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Kleene algebra with tests

Editor: William Pugh Author:

Dexter KozenAuthors Info & Claims

ACM Transactions on Programming Languages and Systems (TOPLAS), Volume 19, Issue 3

Pages 427 - 443

https://doi.org/10.1145/256167.256195

Published: 01 May 1997 Publication History

Abstract

We introduce Kleene algebra with tests, an equational system for manipulating programs. We give a purely equational proof, using Kleene algebra with tests and commutativity conditions, of the following classical result: every while program can be simulated by a while program can be simulated by a while program with at most one while loop. The proof illustrates the use of Kleene algebra with tests and commutativity conditions in program equivalence proofs.

References

[1]

AHO, A. V., HOPCROFT, J. E., AND ULLMAN, J. D. 1975. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Mass.

[2]

ANDERAA, S. 1965. On the algebra of regular expressions. Appl. Math., 1-18.

[3]

ARCHANGELSKY, I<. V. 1992. A new finite complete solvable quasiequational calculus for algebra of regular languages. Manuscript, Kiev State University.

[4]

BACKHOUSE, R. C. 1975. Closure algorithms and the star-height problem of regular languages. Ph.D. thesis, Imperial College, London, U.K.

[5]

BERSTEL, J. 1979. Transductions and Context-free Languages. Teubner, Stuttgart, Germany.

[6]

BLOOM, S. L. AND t~SIK, Z. 1993. Equational axioms for regular sets. Math. Struct. Comput. Sci. 3, 1-24.

[7]

BOFFA, iV{. 1990. Une remarque sur les syst6mes complets d'identit6s rationnelles. Informatique Thdoretique et Applications//Theoretical Informatics and Applications 24, 4, 419-423.

[8]

BOHM, C. AND JACOPINI, C. 1966. Flow diagrams, Turing machines, and languages with only two formation rules. Commun. A CM 9, 5 (May), 366-371.

[9]

COHEN, E. 1994a. Hypotheses in Kleene algebra. Available from ftp:////ftp, b ellcore, corn//pub//er nie//research//homepage, ht ml

[10]

COHEN, E. 1994b. Lazy caching. Available from ftp:////ftp, b ellcore, corn//pub//er nie//research//homepage, ht ml

[11]

COHEN, E. 1994c. Using Kleene algebra to reason about concurrency control. Available from ftp://ftp, b ellcore, corn/pub / er nie/research / homepage, ht ml

[12]

COHEN, E., KOZEN, D. C., AND SMITH, F. 1996. The complexity of Kleene algebra with tests. Tech. Rep. TR96-1598, Cornell University. July.

[13]

CONWAY, J. H. 1971. Regular Algebra and Finite Machines. Chapman and Hall, London, U.K.

[14]

FISCHER, M. J. AND LADNER, R. E. 1979. Propositional dynamic logic of regular programs. J. Comput. Syst. Sci. 18, 2, 194-211.

[15]

GIBBONS, A. AND RYTTER, W. 1986. On the decidability of some problems about rational subsets of free partially commutative monoids. Theor. Comput. Sci. 48, 329-337.

[16]

GORSHKOV, P. V. 1989. Rational data structures and their applications. Cybernetics 25, 6 (Nov.- Dec.), 760-765.

[17]

HAREL, D. 1980. On folk theorems. Commun. ACM 23, 7 (July), 379-389.

[18]

HIROSE, K. AND OYA, M. 1972. General theory of flowcharts. Comment. Math. Univ. St. Pauli 21, 2, 55-71.

[19]

IWANO, K. AND STEIGLITZ, K. 1990. A semiring on convex polygons and zero-sum cycle problems. SIAM J. Comput. 19, 5, 883-901.

[20]

KLEENE, S. C. 1956. Representation of events in nerve nets and finite automata. In Automata Studies, C. E. Shannon and J. McCarthy, Eds. Princeton University Press, Princeton, N.J., 3-41.

[21]

KOZEN, D. C. 1981. On induction vs. *-continuity. In Proc. Workshop on Logic of Programs, Kozen, Ed. Lecture Notes in Computer Science, vol. 131. Springer-Verlag, New York, 167-176.

[22]

KOZEN, D. C. 1990. On Kleene algebras and closed semirings. In Proc. Math. Found. Comput. Sci., Rovan, Ed. Lecture Notes in Computer Science, vol. 452. Springer-Verlag, Banska-Bystrica, Slovakia, 26-47.

