An algebraic approach to problem solution and problem semantics

  • A. Bertoni
  • G. Mauri
  • M. Torelli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 53)


The usual approach to the synthesis of algorithms for the solution of problems in combinatorial mathematics consists of two steps.

1 — Description: the problem is embedded in a general structure which is rich enough to permit a mathematical modelling of the problem.

2 — Solution: the problem is solved by means of techniques "as simple as possible", with respect to some given notion of complexity.

We give a formalization of this approach in the framework of category theory, which is general enough to get rid of unessential details. In particular such a framework will be provided by the category of ordered complete Σ-algebras, and we will describe the relation between description and solution by means of a variant of so called "Mezei-Wright like results" [10], relating the concept of least fixed point to that of a suitable natural transformation between functors.


Formal Power Series Natural Transformation Category Theory Context Free Language Surjective Morphism 


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  1. 1.
    Birkhoff,G., Lattice theory, Am.Math. Soc. Coll. pub.25, New YorkGoogle Scholar
  2. 2.
    Arbib,M.A., Manes,E.G., Basic concepts of category theory applicable to computation and control, Proc. First Internat. Symp., San Francisco (1974)Google Scholar
  3. 3.
    MacLane, S., Categories for the working mathematician, Springer Verlag, New York-Berlin (1971)Google Scholar
  4. 4.
    Bertoni,A., Equations of formal power series over non commutative semirings, Proc. MFCS Symp., High Tatras (1973)Google Scholar
  5. 5.
    Goguen, J.A., Thatcher,J.W., Wagner,E.G., Wright,J.B., Factorizations, congruences, decomposition of automata and systems, IBM Research Report RC 4934 (1974)Google Scholar
  6. 6.
    Chomsky, N., Schuetzenberger, M.P., The algebraic theory of context free languages, Computation, Programming and Formal Systems, North Holland, Amsterdam (1963)Google Scholar
  7. 7.
    Riordan, J., An introduction to combinatorial analysis, Wiley and Sons, New York-Toronto (1958)Google Scholar
  8. 8.
    Boole, G., Finite differences, Dover Publications, New York (1960)Google Scholar
  9. 9.
    Nivat,M., On the interpretation of recursive polyadic program schemata, Ist. di Alta Matematica, Symposia Matematica Vol XV, Bologna (1975)Google Scholar
  10. 10.
    Mezei, J., Wright, J.B., Algebraic automata and context free sets, Information and Control, 11 (1967), 3–29CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1977

Authors and Affiliations

  • A. Bertoni
    • 1
  • G. Mauri
    • 1
  • M. Torelli
    • 1
  1. 1.Istituto di CiberneticaUniversità di MilanoMilanoItaly

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