Assertional categories

  • Ernie Manes
Part I Categorical And Algebraic Methods
Part of the Lecture Notes in Computer Science book series (LNCS, volume 298)


Assertional categories provide a general algebraic framework for the denotation of programs. While the axioms deal exclusively with the abstract structure of coproducts, it is possible to express Boolean structure, loop-free constructs and predicate transformers and to deduce basic properties associated with propositional dynamic logic. At the foundational level, new "quarks" to build the atomic constructions of programs are espoused, leading to a new categorical duality principle for predicate transformers (based on a semilattice completion by ideals) and to the "grand unification principle" that "composition determines semantics". The first-order theory of assertional categories is more general than dynamic logic in that nondeterminism may include repetition count, that is, f + f need not be f. On the other hand, an adaptation of a theorem of Kozen shows that, at least for iteration-free sentences about predicate transformers, semantics is standard. Several algebraic characterizations of Dijkstra's definition of determinism are offered and one leads to a technique to reduce loop-free expressions to guarded commands. Axioms for iteration include a "uniformity principle" that "related programs have related iterates" and the Segerburg induction axiom follows.


Boolean Algebra Abelian Category Dynamic Logic Fixed Point Equation Propositional Dynamic Logic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. M. A. Arbib and E. G. Manes, Functorial iteration, Notices Amer. Math. Soc. 25, 1978, A-381.Google Scholar
  2. M. A. Arbib and E. G. Manes, Partially-additive categories and the semantics of flow diagrams, J. Algebra 62, 1980, 203–227.CrossRefGoogle Scholar
  3. M. A. Arbib and E. G. Manes, The pattern-of-calls expansion is the canonical fix-point for recursive definitions, J. Assoc. Comput. Mach. 29, 1982, 557–602.Google Scholar
  4. J. Backus, Can programming be liberated from the von Neumann Style? A functional style and its algebra of programs, Commun. Assoc. Comput. Mach. 21, 1978, 613–641.Google Scholar
  5. J. Backus, The algebra of functional programs: function level reasoning, linear equations and extended definitions, in J. Diaz and T. Ramos (eds.), Formalization of programming concepts, Lecture Notes in Computer Science 107, Springer-Verlag, 1981, 1–43.Google Scholar
  6. S. L. Bloom, C. C. Elgot and J. B. Wright, Solutions of the iteration equation and extensions of the scalar iteration operation, SIAM J. Comput. 9, 1980, 25–45.CrossRefGoogle Scholar
  7. S. L. Bloom, C. C. Elgot and J. B. Wright, Vector iteration in pointed iterative theories, SIAM J. Comput. 9, 1980a, 525–540.CrossRefGoogle Scholar
  8. S. L. Bloom and Z. Esik, Axiomatizing schemes and their behaviors, J. Comput. Sys. Sci. 31, 1985, 375–393.CrossRefGoogle Scholar
  9. J. H. Conway, Regular Algebra and Finite Machines, Chapman and Hall, London, 1971.Google Scholar
  10. E. W. Dijkstra, A Discipline of Programming, Prentice-Hall, 1976.Google Scholar
  11. S. Eilenberg, Automata, Languages, and Machines, Vol. A, Academic Press, 1974.Google Scholar
  12. S. Eilenberg and S. Mac Lane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58, 1945, 231–294.Google Scholar
  13. C. C. Elgot, Monadic computation and iterative algebraic theories, in H. E. Rose and J. C. Shepherdson (eds.), Proc. Logic Colloq. '73, North-Holland, 1975, 175–230.Google Scholar
  14. C. C. Elgot and J. C. Shepherdson, An equational axiomatization of reducible flowchart schemes, IBM Research Report RC-8221, April 1980; in S. L. Bloom (ed.), Calvin C. Elgot, Selected Papers, Springer-Verlag, 1982, 361–409.Google Scholar
  15. C. J. Everett and S. Ulam, Projective algebra I, Amer. J. Math. 68, 1946, 77–88.Google Scholar
  16. M. J. Fischer and R. E. Ladner, Propositional dynamic logic of regular programs, J. Comput. Sys. Sci. 18, 1979, 194–211.CrossRefGoogle Scholar
  17. P. Freyd, Abelian Categories, Harper and Row, 1964.Google Scholar
