2,044 research outputs found
Probabilistic Operational Semantics for the Lambda Calculus
Probabilistic operational semantics for a nondeterministic extension of pure
lambda calculus is studied. In this semantics, a term evaluates to a (finite or
infinite) distribution of values. Small-step and big-step semantics are both
inductively and coinductively defined. 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.Comment: 35 page
Probabilistic Operational Semantics for the Lambda Calculus
Probabilistic operational semantics for a nondeterministic extension of pure \u3bb-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\u2019s 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
A coherent differential PCF
The categorical models of the differential lambda-calculus are additive
categories because of the Leibniz rule which requires the summation of two
expressions. This means that, as far as the differential lambda-calculus and
differential linear logic are concerned, these models feature finite
non-determinism and indeed these languages are essentially non-deterministic.
In a previous paper we introduced a categorical framework for differentiation
which does not require additivity and is compatible with deterministic models
such as coherence spaces and probabilistic models such as probabilistic
coherence spaces. Based on this semantics we develop a syntax of a
deterministic version of the differential lambda-calculus. One nice feature of
this new approach to differentiation is that it is compatible with general
fixpoints of terms, so our language is actually a differential extension of PCF
for which we provide a fully deterministic operational semantics
A lambda calculus for quantum computation with classical control
The objective of this paper is to develop a functional programming language
for quantum computers. We develop a lambda calculus for the classical control
model, following the first author's work on quantum flow-charts. We define a
call-by-value operational semantics, and we give a type system using affine
intuitionistic linear logic. The main results of this paper are the safety
properties of the language and the development of a type inference algorithm.Comment: 15 pages, submitted to TLCA'05. Note: this is basically the work done
during the first author master, his thesis can be found on his webpage.
Modifications: almost everything reformulated; recursion removed since the
way it was stated didn't satisfy lemma 11; type inference algorithm added;
example of an implementation of quantum teleportation adde
Applying quantitative semantics to higher-order quantum computing
Finding a denotational semantics for higher order quantum computation is a
long-standing problem in the semantics of quantum programming languages. Most
past approaches to this problem fell short in one way or another, either
limiting the language to an unusably small finitary fragment, or giving up
important features of quantum physics such as entanglement. In this paper, we
propose a denotational semantics for a quantum lambda calculus with recursion
and an infinite data type, using constructions from quantitative semantics of
linear logic
Taylor expansion for Call-By-Push-Value
The connection between the Call-By-Push-Value lambda-calculus introduced by Levy and Linear Logic introduced by Girard has been widely explored through a denotational view reflecting the precise ruling of resources in this language. We take a further step in this direction and apply Taylor expansion introduced by Ehrhard and Regnier. We define a resource lambda-calculus in whose terms can be used to approximate terms of Call-By-Push-Value. We show that this approximation is coherent with reduction and with the translations of Call-By-Name and Call-By-Value strategies into Call-By-Push-Value
A Lambda-Calculus Foundation for Universal Probabilistic Programming
We develop the operational semantics of an untyped probabilistic
lambda-calculus with continuous distributions, as a foundation for universal
probabilistic programming languages such as Church, Anglican, and Venture. Our
first contribution is to adapt the classic operational semantics of
lambda-calculus to a continuous setting via creating a measure space on terms
and defining step-indexed approximations. We prove equivalence of big-step and
small-step formulations of this distribution-based semantics. To move closer to
inference techniques, we also define the sampling-based semantics of a term as
a function from a trace of random samples to a value. We show that the
distribution induced by integrating over all traces equals the
distribution-based semantics. Our second contribution is to formalize the
implementation technique of trace Markov chain Monte Carlo (MCMC) for our
calculus and to show its correctness. A key step is defining sufficient
conditions for the distribution induced by trace MCMC to converge to the
distribution-based semantics. To the best of our knowledge, this is the first
rigorous correctness proof for trace MCMC for a higher-order functional
language
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