The max-plus algebra of the natural numbers has no finite equational basis

Theoretical Computer Science

Volumn 293, Issue 1, 2003, Pages 169-188

  Aceto, Luca ^a   Ésik, Zoltán ^b   Ingólfsdóttir, Anna ^a  

^a AALBORG UNIVERSITY   (Denmark)

^b UNIVERSITY OF SZEGED   (Hungary)

Author keywords

Complete axiomatizations; Equational logic; Exponential time complexity; Varieties

Indexed keywords

AUTOMATA THEORY; COMPUTABILITY AND DECIDABILITY; COMPUTATIONAL COMPLEXITY;

NATURAL NUMBERS;

ALGEBRA;

EID: 0037415373     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(02)00236-0     Document Type: Conference Paper

References (27)

1. L. Aceto, Z. Ésik, A. Ingólfsdóttir, On the two-variable fragment of the equational theory of the max-sum algebra of the natural numbers, in: H. Reichel, S. Tison (Eds.), Proc. 17th Internat. Symp. on Theoret. Aspects Comput. Sci., Vol. 1770, Lecture Notes in Comput. Sci., Springer, Berlin, 2000, pp. 267-278. Full version available at the URL http://www.brics.dk/RS/99/22/index.html.
2. Aceto L., Fokkink W.J., Ingólfsdóttir A. A menagerie of non-finitely based process semantics over BPA* - from ready simulation to completed traces. Math. Structures Comput. Sci. 8:1998;193-230.
3. Bergstra J., Bethke I., Ponse A. Process algebra with iteration and nesting. Comput. J. 37:1994;243-258.
4. J. Bergstra, J.W. Klop, Fixed point semantics in process algebras, Report IW 206, Mathematisch Centrum, Amsterdam, 1982.
5. Bloom B., Istrail S., Meyer A.R. Bisimulation can't be traced. J. Assoc. Comput. Mach. 42:1995;232-268.
6. Burris S., Sankappanavar H.P. A Course in Universal Algebra. 1981;Springer, New York.
7. Conway J.H. Regular Algebra and Finite Machines, Mathematics Series. 1971;Chapman & Hall, London, UK.
8. Crvenković S., Dolinka I., Ésik Z. The variety of Kleene algebras with conversion is not finitely based. Theoret. Comput. Sci. 230(1-2):2000;235-245.
9. Ésik Z., Bertol M. Nonfinite axiomatizability of the equational theory of shuffle. Acta Inform. 35:1998;505-539.
10. M.J. Fischer, M.O. Rabin, Super-exponential complexity of Presburger arithmetic, in: Complexity of Computation (Proc. SIAM-AMS Symposium, New York, 1973), SIAM-AMS Proc., Vol. VII, American Mathematical Society, Providence, RI, 1974, pp. 27-41.
11. Fokkink W.J., Zantema H. Basic process algebra with iteration: completeness of its equational axioms. Comput. J. 37:1994;259-267.
12. Grätzer G. Universal Algebra. 2nd Edition:1979;Springer, New York.
13. Larsen K.G., Skou A. Bisimulation through probabilistic testing. Inform. and Comput. 94:1991;1-28.
14. Lyndon R. Identities in two-valued calculi. Trans. Amer. Math. Soc. 71:1951;457-465.
15. McKenzie R. Equational bases for lattice theories. Math. Scand. 27:1970;24-38.
16. McKenzie R. Tarski's finite basis problem is undecidable. Internat. J. Algebra Comput. 6:1996;49-104.
17. McNulty G. A field guide to equational logic. J. Symbol. Comput. 14:1992;371-397.
18. V.L. Murskiǐ, The existence in three-valued logic of a closed class with finite basis, not having a finite complete system of identities, Dokl. Akad. Nauk SSSR 163 (1965) 815-818, Sov. Math. Dokl. 6 (1965) 1020-1024 (in English).
19. V.L. Murskiǐ, The existence in three-valued logic of a closed class with finite basis, not having a finite complete system of identities, Dokl. Akad. Nauk SSSR 163 (1965) 815-818, Sov. Math. Dokl. 6 (1965) 1020-1024 (in English).
20. Murskiǐ V.L. The existence of a finite basis and some other properties of "almost all" finite algebras. Problemy Kibernet. 30:1975;43-56.
21. Oates S., Powell M.B. Identical relations in finite groups. J. Algebra. 1:1965;11-39.
22. R.E. Park, Equational classes of non-associative ordered algebras, Ph.D. Thesis, University of California at Los Angeles, 1976.
23. Park D. Concurrency and automata on infinite sequences. Deussen P. 5th GI Conf., Karlsruhe, Germany. Lecture Notes in Computer Science. Vol. 104:1981;167-183 Springer, Berlin.
24. Schrijver A. Theory of Linear and Integer Programming. 1986;Wiley, New York.
25. W. Taylor, Equational logic, in: B. Csákány, J. Schmidt (Eds.), Contributions to Universal Algebra, North-Holland, Amsterdam, 1977, pp. 465-501; Proc. Colloq. held in Szeged, 1975, Colloquia Mathematica Societatis, János Bolyai, Vol. 17.
26. W. Taylor, Equational logic, in: B. Csákány, J. Schmidt (Eds.), Contributions to Universal Algebra, North-Holland, Amsterdam, 1977, pp. 465-501; Proc. Colloq. held in Szeged, 1975, Colloquia Mathematica Societatis, János Bolyai, Vol. 17.
27. W. Taylor, Equational logic, in [12], Appendix 4, pp. 378-400.