PURRS: Towards Computer Algebra Support for Fully Automatic Worst-Case Complexity Analysis

Fully automatic worst-case complexity analysis has a number of applications in computer-assisted program manipulation. A classical and powerful approach to complexity analysis consists in formally deriving, from the program syntax, a set of constraints expressing bounds on the resources required by the program, which are then solved, possibly applying safe approximations. In several interesting cases, these constraints take the form of recurrence relations. While techniques for solving recurrences are known and implemented in several computer algebra systems, these do not completely fulfill the needs of fully automatic complexity analysis: they only deal with a somewhat restricted class of recurrence relations, or sometimes require user intervention, or they are restricted to the computation of exact solutions that are often so complex to be unmanageable, and thus useless in practice. In this paper we briefly describe PURRS, a system and software library aimed at providing all the computer algebra services needed by applications performing or exploiting the results of worst-case complexity analyses. The capabilities of the system are illustrated by means of examples derived from the analysis of programs written in a domain-specific functional programming language for real-time embedded systems.Comment: 6 page

A new look at the automatic synthesis of linear ranking functions

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Verification of C Programs Via Natural Semantics and Abstract Interpretation (Extended Abstract)

We are witnessing a substantial lack of available tools able to verify the absence of relevant classes of run-time errors in code written in (reasonably rich fragments of) C and C ++. This is despite the progress made in recent years in the fields of program analysis and verification, and despite the huge impact such tools could have on the quality of a good portion of our software universe. It is interesting to observe that, among the dozens of freely available software development tools, hardly any, by analyzing the program semantics, are able to certify the absence of important classes of run-time hazards such as, say, the widely known buffer overflows in C code. The reason is, of course, that C and C ++ are complex languages and the techniques that can be used to dominate this complexity still do not reduce tool development to simple, manageable tasks. Our overall aim for this research is to investigate how known techniques based on natural semantics and abstract interpretation can be extended so as to conveniently formalize and implement a range of analysis and verification tools for modern imperative languages such as C and C ++

A New Look at the Automatic Synthesis of Linear Ranking Functions ✩

The classical technique for proving termination of a generic sequential computer program involves the synthesis of a ranking function for each loop of the program. Linear ranking functions are particularly interesting because many terminating loops admit one and algorithms exist to automatically synthesize it. In this paper we present two such algorithms: one based on work dated 1991 by Sohn and Van Gelder; the other, due to Podelski and Rybalchenko, dated 2004. Remarkably, while the two algorithms will synthesize a linear ranking function under exactly the same set of conditions, the former is mostly unknown to the community of termination analysis and its general applicability has never been put forward before the present paper. In this paper we thoroughly justify both algorithms, we prove their correctness, we compare their worst-case complexity and experimentally evaluate their efficiency, and we present an open-source implementation of them that will make it very easy to include termination-analysis capabilities in automatic program verifiers. Keywords: Static analysis, computer-aided verification, termination analysis