[23]

KOZEN, D. C. 1991. The Design and Analysis of Algorithms. Springer-Verlag, New York.

[24]

KOZEN, D. C. 1994. A completeness theorem for Kleene algebras and the algebra of regular events. Infor. and Comput. 110, 2 (May), 366-390.

[25]

KOZEN, D. C. 1996. Kleene algebra with tests and commutativity conditions. In Proc. Second Int. Workshop Tools and Algorithms for the Construction and Analysis of Systems (TACAS'96), T. Margaria and B. Steffen, Eds. Lecture Notes in Computer Science, vol. 1055. Springer-Verlag, Passau,Germany, 14-33.

[26]

KOZEN, D. C. 1997. On the complexity of reasoning in Kleene algebra. In Proc. 12th Syrup. Logic in Comput. Sci. IEEE, Los Alamitos, Ca. To appear. Cornell Tech. Report 97-1624, March 1997.

[27]

KOZEN, D. C. AND SMITH, F. 1996. Kleene algebra with tests: Completeness and decidability. In Proc. Conf. Computer Science Logic (CSL'96). Lecture Notes in Computer Science. Springer- Verlag, New York. To appear. Cornell University Tech. Report TR96-1582, April, 1996.

[28]

KOZEN, D. C. AND TIURYN, J. 1990. Logics of programs. In Handbook of Theoretical Computer Science, van Leeuwen, Ed. Vol. B. North Holland, Amsterdam, 789-840.

[29]

KROB, D. 1991. A complete system of B-rational identities. Theor. Comput. Sci. 89, 2 (October), 207-343.

[30]

KUICH, W. 1987. The Kleene and Parikh theorem in complete semirings. In Proc. l~th Colloq. Automata, Languages, and Programming, T. Ottmann, Ed. Lecture Notes in Computer Science, vol. 267. EATCS, Springer-Verlag, New York, 212-225.

[31]

KUICH, W. AND SALOMAA, A. 1986. Semirings, Automata, and Languages. Springer-Verlag, Berlin.

[32]

MIRKOWSKA, G. 1972. Algorithmic logic and its applications. Ph.D. thesis, University of Warsaw. in Polish.

[33]

NG, K. C. 1984. Relation algebras with transitive closure. Ph.D. thesis, University of California, Berkeley.

[34]

PRATT, V. 1988. Dynamic algebras as a well-behaved fragment of relation algebras. In Proc. Conf. on Algebra and Computer Science, D. Pigozzi, Ed. Lecture Notes in Computer Science, vol. 425. Springer-Verlag, Ames, Iowa, 77-110.

[35]

PRATT, V. 1990. Action logic and pure induction. In Proc. Logics in AI: European Workshop JELIA '90, J. van Eijck, Ed. Lecture Notes in Computer Science, vol. 478. Springer-Verlag, New York, 97-120.

[36]

REDKO, V. N. 1964. On defining relations for the algebra of regular events. Ukrain. Mat. Z. 16, 120-126. In Russian.

[37]

SAKAROVITCH, J. 1987. Kleene's theorem revisited: A formal path from Kleene to Chomsky. In Trends, Techniques, and Problems in Theoretical Computer Science, A. Kelemenova and J. Keleman, Eds. Lecture Notes in Computer Science, vol. 281. Springer-Verlag, New York, 39-50.

[38]

SALOMAA, t. 1966. Two complete axiom systems for the algebra of regular events. J. Assoc. Comput. Mach. 13, 1 (January), 158-169.

[39]

STOCKMEYER, L. J. AND MEYER, A. R. 1973. Word problems requiring exponential time. In Proc. 5th Syrup. Theory of Computing. ACM, ACM, New York, 1-9.

[40]

TARSKI, A. 1941. On the calculus of relations. J. Symb. Logic 6, 3, 65-106.

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Published In

cover image ACM Transactions on Programming Languages and Systems

ACM Transactions on Programming Languages and Systems Volume 19, Issue 3

May 1997

143 pages

ISSN:0164-0925

EISSN:1558-4593

DOI:10.1145/256167

Editor:
William Pugh
Univ. of Maryland, College Park

Issue’s Table of Contents

Copyright © 1997 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 May 1997

Published in TOPLAS Volume 19, Issue 3

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