Proving Positive Almost Sure Termination Under Strategies

Bournez, Olivier; Garnier, Florent

doi:10.1007/11805618_27

Cited by 12 publications

(26 citation statements)

References 19 publications

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“…Moreover, the completeness results of [4,5], based on real interpretations, are nice from a theoretical point of view, but not constructive, and no algorithmic help exists yet, to exhibit ad-hoc interpretations.…”

Section: Resultsmentioning

confidence: 97%

“…Narrow σ=Id y y t t t t t t t t t Note that on such an example, where the inductive principle is crucial, the real interpretation technique of [4,5], is very hard to apply. Because this technique involves arguments that are local to one rule, and are not modular w.r.t rules, this is also the case for examples where the cycle is generated by more than one rule like {a → c : 1, c → a : 1/2|b : 1/2}, and that we easily handle.…”

Section: One Step Further: Considering Infinitementioning

confidence: 98%

“…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.…”

Section: Introducing the Problemmentioning

confidence: 99%

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Induction for positive almost sure termination

Gnaedig

2007

Proceedings of the 9th ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming

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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...

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Section: Resultsmentioning

confidence: 97%

Section: One Step Further: Considering Infinitementioning

confidence: 98%

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Induction for positive almost sure termination

Gnaedig

2007

Proceedings of the 9th ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming

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“…When the latter holds, techniques akin to sized types have been shown to be applicable [10]. Finally, as already mentioned, the current work can be seen as stemming from the work by Bournez et al [6,4,5]. The added value compared to their work are first of all the notion of multidistribution as way to give an instantaneous description of the state of the underlying system which exhibits both nondeterministic and probabilistic features.…”

Section: Related Workmentioning

confidence: 89%

On probabilistic term rewriting

Avanzini

Lago

Yamada

2020

Science of Computer Programming

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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 way of multidistribution reduction sequences, this way accounting for both the nondeterminism in the choice of the redex and the probabilism intrinsic in firing each rule.

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“…Probabilities make termination more than a simple yes-no question, and the following criteria have been proposed: probabilistic termination -a derivation terminates with some probability > 0 -and almost-sure termination -a derivation terminates with probability = 1, even if infinite derivations may exist (and whose total probability thus amounts to 0). When considering a PARS as a computational system, almostsure termination may be the most interesting, and there exist well-established methods for proving this property [6,10].PARS cover a variety of probabilistic algorithms and programs, scheduling strategies and protocols [5,7,23], and PARS is a well-suited abstraction level for better understanding their termination and correctness properties. Randomized or probabilistic algorithms (e.g., [4,20,21]) come in two groups: Monte Carlo Algorithms that allow a set of alternative outputs (typically only correct with a certain probability or within a certain accuracy), e.g., , Monte Carlo integration and Simulated Annealing [19]; and Las Vegas Algorithms, that provide one (correct) output and that may be simpler and on average more efficient than their deterministic counterparts, e.g., Randomized Quicksort [11], checking equivalence of circular lists [17], probabilistic modular GCD [30].…”

mentioning

confidence: 99%

Confluence and Convergence in Probabilistically Terminating Reduction Systems

Kirkeby

Christiansen

2018

Logic-Based Program Synthesis and Transformation

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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. IntroductionProbabilistic abstract reduction systems, PARS, are general models of systems that develop over time in discrete steps [7]. In each non-final state, the choice of successor state is governed by a probability distribution, which in turn induces a global, probabilistic behaviour of the system. Probabilities make termination more than a simple yes-no question, and the following criteria have been proposed: probabilistic termination -a derivation terminates with some probability > 0 -and almost-sure termination -a derivation terminates with probability = 1, even if infinite derivations may exist (and whose total probability thus amounts to 0). When considering a PARS as a computational system, almostsure termination may be the most interesting, and there exist well-established methods for proving this property [6,10].PARS cover a variety of probabilistic algorithms and programs, scheduling strategies and protocols [5,7,23], and PARS is a well-suited abstraction level for better understanding their termination and correctness properties. Randomized or probabilistic algorithms (e.g., [4,20,21]) come in two groups: Monte Carlo Algorithms that allow a set of alternative outputs (typically only correct with a certain probability or within a certain accuracy), e.g., , Monte Carlo integration and Simulated Annealing [19]; and Las Vegas Algorithms, that provide one (correct) output and that may be simpler and on average more efficient than their deterministic counterparts, e.g., Randomized Quicksort [11], checking equivalence of circular lists [17], probabilistic modular GCD [30]. We focus on results that are relevant for the latter kind of systems, and here the property of convergence is interesting, as it may be a necessary condition for correctness: a system is convergent if it is guaranteed to terminate

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Proving Positive Almost Sure Termination Under Strategies

Cited by 12 publications

References 19 publications

Induction for positive almost sure termination

Induction for positive almost sure termination

On probabilistic term rewriting

Confluence and Convergence in Probabilistically Terminating Reduction Systems

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