ABSTRACT
This paper investigates what is essentially a call-by-value version of PCF under a complexity-theoretically motivated type system. The programming formalism, ATR1, has its first-order programs characterize the poly-time computable functions, and its second-order programs characterize the type-2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATR1-types are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The size-restricted part is motivated by Bellantoni and Cook's and Leivant's implicit characterizations of poly-time. The time-restricting part is an affine version of Barber and Plotkin's DILL. Two semantics are constructed for ATR1. The first is a pruning of the naïve denotational semantics for ATR1. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR1's time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR1 recursions, the solutions of which yield second-order polynomial time bounds. The time-complexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine.
- A. Barber and G. Plotkin, Dual intuitionistic linear logic, Tech. report, LFCS, Univ of Edinburgh, 1997.Google Scholar
- S. Bellantoni and S. Cook, A new recursion-theoretic characterization of the polytime functions, Computational Complexity 2 (1992), 97--110. Google Scholar
Digital Library
- S. Bellantoni, K.-H. Niggl, and H. Schwichtenberg, Characterising polytime through higher type recursion, Annals of Pure and Applied Logic (2000), 17--30.Google Scholar
Cross Ref
- A. Cobham, The intrinsic computational difficulty of functions, Proceedings of the International Conference on Logic, Methodology and Philosophy (Y. Bar Hillel, ed.), North-Holland, 1965, pp. 24--30.Google Scholar
- S. Cook and A. Urquhart, Functional interpretations of feasibly constructive arithmetic, Annals of Pure and Applied Logic 63 (1993), 103--200.Google Scholar
Cross Ref
- M. Felleisen and D. Friedman, Control operators, the SECD-machine, and the lambda calculus, Formal Descriptions of Programming Concepts III, 1987, pp. 193--217.Google Scholar
- C. Frederiksen and N. Jones, Recognition of polynomial-time programs, Tech. Report TOPPS/D-501, DIKU, University of Copenhagen, 2004.Google Scholar
- O. Goldreich, Foundations of cryptography, Vol. I: Basic tools, Cambridge University Press, 2001. Google Scholar
Digital Library
- D. J. Gurr, Semantic frameworks for complexity, Ph.D. thesis, University of Edinburgh, 1990.Google Scholar
- M. Hofmann, Programming languages capturing complexity classes, SIGACT News 31 (2000), 31--42. Google Scholar
Digital Library
- -----, Linear types and non-size increasing polynomial time computation, Information and Computation 183 (2003), 57--85. Google Scholar
Digital Library
- R. Irwin, B. Kapron, and J. Royer, On characterizations of the basic feasible functional, Part I, Journal of Functional Programming 11 (2001), 117--153. Google Scholar
Digital Library
- -----, On characterizations of the basic feasible functional, Part II, unpublished manuscript, 2002.Google Scholar
- B. Kapron, Feasible computation in higher types, Ph.D. thesis, Department of Computer Science, University of Toronto, 1991. Google Scholar
Digital Library
- B. Kapron and S. Cook, A new characterization of type 2 feasibility, SIAM Journal on Computing 25 (1996), 117--132. Google Scholar
Digital Library
- D. Leivant, A foundational delineation of poly-time, Information and Computation 110 (1994), 391--420. Google Scholar
Digital Library
- -----, Ramified recurrence and computational complexity I: Word recurrence and poly-time, Feasible Mathematics II (P. Clote and J. Remmel, eds.), Birkhäuser, 1995, pp. 320--343.Google Scholar
- D. Leivant and J.-Y. Marion, Lambda calculus characterizations of polytime, FundamentæInformaticæ19 (1993), 167--184. Google Scholar
Digital Library
- J. Longley, On the ubiquity of certain total type structures (Extended abstract), Proceedings of the Workshop on Domains VI (M. Escardó and A. Jung, eds.), Electronic Notes in Theoretical Computer Science, vol. 73, Elsevier Science Publishers, 2004, pp. 87--109.Google Scholar
Digital Library
- -----, Notions of computability at higher types, I, Logic Colloquium 2000 (R. Cori, A. Razborov, S. Torcevic, and C. Wood, eds.), Lecture Notes in Logic, vol. 19, A. K. Peters, 2005.Google Scholar
- K. Mehlhorn, Polynomial and abstract subrecursive classes, Journal of Computer and System Science 12 (1976), 147--178.Google Scholar
Digital Library
- B. Pierce, Types and programming languages, MIT Press, 2002. Google Scholar
Digital Library
- G. Plotkin, Call-by-name, call-by-value and the γ-calculus, Theoretical Computer Science 1 (1975), 125--159.Google Scholar
Cross Ref
- -----, LCF considered as a programming language, Theoretical Computer Science 5 (1977), 223--255.Google Scholar
Cross Ref
- J. Royer and J. Case, Subrecursive programming systems: Complexity & succinctness, Birkhääuser, 1994. Google Scholar
Digital Library
- D. Sands, Calculi for time analysis of functional programs, Ph.D. thesis, University of London, 1990.Google Scholar
- A. Schönhage, Storage modification machines, SIAM Journal on Computing 8 (1980), 490--508.Google Scholar
Digital Library
- J. Shultis, On the complexity of higher-order programs, Tech. Report CU-CS-288-85, University of Colorado, Boulder, 1985.Google Scholar
- K. Van Stone, A denotational approach to measuring complexity in functional programs, Ph.D. thesis, School of Computer Science, Carnegie Mellon University, 2003. Google Scholar
Digital Library
- G. Winskel, Formal semantics, MIT Press, 1993.Google Scholar
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