Abstract Interpretation Frameworks

Cousot, Patrick; Cousot, Radhia

doi:10.1093/logcom/2.4.511

Cited by 536 publications

(413 citation statements)

References 29 publications

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“…We show how Galois connections are generated from U-GLB-L-LUB-closed binary relations (cf. [8,23,28]). 2.…”

Section: Outline Of Resultsmentioning

confidence: 99%

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Closed and Logical Relations for Over- and Under-Approximation of Powersets

Schmidt

2004

Static Analysis

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Abstract. We redevelop and extend Dams's results on over-and underapproximation with higher-order Galois connections: (1) We show how Galois connections are generated from U-GLB-L-LUBclosed binary relations, and we apply them to lower and upper powerset constructions, which are weaker forms of powerdomains appropriate for abstraction studies.(2) We use the powerset types within a family of logical relations, show when the logical relations preserve U-GLB-L-LUB-closure, and show that simulation is a logical relation. We use the logical relations to rebuild Dams's most-precise simulations, revealing the inner structure of overand under-approximation. (3) We extract validation and refutation logics from the logical relations, state their resemblance to Hennessey-Milner logic and description logic, and obtain easy proofs of soundness and best precision.Almost all Galois-connection-based static analyses are over-approximating: For Galois connection, (P(C), ⊆) α o , γ (A, A ), an abstract value a ∈ A proclaims a property of all the outputs of a program. For example, even ∈ Parity (see Figure 2 for the abstract domain Parity) asserts, "∀even" -all the program's outputs are even numbers, that is, the output is a set from {S ∈ P(Nat) | S ⊆ γ(even)}.An under-approximating Galois connection, (P(C), ⊇) α u , γ A op , where A op = (A, A ), is the dual. Here, even ∈ Parity op asserts that all even numbers are included in the program's outputs -a strong assertion. Also, we may reuse γ : A → P(C) as the upper adjoint from A op to P(C) op iff γ preserves joins in (A, A ) -another strong demand.Fortunately, there is an alternative view of under-approximation: a ∈ A op asserts an existential property -there exists an output with property a. For example, even ∈ Parity op asserts "∃even" -there is an even number in the program's outputs, which is a set from {S ∈ P(Nat) | S ∩ γ(even) = ∅}. Now, we can generalize both over-and under-approximation to multiple properties, e.g., ∀{even, odd } ≡ ∀(even ∨ odd ) -all outputs are even-or odd-valued; and ∃{even, odd } ≡ ∃even ∧ ∃odd -the output set includes an even value and an odd value. These examples "lift" A and A op into the powerset lattices, P L (A) and P U (A), respectively, and set the stage for the problem studied in this paper.schmidt@cis.ksu.edu.

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“…We show how Galois connections are generated from U-GLB-L-LUB-closed binary relations (cf. [8,23,28]). 2.…”

Section: Outline Of Resultsmentioning

confidence: 99%

“…The following results are assembled from [4,8,14,23,24,28]: Let C and A be complete lattices, and let ρ ⊆ C × A, where c ρ a means c is approximated by a.…”

Section: Closed Binary Relations Generate Galois Connectionsmentioning

confidence: 99%

Closed and Logical Relations for Over- and Under-Approximation of Powersets

Schmidt

2004

Static Analysis

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“…However some applications in which relational data are involved cannot tolerate any permanent distortions and data's integrity needs to be authenticated. In this paper we further strengthen this approach by refining the distortion free watermarking algorithm [9] for relational databases and discuss it in abstract interpretation framework proposed by Patrick Cousot and Radhia Cousot [10], [11], [12].…”

Section: Introductionmentioning

confidence: 99%

Distortion-Free Authentication Watermarking.

Bhattacharya

Cortesi

2013

Communications in Computer and Information Science

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“…Widening operators [15][16][17][18] provide a simple and general characterization for such mechanisms. In its simplest form, a widening operator on a poset (L, ) is defined as a partial function ∇ : L × L L satisfying:…”

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confidence: 99%

Precise Widening Operators for Convex Polyhedra

et al. 2003

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Abstract. Convex polyhedra constitute the most used abstract domain among those capturing numerical relational information. Since the domain of convex polyhedra admits infinite ascending chains, it has to be used in conjunction with appropriate mechanisms for enforcing and accelerating convergence of the fixpoint computation. Widening operators provide a simple and general characterization for such mechanisms. For the domain of convex polyhedra, the original widening operator proposed by Cousot and Halbwachs amply deserves the name of standard widening since most analysis and verification tools that employ convex polyhedra also employ that operator. Nonetheless, there is an unfulfilled demand for more precise widening operators. In this paper, after a formal introduction to the standard widening where we clarify some aspects that are often overlooked, we embark on the challenging task of improving on it. We present a framework for the systematic definition of new and precise widening operators for convex polyhedra. The framework is then instantiated so as to obtain a new widening operator that combines several heuristics and uses the standard widening as a last resort so that it is never less precise. A preliminary experimental evaluation has yielded promising results.

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Abstract Interpretation Frameworks

Cited by 536 publications

References 29 publications

Closed and Logical Relations for Over- and Under-Approximation of Powersets

Closed and Logical Relations for Over- and Under-Approximation of Powersets

Distortion-Free Authentication Watermarking.

Precise Widening Operators for Convex Polyhedra

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