Induction for positive almost sure termination

Abstract

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 can be fully automatized, in particular for rewrite systems on constants, very useful to modelize probabilistic protocols.Probabilistic rewriting has recently been introduced to modelize systems, where probabilistic and non-deterministic phenomena are combined [7]. A lot of models of systems, formalisms or techniques have already been enriched with probabilities, but most of them are restricted to finite state systems. Let us cite automata based models [11,42], Petri Nets [2, 38], process algebra [24], model checking technques [29]. Note also the existence of the PRISM [30], and the APMC [25] tools.Rewriting allows for expressing complex relations on infinite sets of states in a finite way, provided they are countable. Rewriting also provides a nice programming paradigm: the rule-based formalism allows to easily write programs in a declarative way and the underlying algebraic semantics enables automatable or semi-automatable correctness proofs on the programs. Rule-based programming languages and environments are now currently used for any kind of application. Let us cite ASF+SDF [8], Maude [9], CafeOBJ [17], ELAN [3], Stratego [43], or Tom [32]. A probabilistic dimension has recently been introduced in ELAN [6], Tom [18] and Maude [28] to increase the power of the languages to applications like probabilistic protocols.In the context of probabilistic rewriting, the problem of termination naturally arises and in [4], the notions of simple almost sure termination and positive almost sure (PAS in short) termination have been proposed, as well as a method based on interpretations on the reals to ensure the second property. The first termination notion expresses that the probability for a given rewriting derivation to terminate is 1; the second, stronger and more useful from a practical point of view, expresses that the mean length of the derivations from a term is finite.Then, in [5], rewriting strategies have been considered, and sufficient criterions, still based on interpretations on the reals, have been given for PAS termination under strategies.Here, we try to go one step further. In the previously cited paper, the considered strategies defined themselves with probabilities, expressing the ratio of the selection of a rule w.r.t to another. We tackle here the PAS termination problem for position strategies, defined by the position of t...

Termination of rewriting under strategies

“…For instance, it has been recently adapted to the sufficient completeness problem [Gnaedig and Kirchner 2006] and to termination of probabilistic rewriting [Gnaedig 2007]. We expect it to be interesting also to tackle other properties, like ground confluence, or other reduction frameworks like transition systems.…”

“…We could also consider extensions of rewriting like conditional, equational, typed, or constrained rewriting, provided the corresponding narrowing relation can be defined. We have shown in Gnaedig andKirchner [2007, 2006], and Gnaedig [2007] how the method applies to properties other than termination. This is a clear advantage with respect to other existing approaches dedicated to termination.…”

On Probabilistic Term Rewriting

“…Recursiontheoretically, checking (positive) almost-sure termination is harder than checking termination of non-probabilistic programs, where termination is at least recursively enumerable, although undecidable: in a universal probabilistic imperative programming language, the termination questions for almost-sure and positive almost-sure termination on a single input are already 0 2 and 0 2 complete, respectively [24]. Many sound verification methodologies for probabilistic termination have recently been introduced (see, e.g., [6,7,18,14,10]). In particular, the use of ranking martingales has turned out to be quite successful when the analyzed program is imperative, and thus does not have an intricate recursive structure.…”

Probabilistic Termination

“…Their notion is less general than ranking supermartingales as they require variants to be upper bounded. Bournez and Garnier [3] studied positive a.s. termination in the context of term rewriting systems, Gnaedig [17] developed a proof system for positive a.s. termination for rewrite systems under innermost strategies using induction. It is not clear how the author dealt with the integrability problems discussed throughout this paper.…”