.

[23]  John E. Hopcroft.  On the equivalence and containment problems for context-free languages , 2005, Mathematical systems theory.

[24]  Stuart W. Margolis,et al.  Free Inverse Monoids and Graph immersions , 1993, Int. J. Algebra Comput..

[25]  Randolph B. Tarrier,et al.  Groups , 1973, Algebra.

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

[27]  Claas E. Röver,et al.  Groups with Context‐Free Co‐Word Problem , 2005 .

[28]  John R. Stallings,et al.  Topology of finite graphs , 1983 .

[29]  S. Margolis,et al.  Distortion functions and the membership problem for submonoids of groups and monoids , 2004 .

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

[31]  T. Hall,et al.  Journal of Algebra , 1964, Nature.

[32]  Pascal Weil,et al.  PSPACE-complete problems for subgroups of free groups and inverse finite automata , 2000, Theor. Comput. Sci..

[33]  B. M. Fulk.  MATH , 1992 .

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

[35]  Alfred V. Aho.  Indexed Grammars-An Extension of Context Free Grammars , 1967, SWAT.

[36]  Gillian Z. Elston,et al.  On groups whose word problem is solved by a~counter automaton , 2004, Theor. Comput. Sci..

[37]  D. J. Collins.  Review: K. A. Mihajlova, (Problema vhozdenia did pramyh proizvedenij grupp):The Occurrence Problem for Direct Products of Groups , 1971 .

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

[39]  Friedrich Otto,et al.  String-Rewriting Systems , 1993, Text and Monographs in Computer Science.

[40]  Mark Kambites,et al.  Formal Languages and Groups as Memory , 2006, math/0601061.

[41]  J. Sakarovitch,et al.  Finiteness Conditions on Subgroups and Formal Language Theory , 1989 .

[42]  R. Lyndon,et al.  Combinatorial Group Theory , 1977 .