Skip to main content Accessibility help
×
Home

Probabilistic operational semantics for the lambda calculus

  • Ugo Dal Lago (a1) and Margherita Zorzi (a2)

Abstract

Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics, inductively and coinductively defined, are given. Moreover, small-step and big-step semantics are shown to produce identical outcomes, both in call-by-value and in call-by-name. Plotkin’s CPS translation is extended to accommodate the choice operator and shown correct with respect to the operational semantics. Finally, the expressive power of the obtained system is studied: the calculus is shown to be sound and complete with respect to computable probability distributions.

Copyright

References

Hide All
[1] P. Audebaud and C. Paulin-Mohring, Proofs of randomized algorithms in Coq, in Proc. of Mathematics of Program Construction. Lect. Notes Comput. Sci. 4014 49–68 (2006).
[2] P.-L. Curien and H. Herbelin, The duality of computation, in Proc. of International Conference on Functional Programming (2000) 233–243.
[3] U. Dal Lago and M. Zorzi, Probabilistic operational semantics for the lambda calculus. Long Version. Available at http://arxiv.org/abs/1104.0195, 2012.
[4] Danvy, O. and Filinski, A., Representing control : A study of the CPS transformation. Math. Struct. Comput. Sci. 2 (1992) 361391.
[5] Danvy, O. and Nielsen, L.R., CPS transformation of beta-redexes. Inform. Process. Lett. 94 (2005) 217224.
[6] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order. Cambridge University Press (2002).
[7] de’ Liguoro, U. and Piperno, A., Nondeterministic extensions of untyped λ-calculus. Inform. Comput. 122 (1995) 149177.
[8] Di Pierro, A., Hankin, C. and Wiklicky, H., Probabilistic λ-calculus and quantitative program analysis. J. Logic Comput. 15 (2005) 159179.
[9] Edalat, A., Domains for computation in mathematics, physics and exact real arithmetic. Bull. Symbolic Logic 3 (1997) 401452.
[10] A. Edalat and M.H. Escard, Integration in real PCF, in Proc. of IEEE Symposium on Logic in Computer Science. Society Press (1996) 382–393.
[11] M. Gaboardi, Inductive and coinductive techniques in the operational analysis of functional programs : an introduction. Master’s thesis, Universita’ di Milano, Bicocca (2004).
[12] M. Giry, A categorical approach to probability theory, in Categorical Aspects of Topology and Analysis, edited by B. Banaschewski. Springer, Berlin, Heidelberg (1982) 68–85.
[13] Jacobs, B. and Rutten, J., A tutorial on (co)algebras and (co)induction. Bull. EATCS 62 (1996) 222259.
[14] C. Jones, Probabilistic non-determinism. Ph.D. thesis, University of Edinburgh, Edinburgh, Scotland, UK (1989).
[15] C. Jones and G. Plotkin, A probabilistic powerdomain of evaluations, in Proc. of IEEE Symposium on Logic in Computer Science. IEEE Press (1989) 186–195.
[16] Leroy, X. and Grall, H., Coinductive big-step operational semantics. Inform. Comput. 207 (2009) 284304.
[17] E. Moggi, Computational lambda-calculus and monads, in Proc. of IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press (1989) 14–23.
[18] Moggi, E., Notions of computation and monads. Inform. Comput. 93 (1989) 5592.
[19] S. Park, A calculus for probabilistic languages, in Proc. of ACM SIGPLAN International Workshop on Types in Languages Design and Implementation. ACM Press (2003) 38–49.
[20] S. Park, F. Pfenning and S. Thrun, A monadic probabilistic language. Manuscript. Available at http://www.cs.cmu.edu/˜fp/papers/prob03.pdf (2003).
[21] S. Park, F. Pfenning and S. Thrun, A probabilistic language based upon sampling functions, in Proc. of ACM Symposium on Principles of Programming Languages 40 (2005) 171–182.
[22] Plotkin, G.D., Call-by-name, call-by-value and the λ-calculus. Theoret. Comput. Sci. 1 (1975) 125159.
[23] Plotkin, G.D., LCF considered as a programming language. Theoret. Comput. Sci. 5 (1977) 223255.
[24] N. Ramsey and A. Pfeffer, Stochastic lambda calculus and monads of probability distributions, in Proc. of ACM Symposium on Principles of Programming Languages. ACM Press (2002) 154–165.
[25] Rutten, J., Elements of Stream Calculus (An Extensive Exercise In Coinduction). Electron. Notes Theor. Comput. Sci 45 (2001) 358423.
[26] Saheb-Djaromi, N., Probabilistic LCF, in Proc. of International Symposium on Mathematical Foundations of Computer Science. Lect. Notes Comput. Sci. 64 (1978) 442451.
[27] D. Sangiorgi, Introduction to Bisimulation and Coinduction. Cambridge University Press (2012).
[28] Selinger, P. and Valiron, B., A lambda calculus for quantum computation with classical control. Math. Struct. Comput. Sci. 16 (2006) 527552.
[29] C. Wadsworth, Some unusual λ-calculus numeral systems, in To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus and Formalism, edited by J.P. Seldin and J.R. Hindley. Academic Press (1980).

Keywords

Related content

Powered by UNSILO

Probabilistic operational semantics for the lambda calculus

  • Ugo Dal Lago (a1) and Margherita Zorzi (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.