Ranking Functions for Linear-Constraint Loops

Ben-Amram, Amir M.; Genaim, Samir

doi:10.1145/2629488

Cited by 65 publications

(26 citation statements)

References 58 publications

(119 reference statements)

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“…coNP-hardness directly follows from our class of input including that studied by Ben-Amram and Genaim [2014]; let us now show membership in coNP. Our algorithms (section 4, section 6) stop when either they have a linear ranking function, either they reach an unsolvable system of constraints.…”

Section: Algorithm 3 Monodim-multi For Multiple Control Pointsmentioning

confidence: 99%

“…, ρ m (x) , which is shown to be strictly decreasing with respect to lexicographic ordering. Again, the function ρ can be allowed to depend on the program point, and complete automated synthesis methods exist for this class Alias et al [2010], Ben-Amram and Genaim [2014].…”

Section: Introductionmentioning

confidence: 99%

“…Lexicographic linear ranking functions Alias et al [2010], Cook et al [2013], Bradley et al [2005b,a], Ben-Amram and Genaim [2014] are both much more powerful than linear ranking functions (e.g. one can easily prove the termination of the Ackermann-Péter function) and not much harder to obtain.…”

Section: Introductionmentioning

confidence: 99%

“…A comparison of the two dual approaches (constraints vs generators) showed that constraints scaled better for scheduling problems Balev et al [1998]. In contrast, Ben-Amram and Genaim [2014] used the vertices of the integer hull of the transition relation. We improve on this approach by lazily enumerating the generators, without explicitly computing this convex hull.…”

Section: Introductionmentioning

confidence: 99%

“…∀0 ≤ k < n t[k] = 4) can be over-approximated by quantifier instantiation. Ben-Amram and Genaim [2014] call weak ranking functions on programs using rational variables: in Definition 6, replace Z by Q.…”

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confidence: 99%

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Synthesis of ranking functions using extremal counterexamples

GonnordLaure¹,

MonniauxDavid²,

RadanneGabriel³

2015

SIGPLAN Not.

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We present a complete method for synthesizing lexicographic linear ranking functions (and thus proving termination), supported by inductive invariants, in the case where the transition relation of the program includes disjunctions and existentials (large block encoding of control flow). Previous work would either synthesize a ranking function at every basic block head, not just loop headers, which reduces the scope of programs that may be proved to be terminating, or expand large block transitions including tests into (exponentially many) elementary transitions, prior to computing the ranking function, resulting in a very large global constraint system. In contrast, our algorithm incrementally refines a global linear constraint system according to extremal counterexamples: only constraints that exclude spurious solutions are included. Experiments with our tool Termite show marked performance and scalability improvements compared to other systems.

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Section: Algorithm 3 Monodim-multi For Multiple Control Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Synthesis of ranking functions using extremal counterexamples

GonnordLaure¹,

MonniauxDavid²,

RadanneGabriel³

2015

SIGPLAN Not.

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Tight Worst-Case Bounds for Polynomial Loop Programs

Ben-Amram

Hamilton

2019

Lecture Notes in Computer Science

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In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language-representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication-it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate.A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found.A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid.

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Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

2019

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Multiphase ranking functions (MΦRFs) are tuples f 1 , . . . , f d of linear functions that are often used to prove termination of loops in which the computation progresses through a number of "phases". Our work provides new insights regarding such functions for loops described by a conjunction of linear constraints (Single-Path Constraint loops). The decision problem existence of a MΦRF asks to determine whether a given SLC loop admits a MΦRF; and the corresponding bounded decision problem restricts the search to MΦRFs of depth d, where the parameter d is part of the input. The algorithmic and complexity aspects of the bounded problem have been completely settled in a recent work. In this paper we make progress regarding the existence problem, without a given depth bound. In particular, we present an approach that reveals some important insights into the structure of these functions. Interestingly, it relates the problem of seeking MΦRFs to that of seeking recurrent sets (used to prove non-termination). It also helps in identifying classes of loops for which MΦRFs are sufficient. Our research has led to a new representation for single-path loops, the difference polyhedron replacing the customary transition polyhedron. This representation yields new insights on MΦRFs and SLC loops in general. For example, a result on bounded SLC loops becomes straightforward.

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Ranking Functions for Linear-Constraint Loops

Cited by 65 publications

References 58 publications

Synthesis of ranking functions using extremal counterexamples

Synthesis of ranking functions using extremal counterexamples

Tight Worst-Case Bounds for Polynomial Loop Programs

Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

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