Finiteness conditions for fixed point iteration

Nielson, Flemming; Nielson, Hanne Riis

doi:10.1145/141478.141514

Cited by 4 publications

(7 citation statements)

References 7 publications

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“…Consequently, by Theorem 3.8, when the Tabulation Algorithm terminates, the value of X n is the value for node n in the Figure 4. The above five diagrams show the situations handled in lines [14]- [16], [17]- [19], [25], [26]- [28] The Tabulation Algorithm always terminates, and upon termination, X n = MVP n , for all n ∈ N * .…”

Section: An Efficient Algorithm For the Realizable-path Reachability mentioning

confidence: 99%

“…Graph reachability can also be thought of as an implementation of the pointwise computation of fixed points, which has been studied by Cai and Paige [4] and Nielson and Nielson [26,25]. Theorem 3.3, the basis on which we convert dataflow-analysis problems to reachability problems, is similar to Lemma 14 of Cai and Paige; however, the relation that Cai and Paige define for representing distributive functions does not have the subsumption property.…”

Section: Dataflow Analysis Via Graph Reachability and Pointwise Compumentioning

confidence: 99%

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Precise interprocedural dataflow analysis via graph reachability

Reps

Horwitz

Sagiv

1995

Proceedings of the 22nd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '95

940

774

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Section: An Efficient Algorithm For the Realizable-path Reachability mentioning

confidence: 99%

Section: Dataflow Analysis Via Graph Reachability and Pointwise Compumentioning

confidence: 99%

Precise interprocedural dataflow analysis via graph reachability

Reps

Horwitz

Sagiv

1995

Proceedings of the 22nd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '95

940

774

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“…Similarly, the aims of fast fixed-point finding of functionals illustrated in our example on strictness analysis, is shared by the work of Nielson and Nielson [62,61]. Their approach is, however, somewhat orthogonal to the algorithms described here.…”

Section: Further Work 185mentioning

confidence: 94%

Verification of Temporal Properties of Concurrent Systems

Andersen¹

1993

DPB

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2 Dansk SammenfatningDenne afhandling behandler metoder og algoritmer til verifikation af parallelle systemer. Verifikationen af et system foretages ud fra en givet specifikation, der udtaler sig om visse ønskelige aspekter, og en, muligvis ufuldstaendig, beskrivelse af det parallelle system. Specifikationer vil blive angivet i en kraftig modallogik -den modale µ-kalkule -og parallelle systemer beskrives som processer fra en procesalgebra, der følger traditionen fra Milner's CCS og Hoare's CSP. Afhandlingen tager udgangspunkt i en kompositionel metode, der, givet en specifikation til en sammensat proces, dekomponerer denne til speficikationer om delprocesser, således at den oprindelige proces vi1 tilfredsstille sin specifikation, hvis og kun hvis, delprocesserne tilfredsstiller deres delspecifikationer. En sådan kompositionel metode bidrager til at kunne håndtere den store kompleksitet som parallelle systemer ofte besidder.Selvom den modale µ-kalkule er meget udtryksfuld, har den fra et pragmatisk synspunkt nogle mangler, som vi vi1 forsøge at udbedre ved introduktionen af en udvidet µ-kalkule, der, udover nogle yderligere logiske konstruktioner, som tillader nemmere formulering af egenskaber i logikken, også har en mulighed for kompakt repraesentation af specifikationer ved deling af deludtryk gennem simultane fikspunkter. Fra en mindre observation i den kompositionelle metode udnyttes denne mulighed for kompakte repraesentationer til at give effektive globale og lokale algoritmer til automatisk afgørelse af om en endelig proces tilfredsstiller sin specifikation -et problem der benaevnes model-check. Den modale µ-kalkule får sin udtrykskraft fra tilstedevaerelsen af minimale og maksimale fikspunkter; effektiv beregning af fikspunkter indgår således som en vigtig del af algoritmerne til modelcheck.De centrale ideer med deling af vaerdier og opfølgning af aendringer brugt i disse algoritmer er af en generel natur -en observation, der bliver brugt til at give en algoritme for beregning af fikspunkter i endelige fuldstaendige partielle ordninger og gitre.For uendelige tilstandssystemer er modelcheckproblemet generelt uafgør-ligt, så vi er nødt til betragte semi-automatiske eller brugerassisterede systemer. Vi praesenterer en metode baseret på angivelse af passende vel-funderede ordninger for de minimale fikspunkter. Metoden er en slags målorienteret bevissystem: Startende med målet, en proces og en specifikation som processen skal vises at tilfredsstille, konstrueres nye delmål ud fra et saet af regler. Dette 3 kan gentages indtil alle delmål er trivielle. Metoden bevises at vaere sund og fuldstaendig.Endelig praesesenteres en ny måde at angribe detåbne problem for µ-kalkulen, der består i at finde en endelig aksiomatisering. Vi karakteriserer en klasse af kategoriske modeller for en intuitionistisk version af kalkulen og reformulerer problemet som et problem om hvordan disse kategoriske modeller kan skaeres ned til mere traditionelle Kripke-agtige modeller.I det konkluderende kapitel diskuteres kort kompleksiteten...

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“…Termination may, however, also be achieved when D is infinite or has infinite height if restrictions are placed on the expression E. The use of widening [2] in abstract interpretation is an example of this. See also [13] and [6] for a discussion of methods to guarantee termination.…”

Section: Terminationmentioning

confidence: 99%

Higher-order chaotic iteration sequences

Rosendahl

1993

Lecture Notes in Computer Science

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Chaotic iteration sequences is a method for approximating fixpoints of monotonic functions proposed by Patrick and Radhia Cousot. It may be used in specialisation algorithms for Prolog programs and in abstract interpretation when only parts of a fixpoint result is needed to perform program optimisations. In the first part of this paper we reexamine the definition of chaotic iteration sequences and show how a number of other methods to compute fixpoints may be formulated in this framework. In the second part we extend the technique to higherorder functions.The method of expressing solutions to problems in a recursive style has a long history in Computer Science. Often a solution algorithm may use itself on a subproblem and an implementation may be based on recursion. At times it is more natural to express the solution as a fixpoint equation in which well-definedness is only guaranteed using a fixpoint theorem. This is common in abstract interpretation, data flow analysis, "first" and "follow" computation, etc.In strictness analysis [11] we may construct functions over the domain 2 = {0, 1} with 0 ≤ 1. If we want to evaluate the functionfor certain arguments, we will in general need to evaluate the function for all possible arguments and recompute until stability. It is, however, often only necessary to recompute the function for some of the possible arguments. By restricting the iteration to a smaller set of arguments, we may greatly simplify the computational task of evaluating the function. In this paper we consider such recursively defined functions for which recursion is not sufficient in the implementation.

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Finiteness conditions for fixed point iteration

Cited by 4 publications

References 7 publications

Precise interprocedural dataflow analysis via graph reachability

Precise interprocedural dataflow analysis via graph reachability

Verification of Temporal Properties of Concurrent Systems

Higher-order chaotic iteration sequences

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