  18. D. Harel, First-order Dynamic Logic, Lecture Notes in Computer Science 68, Springer-Verlag, 1979.Google Scholar
  19. C. A. R. Hoare, An axiomatic basis for computer programming, Comm. Assoc. Comput. Mach. 12, 1969, 576–580, 583.Google Scholar
  20. B. Jonsson and A. Tarski, Representation problems for relation algebras, abstract 89t, Bull. Amer. Math. Soc. 54, 1948, 80.Google Scholar
  21. D. Kozen, A representation theorem for models of *-free PDL, in J. W. de Bakker and J. Van Leeuwen (eds.), Automata, Languages and Programming, ICALP '80, Lecture Notes in Computer Science 85, Springer-Verlag, 1980, 351–362.Google Scholar
  22. D. Kozen, On induction vs. *-continuity, in E. Engeler (ed.), Logics of Programs, Lecture Notes in Computer Science 131, 1981, 167–176.Google Scholar
  23. F. W. Lawvere, Functorial semantics of algebraic theories, Ph.D. Dissertation, Columbia University, 1963.Google Scholar
  24. R. C. Lyndon, The representation of relation algebras, Ann. Math. 51, 1950, 707–729.Google Scholar
  25. E. G. Manes, Additive domains, in A. Melton (ed.), Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science 239, Springer-Verlag, 1985, 184–195.Google Scholar
  26. E. G. Manes, Guard modules, Algebra Universalis 21, 1985a, 103–110.CrossRefGoogle Scholar
  27. E. G. Manes, A transformational characterization of if-then-else, to appear.Google Scholar
  28. E. G. Manes and M. A. Arbib, Algebraic Approaches to Program Semantics, Springer-Verlag, 1986.Google Scholar
  29. E. G. Manes and D. B. Benson, The inverse semigroup of a sum-ordered semiring, Semigroup Forum 31, 1985, 129–152.Google Scholar
  30. J. C. C. McKinsey, Postulates for the calculus of binary relations, J. Symbolic Logic 5, 1940, 85–97.Google Scholar
  31. R. Milne and C. Strachey, A Theory of Programming Language Semantics, Parts a and b, Chapman and Hall, London, 1976.Google Scholar
  32. B. Mitchell, Theory of Categories, Academic Press, 1965.Google Scholar
  33. E. Nelson, Iterative algebras, Theoret. Comp. Sci. 25, 1983, 67–94.CrossRefGoogle Scholar
  34. R. Parikh, Propositional dynamic logics of programs: a survey, in E. Engeler (ed.), Logic of Programs, Lecture Notes in Computer Science 125, 1981, 102–144.Google Scholar
  35. V. R. Pratt, Models of program logics, Proc. 20th IEEE Symp. Found. Comp. Sci., IEEE 79CH1471-2C, 1979, 115–122.Google Scholar
  36. V. R. Pratt, Dynamic algebras and the nature of induction, Proc. 12th ACM Symposium on Theory of Computing, May 1980, 22–28.Google Scholar
  37. J. Reiterman and V. Trnková, Dynamic algebras which are not Kripke structures, Proc. 9th Symposium on Mathematical Foundations of Computer Science, Aug. 1980, 528–538.Google Scholar
  38. K. Segerburg, A completeness theorem in the modal logic of programs, Notices Amer. Math. Soc. 24, 1977, A-522.Google Scholar
  39. M. E. Steenstrup, Sum-ordered partial semirings, Ph.D. Dissertation, University of Massachusetts at Amherst, 1985.Google Scholar
  40. M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40, 1936, 37–111.MathSciNetGoogle Scholar
  41. J. E. Stoy, Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory, M. I. T. Press, 1977.Google Scholar
  42. N. V. Subrahmanyam, Boolean vector spaces I, Math. Zeit. 83, 1964, 422–433.CrossRefGoogle Scholar
  43. N. V. Subrahmanyam, Boolean vector spaces II, Math. Zeit. 87, 1965, 401–419.CrossRefGoogle Scholar
  44. N. V. Subrahmanyam, Boolean vector spaces, III, Math. Zeit. 100, 1967, 295–313.CrossRefGoogle Scholar
  45. A. Tarski, On the calculus of relations, J. Symbolic Logic 6, 1941, 73–89.Google Scholar
  46. J. Tiuryn, Unique fixed points vs. least fixed points, Theoret. Comput. Sci. 13, 1981, 229–254.Google Scholar
  47. E. G. Wagner, S. L. Bloom and J. W. Thatcher, Why algebraic theories?, in M. Nivat and J. C. Reynolds (eds.), Algebraic Methods in Semantics, Cambridge Univ. Press, 1985, 607–634.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Ernie Manes
    • 1
  1. 1.Department of Mathematics and Statistics Lederle Research Center TowerUniversity of MassachusettsAmherst

Personalised recommendations