On the linear ranking problem for integer linear-constraint loops

Ben-Amram, Amir M.; Genaim, Samir

doi:10.1145/2429069.2429078

Cited by 43 publications

(45 citation statements)

References 39 publications

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“…where (7) follows from the definition of conditional expectation (1), (8) follows from the definition of {lev k = j * }, and (9) holds since Y k (ω) = X k [j * ](ω) for ω with F (ω) > k. Almost identical argument shows that…”

Section: Proof Of Proposition 31mentioning

confidence: 98%

“…In this work we extend lexicographic ranking functions to probabilistic programs, and present lexicographic RSMs for almost-sure termination analysis of probabilistic programs with non-determinism. Theoretical complexity of synthesizing lexicographic ranking functions in non-probabilistic programs was studied in [8,9].…”

Section: Related Workmentioning

confidence: 99%

“…We first recall the general statement of the Radon-Nikodym theorem. Given two measurable spaces 8 (Ω, F , µ) and (Ω, F , ν ), we say that ν dominates µ, written µ << ν if for all A ∈ F , ν (A) = 0 implies µ(A) = 0. Radon-Nikodym theorem states that if both µ and ν are sigma-finite (that is, Ω is a union of countably many sets of finite measure under ν and µ), then µ << ν implies that there exists an almost-surely unique F -measurable function f : Ω → [0, ∞) such that for each A ∈ F , the Lebesgue integral of the function f · 1 A in measurable space (Ω, F , ν ) is equal to µ(A).…”

Section: Proof Of Proposition 31mentioning

confidence: 99%

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Lexicographic ranking supermartingales: an efficient approach to termination of probabilistic programs

Agrawal

Chatterjee

Novotný

2017

Proc. ACM Program. Lang.

151

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Probabilistic programs extend classical imperative programs with real-valued random variables and random branching. The most basic liveness property for such programs is the termination property. The qualitative (aka almost-sure) termination problem given a probabilistic program asks whether the program terminates with probability 1. While ranking functions provide a sound and complete method for non-probabilistic programs, the extension of them to probabilistic programs is achieved via ranking supermartingales (RSMs). While deep theoretical results have been established about RSMs, their application to probabilistic programs with nondeterminism has been limited only to academic examples. For non-probabilistic programs, lexicographic ranking functions provide a compositional and practical approach for termination analysis of realworld programs. In this work we introduce lexicographic RSMs and show that they present a sound method for almost-sure termination of probabilistic programs with nondeterminism. We show that lexicographic RSMs provide a tool for compositional reasoning about almost sure termination, and for probabilistic programs with linear arithmetic they can be synthesized efficiently (in polynomial time). We also show that with additional restrictions even asymptotic bounds on expected termination time can be obtained through lexicographic RSMs. Finally, we present experimental results on abstractions of real-world programs to demonstrate the effectiveness of our approach.Lexicographic Ranking Supermartingales: An Efficient Approach to Termination of Probabilistic Programs 1:3Our contributions. In this work our main contributions range from theoretical foundations of lexicographic RSMs, to algorithmic approaches for them, to experimental results showing their applicability to programs. We describe our main contributions below:

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Section: Proof Of Proposition 31mentioning

confidence: 98%

Section: Related Workmentioning

confidence: 99%

Section: Proof Of Proposition 31mentioning

confidence: 99%

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Lexicographic ranking supermartingales: an efficient approach to termination of probabilistic programs

Agrawal

Chatterjee

Novotný

2017

Proc. ACM Program. Lang.

151

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“…Concerning the existence of linear ranking functions, as the Farkas' Lemma is not true for the integers, the method presented in Section 2 is not valid. The problem, which has been solved very recently in [10], is coNP-complete, and the paper proposes an exponential-time algorithm. Extending the present approach to integer SLC loops is another interesting idea to consider for future work.…”

Section: Discussionmentioning

confidence: 99%

“…Various categories of loops have been identified: for the purposes of this paper we focus on single-path linear constraint (SLC) loops [9].…”

Section: Introductionmentioning

confidence: 99%

Eventual linear ranking functions

Bagnara

Mesnard²

2013

Proceedings of the 15th Symposium on Principles and Practice of Declarative Programming

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We present the first operational account of call by need that connects syntactic theory and implementation practice. Syntactic theory: the storeless operational semantics using syntax rewriting to account for demand-driven computation and for caching intermediate results. Implementational practice: the store-based operational technique using memo-thunks to implement demand-driven computation and to cache intermediate results for subsequent sharing. The implementational practice was initiated by Landin and Wadsworth and is prevalent today to implement lazy programming languages such as Haskell. The syntactic theory was initiated by Ariola, Felleisen, Maraist, Odersky and Wadler and is prevalent today to reason equationally about lazy programs, on par with Baren-dregt et al.'s term graphs. Nobody knows, however, how the theory of call by need compares to the practice of call by need: all that is known is that the theory of call by need agrees with the theory of call by name, and that the practice of call by need optimizes the practice of call by name. Our operational account takes the form of three new calculi for lazy evaluation of lambda-terms and our synthesis takes the form of three lock-step equivalences. The first calculus is a hereditarily compressed variant of Ariola et al.'s call-by-need lambda-calculus and makes "neededness" syntactically explicit. The second calculus distinguishes between strict bindings (which are induced by demand-driven computation) and non-strict bindings (which are used for caching intermediate results). The third calculus uses memo-thunks and an algebraic store. The first calculus syntactically corresponds to a storeless abstract machine, the second to an abstract machine with local stores, and the third to a lazy Kriv-ine machine, i.e., a traditional store-based abstract machine implementing lazy evaluation. The machines are intensionally compatible with extensional reasoning about lazy programs and they are lock-step equivalent. Each machine functionally corresponds to a natural semantics for call by need in the style of Launchbury, though for non-preprocessed λ-terms. Our results reveal a genuine and principled unity of computational theory and computational practice, one that readily applies to variations on the general theme of call by need.

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ComplexityParser: An Automatic Tool for Certifying Poly-Time Complexity of Java Programs

Hainry

Jeandel

Péchoux

et al. 2021

Lecture Notes in Computer Science

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ComplexityParser is a static complexity analyzer for Java programs providing the first implementation of a tier-based typing discipline. The input is a file containing Java classes. If the main method can be typed and, provided the program terminates, then the program is guaranteed to do so in polynomial time and hence also to have heap and stack sizes polynomially bounded. The application uses antlr to generate a parse tree on which it performs an efficient type inference: linear in the input size, provided that the method arity is bounded by some constant.

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On the linear ranking problem for integer linear-constraint loops

Cited by 43 publications

References 39 publications

Lexicographic ranking supermartingales: an efficient approach to termination of probabilistic programs

Lexicographic ranking supermartingales: an efficient approach to termination of probabilistic programs

Eventual linear ranking functions

ComplexityParser: An Automatic Tool for Certifying Poly-Time Complexity of Java Programs

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