Probabilistic operational semantics for the lambda calculus

Probabilistic operational semantics for a nondeterminis- tic 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. Mathematics Subject Classification. 68Q55, 03B70.

[1]  Abbas Edalat,et al.  Domains for Computation in Mathematics, Physics and Exact Real Arithmetic , 1997, Bulletin of Symbolic Logic.

[2]  Chris Hankin,et al.  Probabilistic λ-calculus and Quantitative Program Analysis , 2004 .

[3]  Abbas Edalat,et al.  Integration in Real PCF , 2000, Inf. Comput..

[4]  B. Jacobs,et al.  A tutorial on (co)algebras and (co)induction , 1997 .

[5]  Benoît Valiron,et al.  A Lambda Calculus for Quantum Computation with Classical Control , 2005, TLCA.

[6]  Christine Paulin-Mohring,et al.  Proofs of randomized algorithms in Coq , 2006, Sci. Comput. Program..

[7]  Michèle Giry,et al.  A categorical approach to probability theory , 1982 .

[8]  Olivier Danvy,et al.  CPS transformation of beta-redexes , 2000, Inf. Process. Lett..

[9]  Olivier Danvy,et al.  Representing Control: a Study of the CPS Transformation , 1992, Mathematical Structures in Computer Science.

[10]  Ugo de'Liguoro,et al.  Non Deterministic Extensions of Untyped Lambda-Calculus , 1995, Inf. Comput..

[11]  Roman Fric,et al.  A Categorical Approach to Probability Theory , 2010, Stud Logica.

[12]  Norman Ramsey,et al.  Stochastic lambda calculus and monads of probability distributions , 2002, POPL '02.

[13]  G.D. Plotkin,et al.  LCF Considered as a Programming Language , 1977, Theor. Comput. Sci..

[14]  Sebastian Thrun,et al.  A probabilistic language based upon sampling functions , 2005, POPL '05.

[15]  J. Roger Hindley,et al.  To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism , 1980 .

[16]  C. Jones,et al.  A probabilistic powerdomain of evaluations , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[17]  Ugo de'Liguoro,et al.  Non deterministic extensions of untyped-calculus , 1995 .

[18]  Hugo Herbelin,et al.  The duality of computation , 2000, ICFP '00.

[19]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[20]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[21]  Sungwoo Park,et al.  A calculus for probabilistic languages , 2003, TLDI '03.

[22]  Claire Jones,et al.  Probabilistic non-determinism , 1990 .

[23]  S. Thrun,et al.  A Monadic Probabilistic Language , 2003 .

[24]  Eugenio Moggi,et al.  Computational lambda-calculus and monads , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[25]  N. Saheb-Djahromi,et al.  Probabilistic LCF , 1978, International Symposium on Mathematical Foundations of Computer Science.

[26]  Jan J. M. M. Rutten.  Elements of Stream Calculus (An Extensive Exercise in Coinduction) , 2001, MFPS.

[27]  D. Sangiorgi.  Introduction to Bisimulation and Coinduction , 2011 .

[28]  Hervé Grall,et al.  Coinductive big-step operational semantics , 2009, Inf. Comput..