Proof Relevant Corecursive Resolution

Fu, Peng; Komendantskaya, Ekaterina; Schrijvers, Tom; Pond, Andrew

doi:10.1007/978-3-319-29604-3_9

Cited by 20 publications

(58 citation statements)

References 15 publications

(30 reference statements)

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“…It was pointed out in [35] that Haskell's type class inference can also give rise to irregular corecursion. Such cases may require the more general coinduction hypothesis (e.g.…”

Section: De Nition 14mentioning

confidence: 99%

“…Horn Clause Theories in Type Class Inference Haskell type class inference does not require circular uni ers but may require a cyclic resolution inference [51,35]. Consider, for example, the following mutually de ned data structures in Haskell.…”

Section: Introductionmentioning

confidence: 99%

“…Examples like γ 1 appear in Haskell type class resolution [51], and examples like γ 2 in its experimental extensions [35]. Cycle detection would only cover computations for γ 1 , whereas γ 2 , γ 3 require some form of loop detection 1 .…”

Section: Introductionmentioning

confidence: 99%

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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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“…It was pointed out in [35] that Haskell's type class inference can also give rise to irregular corecursion. Such cases may require the more general coinduction hypothesis (e.g.…”

Section: De Nition 14mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Basold

Komendantskaya

2019

Programming Languages and Systems

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“…Corecursive Resolution Fu et al [6] address the divergence problem that arises for generic nested datatypes. They turn the diverging resolution with user-supplied instances into a terminating resolution in terms of automatically derived instances.…”

Section: Related Workmentioning

confidence: 99%

“…6 Implementation Our prototype implementation is available at https://github.com/gkaracha/quantcs-impl. It incorporates higherkinded datatypes and performs type inference, elaboration into System F (as explained in the next section), and type checking of the generated code.…”

Section: Constraint Solving Algorithmmentioning

confidence: 99%

Quantified class constraints

et al. 2017

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Quantified class constraints have been proposed many years ago to raise the expressive power of type classes from Horn clauses to the universal fragment of Hereditiary Harrop logic. Yet, while it has been much asked for over the years, the feature was never implemented or studied in depth. Instead, several workarounds have been proposed, all of which are ultimately stopgap measures.This paper revisits the idea of quantified class constraints and elaborates it into a practical language design. We show the merit of quantified class constraints in terms of more expressive modeling and in terms of terminating type class resolution. In addition, we provide a declarative specification of the type system as well as a type inference algorithm that elaborates into System F. Moreover, we discuss termination conditions of our system and also provide a prototype implementation.

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Proof-Carrying Plans

Schwaab

Komendantskaya²,

Hill³

et al. 2018

Practical Aspects of Declarative Languages

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As AI applications are deployed in more and more real-world settings, it is becoming increasingly important to verify their safety and security. While declarative languages (of the kind found in automated planners and model checkers) are traditionally used for verifying AI systems, a big challenge is to design methods that generate verified executable programs. A good example of such a "verification to implementation" cycle is given by automated planning languages like PDDL, where plans are found via a model search in a declarative language, but then interpreted or compiled into executable code in an imperative language. In this paper, we show that this method can itself be verified. We present a formal framework and a prototype Agda implementation that represent PDDL plans as executable functions that inhabit types that are given by formulae describing planning problems. By exploiting the well-known Curry-Howard correspondence, type-checking then automatically ensures that the generated program corresponds precisely to the specification of the planning problem.

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Proof Relevant Corecursive Resolution

Cited by 20 publications

References 15 publications

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

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

Quantified class constraints

Proof-Carrying Plans

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