37 research outputs found
Confluence and Convergence in Probabilistically Terminating Reduction Systems
Convergence of an abstract reduction system (ARS) is the property that any
derivation from an initial state will end in the same final state, a.k.a.
normal form. We generalize this for probabilistic ARS as almost-sure
convergence, meaning that the normal form is reached with probability one, even
if diverging derivations may exist. We show and exemplify properties that can
be used for proving almost-sure convergence of probabilistic ARS, generalizing
known results from ARS.Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
Confluence of CHR Revisited:Invariants and Modulo Equivalence
Abstract simulation of one transition system by another is introduced as a
means to simulate a potentially infinite class of similar transition sequences
within a single transition sequence. This is useful for proving confluence
under invariants of a given system, as it may reduce the number of proof cases
to consider from infinity to a finite number. The classical confluence results
for Constraint Handling Rules (CHR) can be explained in this way, using CHR as
a simulation of itself. Using an abstract simulation based on a ground
representation, we extend these results to include confluence under invariant
and modulo equivalence, which have not been done in a satisfactory way before.Comment: Pre-proceedings paper presented at the 28th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2018), Frankfurt
am Main, Germany, 4-6 September 2018 (arXiv:1808.03326
Confluence in Probabilistic Rewriting
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of uniqueness of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence which is shown to imply the desired uniqueness (even for infinite sequences of reduction) and further properties. We then carry over several criteria from the classical case, such as Newman's lemma, to simplify proving confluence in concrete languages. Using these criteria, we obtain simple proofs of confluence for λ1, an affine probabilistic λ-calculus, and for Q*, a quantum programming language for which a related property has already been proven in the literature.Fil: DÃaz Caro, Alejandro. Universidad Nacional de Quilmes; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; ArgentinaFil: MartÃnez, Guido. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentin
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
On Higher-Order Probabilistic Subrecursion
We study the expressive power of subrecursive probabilistic higher-order calculi. More specifically, we show that endowing a very expressive deterministic calculus like Godel's T with various forms of probabilistic choice operators may result in calculi which are not equivalent as for the class of distributions they give rise to, although they all guarantee almost-sure termination. Along the way, we introduce a probabilistic variation of the classic reducibility technique, and we prove that the simplest form of probabilistic choice leaves the expressive power of T essentially unaltered. The paper ends with some observations about the functional expressive power: expectedly, all the considered calculi capture the functions which T itself represents, at least when standard notions of observations are considered
Probabilistic Termination by Monadic Affine Sized Typing
International audienceWe introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer, quantitative notion of types, but also without imposing some affinity constraints. The proposed type system is powerful enough to type classic examples of probabilistically terminating programs such as random walks. The way typable programs are proved to be almost surely terminating is based on reducibility, but requires a substantial adaptation of the technique