Partially commutative inverse monoids

Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.

[1]  Ronald V. Book,et al.  Confluent and Other Types of Thue Systems , 1982, JACM.

[2]  Robert Cori,et al.  Automates et Commutations Partielles , 1985, RAIRO Theor. Informatics Appl..

[3]  Anca Muscholl,et al.  Solvability of Equations in Graph Groups Is Decidable , 2006, Int. J. Algebra Comput..

[4]  Igor Walukiewicz.  Pushdown Processes: Games and Model-Checking , 2001, Inf. Comput..

[5]  Stuart W. Margolis,et al.  E-unitary inverse monoids and the Cayley graph of a group presentation , 1989 .

[6]  I. Walukiewicz.  Pushdown Processes: Games and Model Checking , 1996 .

[7]  Stuart W. Margolis,et al.  Inverse monoids, trees, and context-free languages , 1993 .

[8]  Ernst W. Mayr.  An Algorithm for the General Petri Net Reachability Problem , 1984, SIAM J. Comput..

[9]  Carl Droms,et al.  Graph groups, coherence, and three-manifolds , 1987 .

[10]  Orna Kupferman,et al.  An Automata-Theoretic Approach to Reasoning about Infinite-State Systems , 2000, CAV.

[11]  Mario Petrich,et al.  Inverse semigroups , 1985 .

[12]  Hans Bekic,et al.  Definable Operation in General Algebras, and the Theory of Automata and Flowcharts , 1984, Programming Languages and Their Definition.

[13]  Markus Lohrey.  On the Parallel Complexity of Tree Automata , 2001, RTA.

[14]  Samuel R. Buss.  Alogtime Algorithms for Tree Isomorphism, Comparison, and Canonization , 1997, Kurt Gödel Colloquium.

[15]  Walther Dyck.  Ueber Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann'scher Flächen , 1880 .

[16]  Jens Palsberg,et al.  Complexity Results for 1-Safe Nets , 1993, Theor. Comput. Sci..

[17]  Jean-Camille Birget,et al.  Almost finite expansions of arbitrary semigroups , 1984 .

[18]  Carl Droms,et al.  Subgroups of Graph Groups , 1987 .

[19]  M. Rabin.  Decidability of second-order theories and automata on infinite trees. , 1969 .

[20]  Richard J. Lipton,et al.  Word Problems Solvable in Logspace , 1977, JACM.

[21]  Walter Douglas Munn.  Free Inverse Semigroups , 1974 .

[22]  Pedro V. Silva,et al.  On the rational subset problem for groups , 2006, math/0602454.

[23]  J. Vree,et al.  I. Da Costa , 1999 .

[24]  Markus Lohrey,et al.  Inverse monoids: Decidability and complexity of algebraic questions , 2007, Inf. Comput..

[25]  S. Rao Kosaraju,et al.  Decidability of reachability in vector addition systems (Preliminary Version) , 1982, STOC '82.

[26]  John Meakin,et al.  The Word Problem in the Variety of Inverse Semigroups with Abélian Covers , 1996 .

[27]  Jacobo Torán,et al.  A note on the hardness of tree isomorphism , 1998, Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247).

[28]  J. B. Stephen.  Presentations of inverse monoids , 1990 .

[29]  Volker Diekert,et al.  The Book of Traces , 1995 .

[30]  Stuart W. Margolis,et al.  Algorithmic Problems in Groups, Semigroups and Inverse Semigroups , 1995 .

[31]  Anca Muscholl,et al.  Solvability of Equations in Free Partially Commutative Groups Is Decidable , 2001, ICALP.

[32]  Wolfgang Reisig.  Petri Nets: An Introduction , 1985, EATCS Monographs on Theoretical Computer Science.

[33]  Volker Diekert,et al.  Algorithmic Problems on Inverse Monoids over Virtually Free Groups , 2008, Int. J. Algebra Comput..

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

[35]  Celia Wrathall.  The Word Problem for Free Partially Commutative Groups , 1988, J. Symb. Comput..

[36]  Edward Fredkin,et al.  Trie memory , 1960, Commun. ACM.

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

[38]  David E. Muller,et al.  The Theory of Ends, Pushdown Automata, and Second-Order Logic , 1985, Theor. Comput. Sci..

[39]  Volker Diekert,et al.  Combinatorics on Traces , 1990, Lecture Notes in Computer Science.

[40]  Volker Diekert,et al.  Partially commutative inverse monoids , 2008 .

[41]  Markus Lohrey,et al.  Inverse Monoids: Decidability and Complexity of Algebraic Questions , 2005, MFCS.