A Naïve Time Analysis and its Theory of Cost Equivalence

Techniques for reasoning about extensional properties of functional programs are well understood, but methods for analysing the underlying intensional or operational properties have been much neglected. This paper begins with the development of a simple but useful calculus for time analysis of non-strict functional programs with lazy lists. One limitation of this basic calculus is that the ordinary equational reasoning on functional programs is not valid. In order to buy back some of these equational properties we develop a non-standard operational equivalence relation called cost equivalence, by considering the number of computation steps as an ‘observable’ component of the evaluation process. We define this relation by analogy with Park’s definition of bisimulation in CCS. This formulation allows us to show that cost equivalence is a contextual congruence (and thus is substitutive with respect to the basic calculus) and provides useful proof techniques for establishing cost-equivalence laws. It is shown that basic evaluation time can be derived by demonstrating a certain form of cost equivalence, and we give an axiomatization of cost equivalence which is complete with respect to this application. This shows that cost equivalence subsumes the basic calculus. Finally we show how a new operational interpretation of evaluation demands can be used to provide a smooth interface between this time analysis and more compositional approaches, retaining the advantages of both.