Propositional Interpolation and Abstract Interpretation

D’Silva, Vijay

doi:10.1007/978-3-642-11957-6_11

Cited by 9 publications

(4 citation statements)

References 23 publications

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“…Therefore, we resort to the number of variables as a measure of interpolant size. Labelled interpolation systems support the elimination of nonessential (or peripheral [29]) variables from interpolants [10], as stated by the following lemma.…”

Section: Definition 7 (Labelled Interpolation System For Resolution)mentioning

confidence: 99%

“…The following example demonstrates that this approach may in fact have the opposite effect. We obtain the proof P on the right of Figure 6 by applying RmPivots and ReconstructProof to R. P is smaller than R, but the substitution has eliminated a peripheral resolution step and Itp(L, P ) is forced to introduce x 1 when we resolve on [12], and according to Lemma 2, any labelling L would require Itp(L, P ) to introduce x 1 at some point in P [10].…”

Section: Interpolant Reduction Via Subsumptionmentioning

confidence: 99%

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Reduction of Resolution Refutations and Interpolants via Subsumption

Bloem

Malik

Schlaipfer

et al. 2014

Hardware and Software: Verification and Testing

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Propositional resolution proofs and interpolants derived from them are widely used in automated verification and circuit synthesis. There is a broad consensus that "small is beautiful"-small proofs and interpolants lead to concise abstractions in verification and compact designs in synthesis. Contemporary proof reduction techniques either minimise the proof during construction, or perform a post-hoc transformation of a given resolution proof. We focus on the latter class and present a subsumption-based proof reduction algorithm that extends existing single-pass analyses and relies on a meet-overall paths analysis to identify redundant resolution steps and clauses. We show that smaller refutations do not necessarily entail smaller interpolants, and use labelled interpolation systems to generalise our reduction approach to interpolants. Experimental results support the theoretical claims. 1 Introduction Resolution proofs and interpolants are an integral part of many verificationrelated techniques such as abstraction [24] and model checking [17], vacuity detection [29], synthesis [18, 20], and patch generation [32]. These techniques take advantage of the fact that refutations and interpolants direct the focus to the core of the problem instance (literally and metaphorically). In practice, small refutations provide concise abstractions in model checking [24], and small interpolants enable precise refinement and compact designs in synthesis [20]. Consequently, proof reduction as well as the minimisation of unsatisfiable cores has received ample attention. We roughly group the resulting reduction approaches into two categories: techniques that minimise the proof during construction, and techniques that rely on a post-hoc proof transformation. Algorithms for the extraction of minimal unsatisfiable subsets (such as [25, 4]) typically fall into the former category and rely on iterative calls to a SAT solver.

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Section: Definition 7 (Labelled Interpolation System For Resolution)mentioning

confidence: 99%

Section: Interpolant Reduction Via Subsumptionmentioning

confidence: 99%

Reduction of Resolution Refutations and Interpolants via Subsumption

Bloem

Malik

Schlaipfer

et al. 2014

Hardware and Software: Verification and Testing

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“…The interpolant associated with the final resolvent (the empty clause) constitutes an interpolant I for which Theorem 1 holds. For a characterization of these and other interpolation systems see, e.g., [13].…”

Section: Introductionmentioning

confidence: 99%

“…For every propositional interpolation system that computes an interpolant for φ ⇒ ψ, the dual system is defined by the computation of an interpolant for I for ψ ⇒ φ , with the effect that ¬I is an interpolant for φ ⇒ ψ. It is known that the HKP system is self-dual and that McMillan's system is not [13].…”

Section: Introductionmentioning

confidence: 99%

Lazy Decomposition for Distributed Decision Procedures

Hamadi

Marques-Silva

Wintersteiger

2011

Electron. Proc. Theor. Comput. Sci.

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The increasing popularity of automated tools for software and hardware verification puts ever increasing demands on the underlying decision procedures. This paper presents a framework for distributed decision procedures (for first-order problems) based on Craig interpolation. Formulas are distributed in a lazy fashion, i.e., without the use of costly decomposition algorithms. Potential models which are shown to be incorrect are reconciled through the use of Craig interpolants. Experimental results on challenging propositional satisfiability problems indicate that our method is able to outperform traditional solving techniques even without the use of additional resources.

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