No abstract

No abstract

We consider various ways to represent irrational numbers by subrecursive functions: via Cauchy sequences, Dedekind cuts, trace functions, several variants of sum approximations and continued fractions. Let S be a class of subrecursive functions. The set of irrational numbers that can be obtained with functions from S depends on the representation. We compare the sets obtained by the different representations.

We present a method for certifying that the values computed by an imperative program will be bounded by polynomials in the program's inputs. To this end, we introduce mwp-matrices and define a semantic relation |= C : M, where C is a program and M is an mwp-matrix. It follows straightforwardly from our definitions that there exists M such that |= C : M holds iff every value computed by C is bounded by a polynomial in the inputs. Furthermore, we provide a syntactical proof calculus and define the relation C : M to hold iff there exists a derivation in the calculus where C : M is the bottom line. We prove that C : M implies |= C : M . By means of exhaustive proof search, an algorithm can decide if there exists M such that the relation C : M holds, and thus, our results yield a computational method. General Terms: Theory Additional Key Words and Phrases: Implicit computational complexity, static program anaysis, automatable complexity analysis of imperative programs ACM Reference Format: Jones, N. D. and Kristiansen, L. 2009. A flow calculus of mwp-bounds for complexity analysis. ACM Trans. Comput.

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