The Abstract Domain of Segmented Ranking Functions

Urban, Caterina

doi:10.1007/978-3-642-38856-9_5

Cited by 43 publications

(64 citation statements)

References 28 publications

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“…an affine function) of the program variables f # ∈ F # [22]. We can now use the abstractions S # and F # to build the abstract domain O.…”

Section: Ordinal-valued Ranking Functionsmentioning

confidence: 99%

“…that uses intervals [7] as state abstraction and affine functions as abstract functions f # ∈ F # [22]. We consider the join of the abstract functions:…”

Section: Ordinal-valued Functions the Elements Of The Abstract Domainmentioning

confidence: 99%

“…In the following, we will briefly recall the abstract domain of piecewise-defined ranking functions [22]. Then, we describe our extension of this domain using the ordinal-valued ranking functions we presented in Section 4.…”

Section: Piecewise-defined Ranking Functionsmentioning

confidence: 99%

“…In [22], we present a decidable abstraction for imperative programs by means of piecewisedefined ranking functions over natural numbers. These functions are attached to the program control points and represent an upper bound on the number of program execution steps remaining before termination.…”

Section: Introductionmentioning

confidence: 99%

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An Abstract Domain to Infer Ordinal-Valued Ranking Functions

Urban

Miné

2014

Programming Languages and Systems

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Abstract. The traditional method for proving program termination consists in inferring a ranking function. In many cases (i.e. programs with unbounded non-determinism), a single ranking function over natural numbers is not sufficient. Hence, we propose a new abstract domain to automatically infer ranking functions over ordinals.We extend an existing domain for piecewise-defined natural-valued ranking functions to polynomials in ω, where the polynomial coefficients are natural-valued functions of the program variables. The abstract domain is parametric in the choice of the maximum degree of the polynomial, and the types of functions used as polynomial coefficients.We have implemented a prototype static analyzer for a while-language by instantiating our domain using affine functions as polynomial coefficients. We successfully analyzed small but intricate examples that are out of the reach of existing methods.To our knowledge this is the first abstract domain able to reason about ordinals. Handling ordinals leads to a powerful approach for proving termination of imperative programs, which in particular subsumes existing techniques based on lexicographic ranking functions.

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“…an affine function) of the program variables f # ∈ F # [22]. We can now use the abstractions S # and F # to build the abstract domain O.…”

Section: Ordinal-valued Ranking Functionsmentioning

confidence: 99%

“…that uses intervals [7] as state abstraction and affine functions as abstract functions f # ∈ F # [22]. We consider the join of the abstract functions:…”

Section: Ordinal-valued Functions the Elements Of The Abstract Domainmentioning

confidence: 99%

Section: Piecewise-defined Ranking Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Abstract Domain to Infer Ordinal-Valued Ranking Functions

Urban

Miné

2014

Programming Languages and Systems

Self Cite

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“…A natural extension of this framework is, instead of requiring a single, monolithic ranking function at all program points, to make it dependent on the program point. Other extensions include piecewise linear ranking functions [Urban, 2013] and ordinal-value ranking functions [Urban and Miné, 2014], which subsume lexicographic orders.…”

Section: Introductionmentioning

confidence: 99%

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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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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The Abstract Domain of Segmented Ranking Functions

Cited by 43 publications

References 28 publications

An Abstract Domain to Infer Ordinal-Valued Ranking Functions

An Abstract Domain to Infer Ordinal-Valued Ranking Functions

Synthesis of ranking functions using extremal counterexamples

Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

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