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

On Probabilistic Term Rewriting

International audienceWe 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 non-trivial 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

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

Improving Dependency Tuples for Almost-Sure Innermost Termination of Probabilistic Term Rewriting

Recently, we adapted the well-known dependency pair (DP) framework to a dependency tuple (DT) framework in order to prove almost-sure innermost termination (iAST) of probabilistic term rewriting systems. In this paper, we improve this approach into a complete criterion for iAST by considering positions of subterms. Based on this, we extend the probabilistic DT framework by new transformations. Our implementation in the tool AProVE shows that they increase its power substantially

Attenuation Regulation as a Term Rewriting System

The classical attenuation regulation of gene expression in bacteria is considered. We propose to represent the secondary RNA structure in the leader region of a gene or an operon by a term, and we give a probabilistic term rewriting system modeling the whole process of such a regulation.Comment: to appea

Strategic Port Graph Rewriting: An Interactive Modelling and Analysis Framework

We present strategic portgraph rewriting as a basis for the implementation of visual modelling and analysis tools. The goal is to facilitate the specification, analysis and simulation of complex systems, using port graphs. A system is represented by an initial graph and a collection of graph rewriting rules, together with a user-defined strategy to control the application of rules. The strategy language includes constructs to deal with graph traversal and management of rewriting positions in the graph. We give a small-step operational semantics for the language, and describe its implementation in the graph transformation and visualisation tool PORGY.Comment: In Proceedings GRAPHITE 2014, arXiv:1407.767

The probability of non-confluent systems

We show how to provide a structure of probability space to the set of execution traces on a non-confluent abstract rewrite system, by defining a variant of a Lebesgue measure on the space of traces. Then, we show how to use this probability space to transform a non-deterministic calculus into a probabilistic one. We use as example Lambda+, a recently introduced calculus defined through type isomorphisms.Comment: In Proceedings DCM 2013, arXiv:1403.768