Superposition with Datatypes and Codatatypes

Blanchette, Jasmin Christian; Peltier, Nicolas; Robillard, Simon

doi:10.1007/978-3-319-94205-6_25

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…Theorem 19 The following properties are equivalent: Proof To show that (iii) implies (i), assume that (Inf , Red ) is statically refutationally complete. That is, the property…”

Section: Lemma 18mentioning

confidence: 99%

“…Infinitary inference rules are also useful to reason about the theory of datatypes and codatatypes. Superposition with (co)datatypes [ 19 ] includes n -ary Acycl and Uniq rules, which had to be restricted and complemented with axioms so that they could be implemented in Vampire [ 27 ]. In Zipperposition, it would be possible to support the rules in full generality, eliminating the need for the axioms.…”

Section: Prover Architecturesmentioning

confidence: 99%

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A Comprehensive Framework for Saturation Theorem Proving

et al. 2022

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A crucial operation of saturation theorem provers is deletion of subsumed formulas. Designers of proof calculi, however, usually discuss this only informally, and the rare formal expositions tend to be clumsy. This is because the equivalence of dynamic and static refutational completeness holds only for derivations where all deleted formulas are redundant, but the standard notion of redundancy is too weak: A clause C does not make an instance $$C\sigma $$ C σ redundant. We present a framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution and superposition. The framework modularly extends redundancy criteria derived via a familiar ground-to-nonground lifting. It allows us to extend redundancy criteria so that they cover subsumption, and also to model entire prover architectures so that the static refutational completeness of a calculus immediately implies the dynamic refutational completeness of a prover implementing the calculus within, for instance, an Otter or DISCOUNT loop. Our framework is mechanized in Isabelle/HOL.

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“…Theorem 19 The following properties are equivalent: Proof To show that (iii) implies (i), assume that (Inf , Red ) is statically refutationally complete. That is, the property…”

Section: Lemma 18mentioning

confidence: 99%

Section: Prover Architecturesmentioning

confidence: 99%

A Comprehensive Framework for Saturation Theorem Proving

et al. 2022

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“…The Zipperposition prover [9] performs this enumeration in an extended DISCOUNT loop. Another instance of infinitary inferences is the n-ary Acycl and Uniq rules of superposition with (co)datatypes [14]. Abstractly, a Zipperposition loop prover ZL operates on states T | P | Y | A, where T is organized as a finite set of possibly infinite sequences (ι i ) i of inferences, and P, Y, A are as in DL.…”

Section: Delayedmentioning

confidence: 99%

A Comprehensive Framework for Saturation Theorem Proving

Waldmann

Tourret

Robillard

et al. 2020

Automated Reasoning

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We present a framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution or superposition. The framework relies on modular extensions of lifted redundancy criteria. It allows us to extend redundancy criteria so that they cover subsumption, and also to model entire prover architectures in such a way that the static refutational completeness of a calculus immediately implies the dynamic refutational completeness of a prover implementing the calculus, for instance within an Otter or DISCOUNT loop. Our framework is mechanized in Isabelle/HOL.

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“…The work that comes close to algorithmic proof search is the system CIRC [64], but it cannot handle general coinductive predicates and corecursive programming. Inductive and coinductive data types are also being added to SMT solvers [23,63]. However, both CIRC and SMT solving are inherently based on classical logic and are therefore not suited to situations where proof objects are relevant, like programming, type class inference or (dependent) type theory.…”

Section: Conclusion Related Work and The Futurementioning

confidence: 99%

Coinduction in Uniform: Foundations for Corecursive Proof Search with Horn Clauses

Basold

Komendantskaya

2019

Programming Languages and Systems

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We establish proof-theoretic, constructive and coalgebraic foundations for proof search in coinductive Horn clause theories. Operational semantics of coinductive Horn clause resolution is cast in terms of coinductive uniform proofs; its constructive content is exposed via soundness relative to an intuitionistic rst-order logic with recursion controlled by the later modality; and soundness of both proof systems is proven relative to a novel coalgebraic description of complete Herbrand models.We can also view Horn clauses coinductively. The greatest complete Herbrand model for a set P of Horn clauses is the largest set of nite and in nite ground atomic formulae coinductively entailed by P. For example, the greatest complete Herbrand model for the above two clauses is the set N ∞ = N ∪ {nat (s (s (· · · )))}, obtained by taking a backward closure of the above two inference rules on the set of all nite and in nite ground atomic formulae. The greatest Herbrand model is the largest set of nite ground atomic formulae coinductively entailed by P. In our example, it would be given by N already. Finally, one can also consider the least complete Hebrand model, which interprets entailment inductively but over potentially in nite terms. In the case of nat, this interpretation does not di er from N . However, nite paths in coinductive structures like transition systems, for example, require such semantics.The need for coinductive semantics of Horn clauses arises in several scenarios: the Horn clause theory may explicitely de ne a coinductive data structure or a coinductive relation. However, it may also happen that a Horn clause theory, which is not explicitly intended as coinductive, nevertheless gives rise to in nite inference by resolution and has an interesting coinductive model. This commonly happens in type inference. We will illustrate all these cases by means of examples.

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Superposition with Datatypes and Codatatypes

Cited by 7 publications

References 21 publications

A Comprehensive Framework for Saturation Theorem Proving

A Comprehensive Framework for Saturation Theorem Proving

A Comprehensive Framework for Saturation Theorem Proving

Coinduction in Uniform: Foundations for Corecursive Proof Search with Horn Clauses

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