# Probabilistic operational semantics for the lambda calculus

@article{Lago2012ProbabilisticOS, title={Probabilistic operational semantics for the lambda calculus}, author={Ugo Dal Lago and Margherita Zorzi}, journal={ArXiv}, year={2012}, volume={abs/1104.0195} }

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…

## 92 Citations

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## References

SHOWING 1-10 OF 37 REFERENCES

### Non Deterministic Extensions of Untyped Lambda-Calculus

- Computer ScienceInf. Comput.
- 1995

The main concern of this paper is the interplay between functionality and nondeterminism. We ask whether the analysis of parallelism in terms of sequentiality and nondeterminism, which is usual in…

### Non deterministic extensions of untyped-calculus

- Mathematics
- 1995

The main concern of this paper is the study of the interplay between functionality and non determinism. Indeed the rst question we ask is whether the analysis of parallelism in terms of sequentiality…

### Stochastic lambda calculus and monads of probability distributions

- Computer SciencePOPL '02
- 2002

A translation of stochastic lambda calculus into measure terms is given, which can not only denote discrete probability distributions but can also support the best known modeling techniques.

### A calculus for probabilistic languages

- Computer ScienceTLDI '03
- 2003

A probabilistic calculus by extending the traditional lambda calculus is developed, which is founded upon sampling functions, which map the unit interval to probability domains and achieves a unified representation scheme for all types of probability distributions.

### A Monadic Probabilistic Language

- Computer Science
- 2003

A monadic probabilistic language based upon the mathematical notion of sampling function is proposed, which provides a unified representation scheme for probability distributions, enjoys rich expressiveness, and offers high versatility in encoding probability distributions.

### Computational lambda-calculus and monads

- Computer Science[1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science
- 1989

The author gives a calculus based on a categorical semantics for computations, which provides a correct basis for proving equivalence of programs, independent from any specific computational model.

### A lambda calculus for quantum computation with classical control

- Computer ScienceMathematical Structures in Computer Science
- 2006

A functional programming language for quantum computers by extending the simply-typed lambda calculus with quantum types and operations, and gives a type system using affine intuitionistic linear logic.

### The duality of computation

- Computer Science, MathematicsICFP '00
- 2000

The μ -calculus is presented, a syntax for λ-calculus + control operators exhibiting symmetries such as program/context and call-by-name/call- by-value, derived from implicational Gentzen's sequent calculus LK.