Ranking Templates for Linear Loops

Leike, Jan; Heizmann, Matthias

doi:10.2168/lmcs-11(1:16)2015

Cited by 33 publications

(30 citation statements)

References 45 publications

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“…The complexity and algorithmic aspects has been completely settled in [8]. MΦRFs for general loops have been considered in [26,29], where both use non-linear constraint solving. In [4] the notion of "eventual linear ranking functions," which are MΦRFs of depth 2 has been studied.…”

Section: Discussionmentioning

confidence: 99%

“…which is terminating [26]. It is easy to show (by induction) that the polyhedron passed to FindMLRF in the i-th call is…”

Section: Cases In Which Algorithm 1 Does Not Terminatementioning

confidence: 96%

“…. , ρ k be an MΦRF for Q, of optimal depth k ≤ d. Without lose of generality we may assume that ρ k is non-negative on all proj x (Q), this follows immediately from the definition of nested MΦRF [26], which is a special case of MΦRF in which the last component is nonnegative, and the fact that for the case of SLC loops existence of a MΦRF implies the existence of a nested MΦRF [8] of the same optimal depth. Clearly τ ′ = ρ 1 , .…”

Section: Deciding Existence Of Mφrfsmentioning

confidence: 99%

“…When the algorithm does not terminate, the iterates F i (Q) converge to Q ω = ∩ i≥0 F i (Q). For example, for Loop (26), which is terminating, we have Q ω = ∅, and for Loop (27), which is not terminating, we have Q ω = {x 1 ≥ 1, x ′ 2 = 2x 2 , x ′ 1 = 3x 1 − 2} which is a monotonic recurrent set. Some natural questions arise at this point: (i) is it true that Q ω = ∅ iff Q is terminating?…”

Section: Cases In Which Algorithm 1 Does Not Terminatementioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

“…which is terminating [26]. It is easy to show (by induction) that the polyhedron passed to FindMLRF in the i-th call is…”

Section: Cases In Which Algorithm 1 Does Not Terminatementioning

confidence: 96%

Section: Deciding Existence Of Mφrfsmentioning

confidence: 99%

Section: Cases In Which Algorithm 1 Does Not Terminatementioning

confidence: 99%

See 2 more Smart Citations

Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

2019

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“…Complete solutions and complexity. Complete solutions for MΦRFs (over the rationals) appear in [18,20]. Both use non-linear constraint solving, and therefore do not achieve a polynomial time complexity.…”

Section: Introductionmentioning

confidence: 99%

On Multiphase-Linear Ranking Functions

Ben-Amram

Genaim

2017

Computer Aided Verification

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Multiphase ranking functions (MΦRFs) were proposed as a means to prove the termination of a loop in which the computation progresses through a number of "phases", and the progress of each phase is described by a different linear ranking function. Our work provides new insights regarding such functions for loops described by a conjunction of linear constraints (single-path loops). We provide a complete polynomial-time solution to the problem of existence and of synthesis of MΦRF of bounded depth (number of phases), when variables range over rational or real numbers; a complete solution for the (harder) case that variables are integer, with a matching lower-bound proof, showing that the problem is coNP-complete; and a new theorem which bounds the number of iterations for loops with MΦRFs. Surprisingly, the bound is linear, even when the variables involved change in non-linear way. We also consider a type of lexicographic ranking functions, LLRF , more expressive than types of lexicographic functions for which complete solutions have been given so far. We prove that for the above type of loops, lexicographic functions can be reduced to MΦRFs, and thus the questions of complexity of detection and synthesis, and of resulting iteration bounds, are also answered for this class. * amirben@mta.ac.il

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Learning Büchi Automata and Its Applications

Turrini

Chen

et al. 2019

Engineering Trustworthy Software Systems

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Ranking Templates for Linear Loops

Cited by 33 publications

References 45 publications

Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

On Multiphase-Linear Ranking Functions

Learning Büchi Automata and Its Applications

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