On probabilistic term rewriting

open3siThis work is partially supported by the ANR projects 14CE250005 ELICA and 16CE250011 REPAS, the FWF project Y757, the JSPS-INRIA bilateral joint research project “CRECOGI”, the ERC Consolidator Grant DLV-818616 DIAPASoN, and JST ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603).We study the termination problem for probabilistic term rewrite systems. We prove that the interpretation method is sound and complete for a strengthening of positive almost sure termination, when abstract reduction systems and term rewrite systems are considered. Two instances of the interpretation method—polynomial and matrix interpretations—are analyzed and shown to capture interesting and nontrivial examples when automated. We capture probabilistic computation in a novel way by means of multidistribution reduction sequences, thus accounting for both the nondeterminism in the choice of the redex and the probabilism intrinsic in firing each rule.openAvanzini M.; Dal Lago U.; Yamada A.Avanzini M.; Dal Lago U.; Yamada A

Application d'une méthode de preuve probabiliste pour prouver la terminaison en temps moyen fini du protocole CSMA/CA 802.11b

Nous présentons une méthode de preuve qui permet de montrer la terminaison en temps moyen fini d'un algorithme probabiliste et distribué utilisé par le protocole WI-FI 802.11b

Intersection types and (positive) almost-sure termination

Randomized higher-order computation can be seen as being captured by a λ-calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any λ-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight-not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that

Probabilistic Rewriting: On Normalization, Termination, and Unique Normal Forms

While a mature body of work supports the study of rewriting systems, even infinitary ones, abstract tools for Probabilistic Rewriting are still limited. Here, we investigate questions such as uniqueness of the result (unique limit distribution) and we develop a set of proof techniques to analyze and compare reduction strategies. The goal is to have tools to support the operational analysis of probabilistic calculi (such as probabilistic lambda-calculi) whose evaluation is also non-deterministic, in the sense that different reductions are possible. In particular, we investigate how the behavior of different rewrite sequences starting from the same term compare w.r.t. normal forms, and propose a robust analogue of the notion of "unique normal form". Our approach is that of Abstract Rewrite Systems, i.e. we search for general properties of probabilistic rewriting, which hold independently of the specific structure of the objects.Comment: Extended version of the paper in FSCD 2019, International Conference on Formal Structures for Computation and Deductio

Proving Positive Almost Sure Termination Under Strategies

In last RTA, we introduced the notion of probabilistic rewrite systems and we gave some conditions entailing termination of those systems within a finite mean number of reduction steps. Termination was considered under arbitrary unrestricted policies. Policies correspond to strategies for non-probabilistic rewrite systems. This is often natural or more useful to restrict policies to a subclass. We introduce the notion of positive almost sure termination under strategies, and we provide sufficient criteria to prove termination of a given probabilitic rewrite system under strategies. This is illustrated with several examples

Intersection Types and (Positive) Almost-Sure Termination

Randomized higher-order computation can be seen as being captured by a lambda calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any lambda-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight -- not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that

Induction for Positive Almost Sure Termination - Extended version -

In this paper, we propose an inductive approach to prove positive almost sure termination of probabilistic rewriting under the innermost strategy. We extend to the probabilistic case a technique we proposed for termination of usual rewriting under strategies. The induction principle consists in assuming that terms smaller than the starting terms for an induction ordering are positively almost surely terminating. The proof is developed in generating proof trees, modelizing rewriting trees, in alternatively applying abstraction steps, expressing the application of the induction hypothesis, and narrowing steps, simulating the possible rewriting steps after abstraction. This technique is fully automatable for rewrite systems on constants, very useful to modelize probabilistic protocols