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Long-run cost analysis by approximation of linear operators over dioids

Published online by Cambridge University Press:  03 June 2010

DAVID CACHERA ,
THOMAS JENSEN ,
ARNAUD JOBIN  and
PASCAL SOTIN
DAVID CACHERA
Affiliation:
IRISA/ENS Cachan (Bretagne), Campus de Beaulieu, F-35042 Rennes cedex, France Email: david.cachera@irisa.fr
THOMAS JENSEN
Affiliation:
IRISA/CNRS, Campus de Beaulieu, F-35042 Rennes cedex, France Email: thomas.jensen@irisa.fr
ARNAUD JOBIN
Affiliation:
Université Rennes 1, Campus de Beaulieu, F-35042 Rennes cedex, France Email: arnaud.jobin@irisa.fr
PASCAL SOTIN
Affiliation:
CNRS/DGA, F-35042 Rennes cedex, France Email: pascal.sotin@irisa.fr
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Abstract

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In this paper we present a semantics-based framework for analysing the quantitative behaviour of programs with respect to resource usage. We start from an operational semantics in which costs are modelled using a dioid structure. The dioid structure of costs allows the definition of the quantitative semantics as a linear operator. We then develop a theory of approximation of such a semantics, which is akin to what is offered by the theory of abstract interpretation for analysing qualitative properties, in order to compute effectively global cost information from the program. We focus on the notion of long-run cost, which models the asymptotic average cost of a program. The abstraction of the semantics has to take two distinct notions of order into account: the order on costs and the order on states. We prove that our abstraction technique provides a correct approximation of the concrete long-run cost of a program.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Issue 4 , August 2010 , pp. 589 - 624
Copyright
Copyright © Cambridge University Press 2010

References

De Alfaro, L. (1998) How to Specify and Verify the Long-Run Average Behavior of Probabilistic Systems. In: Proceedings 13th Symposium on Logic in Computer Science (LICS'98), IEEE Computer Society Press 174–183.Google Scholar
Argoud, M., Jeannet, B. and Lalire, G. The concurinterproc analyzer. Available at http://pop-art.inrialpes.fr/interproc/concurinterprocweb.cgi.Google Scholar
Aspinall, D., Beringer, L., Hofmann, M., Loidl, H.-W. and Momigliano, A. (2007) A program logic for resources. Theor. Comput. Sci. 389 (3)411–445.Google Scholar
Aziz, B. (2006) A Semiring-based Quantitative Analysis of Mobile Systems. Electronic Notes in Theoretical Computer Science 157 (1)3–21.CrossRefGoogle Scholar
Baccelli, F., Cohen, G., Olsder, G. and Quadrat, J.-P. (1992) Synchronization and Linearity, Wiley.Google Scholar
Bistarelli, S. (2004) Semirings for Soft Constraint Solving and Programming. Springer-Verlag Lecture Notes in Computer Science 2962.CrossRefGoogle Scholar
Bistarelli, S., Codognet, P. and Rossi, F. (2002) Abstracting soft constraints: framework, properties, examples. Artif. Intell. 139 (2)175–211.CrossRefGoogle Scholar
Bistarelli, S., Montanari, U. and Rossi, F. (1997) Semiring-Based Constraint Satisfaction and Optimization. Journal of the ACM 44 (2)201–236.CrossRefGoogle Scholar
Brazdil, T., Esparza, J. and Kucera, A. (2005) Analysis and Prediction of the Long-Run Behavior of Probabilistic Sequential Programs with Recursion (Extended Abstract). In: FOCS '05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society 521–530.Google Scholar
Cachera, D., Jensen, T., Jobin, A. and Sotin, P. (2008) Long-run cost analysis by approximation of linear operators over dioids. In: Meseguer, J. and Roşu, G. (eds.) Proceedings of the 12th International Conference on Algebraic Methodology and Software Technology. Springer-Verlag Lecture Notes in Computer Science 5140 122–138.CrossRefGoogle Scholar
Cochet-Terrasson, J., Cohen, G., Gaubert, S., Mc Gettrick, M. and Quadrat, J.-P. (1998) Numerical Computation of Spectral Elements in Max-Plus Algebra. In: Proceedings of the IFAC Conference on System Structure and Control.CrossRefGoogle Scholar
Cousot, P. and Cousot, R. (1977) Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints. In: Proceedings 4th Symposium on Principles of Programming Languages (POPL'77) 238–252.Google Scholar
Davey, B. A. and Priestley, H. A. (1990) Introduction to Lattices and Order, Cambridge University Press.Google Scholar
De Nicola, R., Ferrari, G., Montanari, U., Pugliese, R. and Tuosto, E. (2005) A Basic Calculus for Modelling Service Level Agreements. In: Proceedings International Conference on Coordination Models and Languages Springer-Verlag Lecture Notes in Computer Science 3454 33–48.Google Scholar
Di Pierro, A., Hankin, C. and Wiklicky, H. (2005a) Probabilistic λ-calculus and Quantitative Program Analysis. J. Logic and Computation 15 (2)159–179.CrossRefGoogle Scholar
Di Pierro, A., Hankin, C. and Wiklicky, H. (2005b) Measuring the confinement of probabilistic systems. Theor. Comput. Sci. 340 (1)3–56.Google Scholar
Di Pierro, A. and Wiklicky, H. (2000) Concurrent Constraint Programming: Towards Probabilistic Abstract Interpretation. In: Principles and Practice of Declarative Programming 127–138.CrossRefGoogle Scholar
Dudnikov, P. and Samborskii, S. (1992) Endomorphisms of finitely generated free semimodules. In: Maslov, V. P. and Samborskii, S. N. (eds.) Idempotent analysis. Advances in Soviet Mathematics 13, American Mathematical Society 65–85.Google Scholar
Dudnikov, P. and Samborskii, S. (1987) Endomorphisms of semimodules over semirings with an idempotent operation – preprint of the Mathematical Institute of the Ukrainian Academy of Sciences (in Russian). (English translation in Math. USSR Izvestiya (1992) 38 (1)91–105).CrossRefGoogle Scholar
Gondran, M. and Minoux, M. (2008) Graphs, Dioids and semirings, New Models and Algorithms, Springer-Verlag.Google Scholar
Hirsch, D. and Tuosto, E. (2005) SHReQ: Coordinating Application Level QoS. In: SEFM '05: Proceedings of the Third IEEE International Conference on Software Engineering and Formal Methods, IEEE Computer Society 425–434.Google Scholar
Miné, A. (2004) Weakly Relational Numerical Abstract Domains, Ph.D. thesis, École Polytechnique, Palaiseau, France.Google Scholar
Santini, S. (2008) Managing quality of service with soft constraints. In: AAAI'08: Proceedings of the 23rd national conference on Artificial intelligence, AAAI Press 1869–1870.Google Scholar
Sotin, P., Cachera, D. and Jensen, T. (2006) Quantitative Static Analysis over Semirings: Analysing Cache Behaviour for Java Card. In: Di Pierro, A. and Wiklicky, H. (eds.) QAPL06, Quantitative Aspects of Programming Languages.CrossRefGoogle Scholar