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**1 - 2**of**2**### Programming languages

"... Abstract The classical approach to automatic cost analysis consists of two phases. Given a program and some measure of cost, the analysis first produces cost relations (CRs), i.e., recursive equations which capture the cost of the program in terms of the size of its input data. Second, CRs are conve ..."

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Abstract The classical approach to automatic cost analysis consists of two phases. Given a program and some measure of cost, the analysis first produces cost relations (CRs), i.e., recursive equations which capture the cost of the program in terms of the size of its input data. Second, CRs are converted into closed-form, i.e., without recurrences. Whereas the first phase has received considerable attention, with a number of cost analyses available for a variety of programming languages, the second phase has been comparatively less studied. This article presents, to our knowledge, the first practical framework for the generation of closed-form upper bounds for CRs which (1) is fully automatic, (2) can handle the distinctive features of CRs, originating from cost analysis of realistic programming languages, (3) is not restricted to simple complexity classes, and (4) produces reasonably accurate solutions. A key idea in our approach is to view CRs as programs, which allows applying semantic-based static analyses and transformations to bound them, namely our method is based on the inference of ranking functions and loop invariants and on the use of partial evaluation. Keywords Cost analysis · Closed-form upper bounds · Resource analysis ·

### A On the Inference of Resource Usage Upper and Lower Bounds

"... Cost analysis aims at determining the amount of resources required to run a program in terms of its input data sizes. The most challenging step is to infer the cost of executing the loops in the program. This requires bounding the number of iterations of each loop and finding tight bounds for the co ..."

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Cost analysis aims at determining the amount of resources required to run a program in terms of its input data sizes. The most challenging step is to infer the cost of executing the loops in the program. This requires bounding the number of iterations of each loop and finding tight bounds for the cost of each of its iterations. This article presents a novel approach to infer upper and lower bounds from cost relations. These relations are an extended form of standard recurrence equations which can be non-deterministic, contain inexact size constraints and have multiple arguments that increase and/or decrease. We propose novel techniques to automatically transform cost relations into worst-case and best-case deterministic oneargument recurrence relations. The solution of each recursive relation provides a precise upper-bound and lower-bound forexecuting a corresponding loop in the program. Importantly, since the approach isdeveloped at the level of the cost equations, our techniques are programming language independent.