Abstract
We present three results on divisibility monoids. These divisibility monoids were introduced in [11] as an algebraic generalization of Mazurkiewicz trace monoids. (1) We give a decidable class of presentations that gives rise precisely to all divisibility monoids. (2) We show that any divisibility monoid is an automatic monoid [5]. This implies that its word problem is solvable in quadratic time. (3) We investigate when a divisibility monoid satisfies Kleeneās Theorem. It turns out that this is the case iff the divisibility monoid is a rational monoid [25] iff it is width-bounded. The two latter results rest on a normal form for the elements of a divisibility monoid that generalizes the Foata normal form known from the theory of Mazurkiewicz traces.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Berry. Stable models of typed Ī»-calculi. In 5th ICALP, Lecture Notes in Comp. Science vol. 62, pages 72ā89. Springer, 1978.
G. Birkhoff. Lattice Theory. Colloquium Publications vol. 25. American Mathematical Society, Providence, 1973.
B. Bosbach. Representable divisibility semigroups. Proc. Edinb. Math. Soc., II. Ser., 34(1):45ā64, 1991.
J.R. BĆ¼chi. On a decision method in restricted second order arithmetics. In E. Nagel et al., editor, Proc. Intern. Congress on Logic, Methodology and Philosophy of Science, pages 1ā11. Stanford University Press, Stanford, 1960.
C. M. Campbell, E. F. Robertson, N. RuÅ”kuc, and R. M. Thomas. Automatic semigroups. Theoretical Computer Science, 250:365ā391, 2001.
R. Corran. On monoids related to braid groups. PhD thesis, University of Sydney, 2000.
P. Dehornoy and L. Paris. Gaussian groups and Garside groups, two generalizations of Artin groups. Proc. London Math. Soc., 79:569ā604, 1999.
V. Diekert. Combinatorics on Traces. Lecture Notes in Comp. Science vol. 454. Springer, 1990.
V. Diekert and G. Rozenberg. The Book of Traces. World Scientific Publ. Co., 1995.
M. Droste. Recognizable languages in concurrency monoids. Theoretical Comp. Science, 150:77ā109, 1995.
M. Droste and D. Kuske. On recognizable languages in divisibility monoids. In G. Ciobanu and Gh. Paun, editors, FCT99, Lecture Notes in Comp. Science vol. 1684, pages 246ā257. Springer, 1999.
M. Droste and D. Kuske. Recognizable languages in divisibility monoids. Mathematical Structures in Computer Science, 2000. To appear.
C. Duboc. Commutations dans les monoĆÆdes libres: un cadre thĆ©orique pour lāĆ©tude du parallelisme. ThĆØse, FacultĆ© des Sciences de lāUniversitĆ© de Rouen, 1986.
C.C. Elgot and G. Mezei. On relations defined by generalized finite automata. IBM J. Res. Develop., 9:47ā65, 1965.
D.B.A. Epstein, J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, and W.P. Thurston. Word Processing In Groups. Jones and Bartlett Publishers, Boston, 1992.
The GAP Group, Aachen, St Andrews. GAP-Groups, Algorithms, and Programming, Version 4.2, 1999. (http://www-gap.dcs.st-and.ac.uk/~gap).
J.F.P. Hudson. Regular rewrite systems and automatic structures. In J. Almeida, G.M.S. Gomes, and P.V. Silva, editors, Semigroups, Automata and Languages, pages 145ā152, Singapure, 1996. World Scientific.
S.C. Kleene. Representation of events in nerve nets and finite automata. In C.E. Shannon and J. McCarthy, editors, Automata Studies, Annals of Mathematics Studies vol. 34, pages 3ā40. Princeton University Press, 1956.
D. Kuske. Contributions to a Trace Theory beyond Mazurkiewicz Traces. Technical report, TU Dresden, 1999.
D. Kuske. On rational and on left divisibility monoids. Technical Report MATHAL-3-1999, TU Dresden, 1999.
A. Mazurkiewicz. Concurrent program schemes and their interpretation. Technical report, DAIMI Report PB-78, Aarhus University, 1977.
E. OchmaÅski. Regular behaviour of concurrent systems. Bull. Europ. Assoc. for Theor. Comp. Science, 27:56ā67, 1985.
M. Peletier and J. Sakarovitch. Easy multiplications. II. Extensions of rational semigroups. Information and Computation, 88:18ā59, 1990.
F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264ā286, 1930.
J. Sakarovitch. Easy multiplications. I. The realm of Kleeneās Theorem. Information and Computation, 74:173ā197, 1987.
M.P. SchĆ¼tzenberger. On the definition of a family of automata. Inf. Control, 4:245ā270, 1961.
G. Winskel. Event structures. In W. Brauer, W. Reisig, and G. Rozenberg, editors, Petri nets: Applications and Relationships to Other Models of Concurrency, Lecture Notes in Comp. Science vol. 255, pages 325ā392. Springer, 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kuske, D. (2001). Divisibility Monoids: Presentation, Word Problem, and Rational Languages. In: Freivalds, R. (eds) Fundamentals of Computation Theory. FCT 2001. Lecture Notes in Computer Science, vol 2138. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44669-9_23
Download citation
DOI: https://doi.org/10.1007/3-540-44669-9_23
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42487-1
Online ISBN: 978-3-540-44669-9
eBook Packages: Springer Book Archive