Abstract
This paper deals with a philosophical question that arises within the theory of computational complexity: how to understand the notion of INTRINSIC complexity or difficulty, as opposed to notions of difficulty that depend on the particular computational model used. The paper uses ideas from Blum's abstract approach to complexity theory to develop an extensional approach to this question. Among other things, it shows how such an approach gives detailed confirmation of the view that subrecursive hierarchies tend to rank functions in terms of their intrinsic, and not just their model-dependent, difficulty, and it shows how the approach allows us to model the idea that intrinsic difficulty is a fuzzy concept.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arbib, M. and M. Blum, 1965, Machine dependence of degrees of difficulty, Proceedings of the American Mathematical Society 16, 442–447.
Baker, T. P., 1978, “Natural” properties of flowchart step-counting measures, Journal of Computer and System Sciences 16, 1–22.
Bass, L. and P. Young, 1973, Ordinal hierarchies and naming complexity classes, Journal of the Association for Computing Machinery 20, 668–686.
Blum, M., 1967, A machine-independent theory of the complexity of recursive functions, Journal of the Association for Computing Machinery 14, 322–336.
Calude, C., 1988, Theories of Computational Complexity, North-Holland, Amsterdam, New York, Oxford, Tokyo.
Cichon, E. A. and S. S. Wainer, 1983, The slow-growing and the Grzegorczyk hierarchies, The Journal of Symbolic Logic 48, 399–408.
Cleave, J. P., 1963, A hierarchy of primitive recursive functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9, 331–345.
Cobham, A., 1965, The intrinsic computational difficulty of functions, Proceedings of the 1964 Congress of Logic, Methodology and Philosophy of Science North-Holland Publishing Company, 24–30.
Epstein, R. and W. A. Carnielli, 1989, Computability: Computable Functions, Logic, and the Foundations of Mathematics Wadsworth.
Feferman, S., 1962, Classification of recursive functions by means of hierarchies, Transactions of the American Mathematical Society 104, 101–122.
Glymour, C., 1992, Thinking Things Through: an Introduction to Philosophical Issues and Achievements, MIT Press.
Grzegorczyk, A., 1953, Some classes of recursive functions, Rozprawy Matematyczne 4, 1–45.
Hartmanis, J., 1973, On the problem of finding natural computational complexity measures, Cornell Computer Science Technical Report, 73–179.
Hartmanis, J. and J. E. Hopcroft, 1971, An overview of the theory of computational complexity, Journal of the Association for Computing Machinery 18, 444–475.
Hartmanis, J. and J. Simon, 1976, On the structure of feasible computations, in M. Rubinoff and M. C. Yovits, eds., Advances in Computers 14, Academic Press, New York, 1–43.
Johnson, D. S., 1990, A catalog of complexity classes, in Leeuwen, van J. (ed.), Handbook of Theoretical Computer Science, vol. A, Algorithms and Complexity, Elsevier, Amsterdam, New York, Oxford, Tokyo, 69–161.
Kroon, F. W. and W. A. Burkhard, 1990, On a complexity-based way of constructivizing the recursive functions, Studia Logica 49, 133–149.
Lewis, D., 1972, General semantics, in D. Davidson and G. Harman (eds.), Semantics of Natural Language, Reidel, Dordrecht, 169–218.
Löb, M. H. and S. S. Wainer, 1970, Hierarchies of number-theoretic functions I, II, Archiv für Mathematische Logik und Grundlagenforschung 13, 39–51 and 97–113.
Meyer, A. R. and D. M. Ritchie, 1972, A classification of recursive functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 18, 71–82.
Pratt, V. R. and L. J. Stockmeyer, 1976, A characterization of the power of vector machines, Journal of Computing System Sciences 12, 198–221.
Ritchie, R. W., 1963, Classes of predictably computable functions, Transactions of the American Mathematical Society 106, 139–173.
Robbin, J. W., 1965, Subrecursive Hierarchies, Ph.D. dissertation, Princeton University.
Rogers, Jr. H., 1967, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York.
Seiferas, J., 1990, Machine-independent complexity theory, in van Leeuwen. Handbook of Theoretical Computer Science, vol. A, 165–186.
Shapiro, S., 1981, Understanding Church's Thesis, Journal of Philosophical Logic 10, 353–366.
Stockmeyer, L. and A. K. Chandra, 1979, Intrinsically difficult problems, Scientific American, 124–134.
Stockmeyer, L., 1987, Classifying the computational complexity of problems, The Journal of Symbolic Logic 52, 1–43.
van Emde Boas, P., 1990, Machine models and simulations, in van Leeuwen, J. (ed.), Handbook of Theoretical Computer Science, vol. A, 1–66.
Verbeek, R., 1978, Primitiv-Rekursiv Grzegorczyk-Hierarchien Universität Bonn, Informatik Berichte, Bonn.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kroon, F.W. The intrinsic difficulty of recursive functions. Stud Logica 56, 427–454 (1996). https://doi.org/10.1007/BF00372775
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00372775