Coaxioms: flexible coinductive definitions by inference systems

Dagnino, Francesco

doi:10.23638/lmcs-15(1:26)2019

“…In some cases, the desired set of derivable judgments is an intermediate fixed point of the inference operator other than the least/greatest one. Generalized inference systems [Ancona et al 2017;Dagnino 2019] allow for the characterization of (some) intermediate fixed points. Specifically, a generalized inference system is a pair (I, I co ) of inference systems whose interpretation is the set of judgments having an arbitrary (finite-or infinite-depth) derivation tree using the rules in I but such that all the judgments in this derivation tree also have a finite-depth derivation tree using the rules in I ∪ I co .…”

Section: Generalized Inference Systems In a Nutshellmentioning

confidence: 99%

“…We will use generalized inference systems to provide compact definitions of fair subtyping (Section 3.2) and of the typing rules (Section 6). The reader interested in the metatheory of generalized inference systems may refer to Ancona et al [2017] and Dagnino [2019].…”

Section: Generalized Inference Systems In a Nutshellmentioning

confidence: 99%

Fair termination of binary sessions

Ciccone

¹

,

Padovani

²

2022

Proc. ACM Program. Lang.

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A binary session is a private communication channel that connects two processes, each adhering to a protocol description called session type . In this work, we study the first type system that ensures the fair termination of binary sessions. A session fairly terminates if all of the infinite executions admitted by its protocol are deemed unrealistic because they violate certain fairness assumptions . Fair termination entails the eventual completion of all pending input/output actions, including those that depend on the completion of an unbounded number of other actions in possibly different sessions. This form of lock freedom allows us to address a large family of natural communication patterns that fall outside the scope of existing type systems. Our type system is also the first to adopt fair subtyping , a liveness-preserving refinement of the standard subtyping relation for session types that so far has only been studied theoretically. Fair subtyping is surprisingly subtle not only to characterize concisely but also to use appropriately, to the point that the type system must carefully account for all usages of fair subtyping to avoid compromising its liveness-preserving properties.

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“…We give a formal account of proof trees, which is useful to state and to prove following technical results. The account follows [18,20], but it is adjusted to our specific setting.…”

Section: Properties Of Proof Treesmentioning

confidence: 99%

“…Divergence propagation rules are very similar to those used in [8,9] to define a big-step judgment which directly includes divergence as result. However, this direct definition relies on a non-standard notion of inference system, allowing corules [7,20], whereas for the trace semantics presented in this work standard coinduction is enough, since all rules are productive, that is, they always add an element to the trace.…”

Section: Equivalence Of Traces Wrong and Pev Semanticsmentioning

confidence: 99%

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Soundness conditions for big-step semantics

Dagnino¹,

Bono²,

Zucca³

et al. 2020

Preprint

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0

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We propose a general proof technique to show that a predicate is sound, that is, prevents stuck computation, with respect to a big-step semantics. This result may look surprising, since in big-step semantics there is no difference between non-terminating and stuck computations, hence soundness cannot even be expressed. The key idea is to define constructions yielding an extended version of a given arbitrary big-step semantics, where the difference is made explicit. The extended semantics are exploited in the meta-theory, notably they are necessary to show that the proof technique works. However, they remain transparent when using the proof technique, since it consists in checking three conditions on the original rules only, as we illustrate by several examples.2. We provide a general proof technique by identifying three sufficient conditions on the original big-step rules to prove soundness.Keypoint (2)'s three sufficient conditions are local preservation, ∃-progress, and ∀-progress. For proving the result that the three conditions actually ensure soundness, the setting up of the extended semantics from the given one is necessary, since otherwise, as said above, we could not even express the property.However, the three conditions deal only with the original rules of the given big-step semantics. This means that, practically, in order to use the technique there is no need to deal with the extended semantics. This implies, in particular, that our approach does not increase the original number of rules. Moreover, the sufficient conditions are checked only on single rules, which makes explicit the proof fragments typically needed in a proof of soundness. Even though this is not exploited in this paper, this form of locality means modularity, in the sense that adding a new rule implies adding the corresponding proof fragment only.As an important by-product, in order to formally define and prove correct the keypoints (1) and ( 2), we propose a formalisation of "what is a big-step semantics" which captures its essential features. Moreover, we support our approach by presenting several examples, demonstrating that: on the one hand, their soundness proof can be easily rephrased in terms of our technique, that is, by directly reasoning on big-step rules; on the other hand, our technique is essential when the property to be checked (for instance, the soundness of a type system) is not preserved by intermediate computation steps, whereas it holds for the final result. On a side note, our examples concern type systems, but the meta-theory we present in this work holds for any predicate.We describe now in more detail the constructions of keypoint (1). Starting from an arbitrary big-step judgment c ⇒ r that evaluates configurations c into results r , the first construction produces an enriched judgement c ⇒ tr t where t is a trace, that is, the (finite or infinite) sequence of all the (sub)configurations encountered during the evaluation. In this way, by interpreting coinductively the rules of the extended semantics, an infinit...

show abstract

Soundness Conditions for Big-Step Semantics

Dagnino

¹

,

Bono

²

,

Zucca

³

et al. 2020

Programming Languages and Systems

Self Cite

View full text Add to dashboard Cite

We propose a general proof technique to show that a predicate is sound, that is, prevents stuck computation, with respect to a big-step semantics. This result may look surprising, since in big-step semantics there is no difference between non-terminating and stuck computations, hence soundness cannot even be expressed. The key idea is to define constructions yielding an extended version of a given arbitrary big-step semantics, where the difference is made explicit. The extended semantics are exploited in the meta-theory, notably they are necessary to show that the proof technique works. However, they remain transparent when using the proof technique, since it consists in checking three conditions on the original rules only, as we illustrate by several examples.

show abstract

Coaxioms: flexible coinductive definitions by inference systems

Cited by 9 publications

References 50 publications

Fair termination of binary sessions

Fair termination of binary sessions

Soundness conditions for big-step semantics

Soundness Conditions for Big-Step Semantics

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