On Applicative Similarity, Sequentiality, and Full Abstraction

Crubillé, Raphaëlle; Lago, Ugo Dal; Sangiorgi, Davide; Vignudelli, Valeria

doi:10.1007/978-3-319-23506-6_7

Cited by 9 publications

(9 citation statements)

References 22 publications

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“…Even in cases where applicative bisimilarity is fully abstract for contextual equivalence, the corresponding simulation may not be fully abstract for the contextual preorder. Pure call-by-value is such an example [6,7]. In contrast, in all calculi in the paper, full abstraction for environmental bisimilarity carries over to the corresponding simulation, with a similar proof.…”

Section: Additional Related Workmentioning

confidence: 90%

Environmental bisimulations for probabilistic higher-order languages

Sangiorgi

Vignudelli

2016

SIGPLAN Not.

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Environmental bisimulations for probabilistic higher-order languages are studied. In contrast with applicative bisimulations, environmental bisimulations are known to be more robust and do not require sophisticated techniques such as Howe’s in the proofs of congruence. As representative calculi, call-by-name and call-by-value λ- calculus, and a (call-by-value) λ-calculus extended with references (i.e., a store) are considered. In each case full abstraction results are derived for probabilistic environmental similarity and bisimilarity with respect to contextual preorder and contextual equivalence, respectively. Some possible enhancements of the (bi)simulations, as ‘up-to techniques’, are also presented. Probabilities force a number of modifications to the definition of environmental bisimulations in non-probabilistic languages. Some of these modifications are specific to probabilities, others may be seen as general refinements of environmental bisimulations, applicable also to non-probabilistic languages. Several examples are presented, to illustrate the modifications and the differences.

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Section: Additional Related Workmentioning

confidence: 90%

Environmental bisimulations for probabilistic higher-order languages

Sangiorgi

Vignudelli

2016

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“…We first present the syntax and the operational semantics of the probabilistic λ-calculus Λ ⊕ , on top of which we shall consider the contextual equivalence and the contextual preorder relations. Then, we recall Larsen and Skou's probabilistic (bi)similarity on labelled Markov chains [18] and, in the spirit of Abramsky's work on applicative (bi)similarity [1] and following [6,7,15,16], we apply it to the operational semantics of Λ ⊕ , getting the probabilistic applicative (bi)similarity.…”

Section: Preliminariesmentioning

confidence: 99%

“…This approach has been lifted to the probabilistic λ-calculus in a series of works by Dal Lago et al [6,7,16], introducing the notion of probabilistic applicative bisimilarity (PAB) for lazy semantics. In particular, PAB is proven to be sound with the contextual equivalence in both cbv and cbn, but only cbv PAB is fully abstract.…”

Section: Introductionmentioning

confidence: 99%

“…On a more technical side, we stress that our proofs of soundness and completeness follow a different reasoning than the one used in probabilistic lazy semantics [6,7,15]. First, the soundness (∼ ⊆ = cxt ) does not need an Howe lifting [14], as we prove a Context Lemma (Lemma 9) for = cxt and an applicative property of ∼ (Lemma 15), the latter using the notion of probabilistic assignments as in [16].…”

Section: Introductionmentioning

confidence: 99%

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The Benefit of Being Non-Lazy in Probabilistic λ-calculus

Curzi

Pagani

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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We consider the probabilistic applicative bisimilarity (PAB)a coinductive relation comparing the applicative behaviour of probabilistic untyped λ-terms according to a specific operational semantics. This notion has been studied by Dal Lago et al. with respect to the two standard parameter passing policies, call-by-value (cbv) and call-by-name (cbn), using a lazy reduction strategy not reducing within the body of a function. In particular, PAB has been proven to be fully abstract with respect to the contextual equivalence in cbv [6] but not in lazy cbn [16].We overcome this issue of cbn by relaxing the laziness constraint: we prove that PAB is fully abstract with respect to the standard head reduction contextual equivalence. Our proof is based on Leventis' Separation Theorem [19], using probabilistic Nakajima trees as a tree-like representation of the contextual equivalence classes.Finally, we prove also that the inequality full abstraction fails, showing that the probabilistic applicative similarity is strictly contained in the contextual preorder.CCS Concepts: • Software and its engineering → Semantics; • Theory of computation → Program semantics.

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“…We add a parallel disjunction construct to the calculus, being inspired by Plotkin's parallel disjunction [29]. Noticeably, it has been recently shown [8] that adding parallel disjunction to a (non-linear) λ-calculus increases the expressive power of contexts to the point of enabling coincidence between the contextual preorder and applicative similarity. The choice of studying a very general calculus is motivated by our desire to be as general as possible.…”

Section: Introductionmentioning

confidence: 99%

Metric Reasoning About $$\lambda $$-Terms: The General Case

Crubillé

Lago

2017

Programming Languages and Systems

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In any setting in which observable properties have a quantitative flavour, it is natural to compare computational objects by way of metrics rather than equivalences or partial orders. This holds, in particular, for probabilistic higher-order programs. A natural notion of comparison, then, becomes context distance, the metric analogue of Morris' context equivalence. In this paper, we analyze the main properties of the context distance in fully-fledged probabilistic λ-calculi, this way going beyond the state of the art, in which only affine calculi were considered. We first of all study to which extent the context distance trivializes, giving a sufficient condition for trivialization. We then characterize context distance by way of a coinductively defined, tuple-based notion of distance in one of those calculi, called Λ ⊕ ! . We finally derive pseudometrics for call-by-name and call-by-value probabilistic λ-calculi, and prove them fully-abstract.The easiest way to render the pure λ-calculus a universal probabilistic computation model [10] consists in endowing it with a binary construct ⊕ for probabilistic choice. The term M ⊕ N evolves as either M or N , each with probability 1 2 . The obtained calculus can be given meaning by an operational semantics which puts terms in correspondence with distributions of values. The natural notion of observation, at least in an untyped setting like the one we will consider in this paper, is thus the probability of convergence of the observed term M , which will be denoted as M . One could then define a notion of context equivalence following Morris' pattern, and stipulate that two terms M and N should be equivalent whenever they terminate with exactly the same probability when put in any context:

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On Applicative Similarity, Sequentiality, and Full Abstraction

Cited by 9 publications

References 22 publications

Environmental bisimulations for probabilistic higher-order languages

Environmental bisimulations for probabilistic higher-order languages

The Benefit of Being Non-Lazy in Probabilistic λ-calculus

Metric Reasoning About $$\lambda $$-Terms: The General Case

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