Integer Polyhedra for Program Analysis

Charles, Philip J.; Howe, Jacob M.; King, Andy

doi:10.1007/978-3-642-02158-9_9

Cited by 9 publications

(6 citation statements)

References 21 publications

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“…Each set of inequalities is first matched against the PTIME cases of Section 4.1. If this matching fails, the integer hull is computed using the algorithm described by Charles et al [2009]. Note that this algorithm supports only bounded polyhedra, the integer hull of an unbounded polyhedron is computed by considering a corresponding bounded one [Schrijver 1986, Theorem 16.1, p. 231].…”

Section: Inductionmentioning

confidence: 99%

Ranking Functions for Linear-Constraint Loops

Ben-Amram

Genaim

2014

J. ACM

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In this article, we study the complexity of the problems: given a loop, described by linear constraints over a finite set of variables, is there a linear or lexicographical-linear ranking function for this loop? While existence of such functions implies termination, these problems are not equivalent to termination. When the variables range over the rationals (or reals), it is known that both problems are PTIME decidable. However, when they range over the integers, whether for single-path or multipath loops, the complexity has not yet been determined. We show that both problems are coNP-complete. However, we point out some special cases of importance of PTIME complexity. We also present complete algorithms for synthesizing linear and lexicographical-linear ranking functions, both for the general case and the special PTIME cases. Moreover, in the rational setting, our algorithm for synthesizing lexicographical-linear ranking functions extends existing ones, because our definition for such functions is more general, yet it has PTIME complexity.

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Section: Inductionmentioning

confidence: 99%

Ranking Functions for Linear-Constraint Loops

Ben-Amram

Genaim

2014

J. ACM

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“…We give an equivalent definition. For a given polyhedron, we can compute its integral hull (the corresponding integral polyhedron) using Hartmann's algorithm [7]. However, the number of inequalities needed can grow exponentially [17].…”

Section: Integer Lasso Programsmentioning

confidence: 99%

Synthesis for Polynomial Lasso Programs

Leike

Tiwari

2014

Lecture Notes in Computer Science

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1 This is an error-corrected version of the original thesis, updated last on July 11, 2018. AbstractThe scope of this work is the constraint-based synthesis of termination arguments for the restricted class of programs called linear lasso programs. A termination argument consists of a ranking function as well as a set of supporting invariants.We extend existing methods in several ways. First, we use Motzkin's Transposition Theorem instead of Farkas' Lemma. This allows us to consider linear lasso programs that can additionally contain strict inequalities. Existing methods are restricted to non-strict inequalities and equalities.Second, we consider several kinds of ranking functions: affine-linear, piecewise and lexicographic ranking functions. Moreover, we present a novel kind of ranking function called multiphase ranking function which proceeds through a fixed number of phases such that for each phase, there is an affine-linear ranking function. As an abstraction to the synthesis of specific ranking functions, we introduce the notion ranking function template. This enables us to handle all ranking functions in a unified way.Our method relies on non-linear algebraic constraint solving as a subroutine which is known to scale poorly to large problems. As a mitigation we formalize an assessment of the difficulty of our constraints and present an argument why they are of an easier kind than general non-linear constraints.We prove our method to be complete: if there is a termination argument of the form specified by the given ranking function template with a fixed number of affine-linear supporting invariants, then our method will find a termination argument.To our knowledge, the approach we propose is the most powerful technique of synthesis-based discovery of termination arguments for linear lasso programs and encompasses and enhances several methods having been proposed thus far [4,18,27].

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“…4.1 and 4.2. If this matching fails, it computes the integer hull using Hartmann's algorithm [28] as explained by Charles et al [18]. Note that this algorithm supports only bounded polyhedra, the integer hull of an unbounded polyhedron is computed by considering a corresponding bounded one [41,Th.…”

Section: Prototype Implementationmentioning

confidence: 99%

On the linear ranking problem for integer linear-constraint loops

Ben-Amram

Genaim

2013

Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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In this paper we study the complexity of the Linear Ranking problem: given a loop, described by linear constraints over a finite set of integer variables, is there a linear ranking function for this loop? While existence of such a function implies termination, this problem is not equivalent to termination. When the variables range over the rationals or reals, the Linear Ranking problem is known to be PTIME decidable. However, when they range over the integers, whether for single-path or multipath loops, the complexity of the Linear Ranking problem has not yet been determined. We show that it is coNP-complete. However, we point out some special cases of importance of PTIME complexity. We also present complete algorithms for synthesizing linear ranking functions, both for the general case and the special PTIME cases.

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Integer Polyhedra for Program Analysis

Cited by 9 publications

References 21 publications

Ranking Functions for Linear-Constraint Loops

Ranking Functions for Linear-Constraint Loops

Synthesis for Polynomial Lasso Programs

On the linear ranking problem for integer linear-constraint loops

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