Exact computation sequences

  • Alex Pelin
  • Jean H. Gallier
Algebraic Theory Of Semantics
Part of the Lecture Notes in Computer Science book series (LNCS, volume 214)


Exact computation sequences are sequences of the form
$$< L_0 ,A_0 > \mathop - \limits^{S_1 } > < L_1 ,A_1 > \mathop - \limits^{S_2 } > ...\mathop - \limits^{S_n } > < L_n ,\emptyset > ,$$
, where L0 is a free algebra, A0 is a set of conditional equations over L0, Si is a "step function", L i =S i (L i −1), and A i =S i (A i −1). Each step function is the top-down reduction extension of a set of confluent and noetherian rewrite rules. These sequences are used in solving the word problem for free algebras, since for any pair of terms t1,t2 in L,
$$t_1 = _{A_0 } t_2 iff S_n o...oS_1 \left( {t_1 } \right) = S_n o...oS_1 \left( {t_2 } \right).$$

We analyze properties of exact computation sequences such as: determining the relation between the sets <L i−1 ,A i−1 > and S i , and the output pair <L i ,A i >, and we present criteria for choosing the equations E i in <L i −1,A i −1> which are used to generate the reductions S i . We also give examples showing how to construct exact computation sequences for several axiom systems by applying the properties and the criteria presented in the article.


Word Problem Free Algebra Standard Language Automate Deduction Abstract Data Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

6. Bibliography

  1. 1.
    Barros L., and Remy J.L., Ecologiste: A system to make complete and consistent specifications easier, Proc. Workshop Rewrite Rule Laboratory, GE Research and Development, Schenectady, NY, 1983Google Scholar
  2. 2.
    Boyer R., Moore J.S., Computational Logic, Academic Press, 1979.Google Scholar
  3. 3.
    Fribourg L., A Narrowing Procedure for Theories with Constructors, 7th International Conference on Automated Deduction, R.E. Shostak, editor, Springer Verlag Lecture Notes in Computer Science, Vol. 170, 259–281.Google Scholar
  4. 4.
    Huet G., Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems, JACM, 1980, 767–821.Google Scholar
  5. 5.
    Huet G., Hullot J.M., Proofs by Induction in Equational Theories with Constructors, JCSS, Vol. 25, 1982, 239–266.Google Scholar
  6. 6.
    Huet G., Oppen D., Equations and Rewrite Rules: A Survey, Formal Language Theory: Perspectives and Open Problems, R. Book, editor, Academic Press, 1980.Google Scholar
  7. 7.
    Jouannaud J.P., Church Rosser Computations with Equational Term Rewriting Systems. Technical Report, CRIN, Nancy, France, January, 1983.Google Scholar
  8. 8.
    Jouannaud J.P., and Kirchner H., Completion of a set of rules modulo a set of equations, Technical Report SRI-CSL and CRIN 84-R-046, Nancy, 1984.Google Scholar
  9. 9.
    Jouannaud J.P., Munoz M., Termination of a set of Rules Model a Set of Equations, 7th International Conference On Automated Deduction, editor R.E. Shostak, Springer Verlag Lecture Notes in Computer Science, Vol. 170, 175–193.Google Scholar
  10. 10.
    Kirchner H, A general inductive completion algorithm and applications to abstract data types, Proc 7th International Conference On Automated Deduction, editor R.E. Shostak, Springer Verlag Lecture Notes in Computer Science, Vol. 170.Google Scholar
  11. 11.
    Knuth D., Bendix P., Simple Word Problems in Universal Algebras, in Computational Problem in Abstract Algebra, Ed. J. Leech, Pergamon Press, 1970, 263–297.Google Scholar
  12. 12.
    Musser D.L., On Proving Inductive Properties of Abstract Data Types, Proc. 7th POPL, Las Vegas, 1980.Google Scholar
  13. 13.
    Paul E., Preuve par induction dans les theories equationelles en presence de relations entre les constructeurs, Proc. 9th Colloquium on trees in algebra and programming, Bordeaux, France, 1984.Google Scholar
  14. 14.
    Pelin A., Gallier J.H., Solving Word Problems in Free Algebras Using Complexity Functions 7th International Conference on Automated Deduction, editor R.E. Shostak, Springer Verlag Lecture Notes in Computer Science, Vol. 170, 476–495.Google Scholar
  15. 15.
    Pelin A., Gallier J.H., Computing Normal Forms Using Complexity Functions Over Nk, Submitted to TCS, 1985.Google Scholar
  16. 16.
    Remy J.L., Etude des Systemes de reecriture conditionels et applications aux types abstraits algebriques, These d'Etat, Universite de Nancy, 1984.Google Scholar
  17. 17.
    Remy J.L., and Zhang H., REVEURS: a system for validating conditional algebraic specifications of algebraic abstract data types, Proc. 5th ECAI, Pisa, Italy, 1984.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Alex Pelin
    • 1
  • Jean H. Gallier
    • 2
  1. 1.Department of Mathematical Sciences Tamiami CampusFlorida International UniversityMiami
  2. 2.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphia

Personalised recommendations