Final algebras, cosemicomputable algebras, and degrees of unsolvability

  • Lawrence S. Moss
  • José Meseguer
  • Joseph A. Goguen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 283)


This paper studies some computability notions for abstract data types, and in particular compares cosemicomputable many-sorted algebras with a notion of finality to model minimal-state realizations of abstract (software) machines. Given a finite many-sorted signature Σ and a set V of visible sorts, for every Σ-algebra A with co-r.e. behavior and nontrivial, computable V-behavior, there is a finite signature extension Σ′ of Σ (without new sorts) and a finite set E of Σ′-equations such that A is isomorphic to a reduct of the final (Σ′, E)-algebra relative to V. This uses a theorem due to Bergstra and Tucker [3]. If A is computable, then A is also isomorphic to the reduct of the initial (Σ′, E)-algebra. We also prove some results on congruences of finitely generated free algebras. We show that for every finite signature Σ, there are either countably many Σ-congruences on the free Σ-algebra or else there is a continuum of such congruences. There are several necessary and sufficient conditions which separate these two cases. We introduce the notion of the Turing degree of a minimal algebra. Using the results above prove that there is a fixed one-sorted signature such that for every r.e. degree d, there is a finite set E of Σ-equations such the initial (Σ, E)-algebra has degree d. There is a two-sorted signature Σ0 and a single visible sort such that for every r.e. degree d there is a finite set E of Σ-equations such that the initial (Σ, E, V)-algebra is computable and the final (Σ, E, V)-algebra is cosemicomputable and has degree d.


Function Symbol Recursive Function Critical Pair Constant Symbol 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 References

  1. [1]
    Bergstra, J.A. and J.-J. Meyer, I/O-Computable data structures. SIGPLAN Notices, vol. 16, no. 4 (1981), pp. 27–32.Google Scholar
  2. [2]
    Bergstra, J.A. and J.V. Tucker, Algebraic specifications of computable and semicomputable data structures. Research Report IW 115, Mathematical Centre, Dept. of Computer Science, Amsterdam, 1979. Revised as University of Leeds CTCS Report 2.86. To appear in Theoretical Computer Science.Google Scholar
  3. [3]
    Bergstra, J.A. and J.V. Tucker, Initial and final algebra semantics for data type specifications: two characterisation theorems. SIAM J. Comput., vol 12 (1983), pp. 366–387.Google Scholar
  4. [4]
    Bergstra, J.A. and J.V. Tucker, Characterization of computable data types by means of a finite equational specification method. In J. W. de Bakker and J. van Leeuwen (eds.), Automata, Languages and Programming, Seventh Colloquium, Noordwijkerhout, Springer Lecture Notes in Computer Science, vol. 81 (1980), pp. 76–90.Google Scholar
  5. [5]
    Giarrantana, V., F. Gimona, and U. Montanari, Observability concepts in abstract data type specification. Mathematical Foundations of Computer Science '76, Springer Lecture Notes in Computer Science, vol. 45, pp. 576–587, 1976.Google Scholar
  6. [6]
    Goguen, J.A., Realization is universal. Mathematical System Theory, vol 6 (1973), pp. 359–374.Google Scholar
  7. [7]
    Goguen, J.A. and J. Meseguer, Completeness of many-sorted equational logic. Algebra Universalis vol. 11 (1985), no. 3, pp. 307–334. Extended abstract appeared in SIGPLAN Notices, July 1981, vol. 16, no. 7, pp. 24–37.Google Scholar
  8. [8]
    Guttag, J.V., The Specification and Application to Programming of Abstract Data Types, Ph.D. Thesis, University of Toronto, 1975. Computer Science Department, Report CSRG-59.Google Scholar
  9. [9]
    Huet, G. and D.C. Oppen, Equations and rewrite rules. In R. Book (ed.), Formal Language Theory, Academic Press, 1980, 350:405.Google Scholar
  10. [10]
    Majster, M.E., Data types, abstract data types and their specification problem. Theoretical Computer Science, vol. 8 (1979), pp. 89–127.Google Scholar
  11. [11]
    Malcev, A.I., Constructive algebras I, Russian Mathematical Surveys 16(3), 1961, pp. 77–129.Google Scholar
  12. [12]
    Meseguer, J. and J.A. Goguen, Initiality, induction, and computability. In M. Nivat and J. Reynolds (eds.), Algebraic Methods in Semantics, pp. 459–541, Cambridge University Press, 1985. Also appeared as SRI CSL Tech. Rep. 140, 1983.Google Scholar
  13. [13]
    Rabin, M., Computable algebra: general theory and theory of computable fields. Transactions of the American Mathematical Society 95(1960), pp. 341–360.Google Scholar
  14. [14]
    Rogers, H., The Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.Google Scholar
  15. [15]
    Wand, M., Final algebra semantics and data type extension. J. Comp. Sys. Sciences, vol. 19 (1979), pp. 27–44.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Lawrence S. Moss
    • 1
  • José Meseguer
    • 2
  • Joseph A. Goguen
    • 2
  1. 1.Center for the Study of Language and InformationUSA
  2. 2.SRI International and Center for the Study of Language and InformationUSA

Personalised recommendations