SyTeCi: automating contextual equivalence for higher-order programs with references

Jaber, Guilhem

doi:10.1145/3371127

“…It should be observed that such an equivalence check is less problematic for first-order programs than it is for higher-order one (the authors of [18] are able to actually check observational equivalence through an SMT solver). On the other hand, various notions of equivalence which are included in contextual equivalence and sometimes coincide with it (e.g., applicative bisimilarity, denotational semantics, or logical relations themselves) have been developed for higher-order languages, and this starts to give rise to actual automatic tools for deciding contextual equivalence [38]. We give in Figure 5 the typing rule for conditionals.…”

Section: Typing Conditionalsmentioning

confidence: 99%

On the Versatility of Open Logical Relations: Continuity, Automatic Differentiation, and a Containment Theorem

Barthe¹,

Crubillé²,

Lago³

et al. 2021

Preprint

0

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Logical relations are one among the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on terms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called open logical relation, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed λ-calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any collection of real-valued firstorder functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.

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“…We believe our work opens up the way to automated reasoning about contextual equivalence for all four languages in a common framework, following the approach proposed in [18].…”

Section: Related Work and Conclusionmentioning

confidence: 99%

“…We call the technique Kripke Normal-Form Bisimulations (KNFB), because it combines the flavour of open/normal-form bisimulations [17], [8] with Kripke-style relational reasoning [16]. An important feature of KNFBs is that universal quantifications range over relatively simple first-order entities, which opens up the way to automated reasoning in each of the four cases, following [18]. By virtue of the correspondence with operational game semantics, the technique is sound and complete.…”

Section: Introductionmentioning

confidence: 99%

Compositional relational reasoning via operational game semantics

Jaber

¹

,

Murawski

²

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We show how to use operational game semantics as a guide to develop relational techniques for establishing contextual equivalences with respect to contexts drawn from a hierarchy of four call-by-value higher-order languages: with either general or ground-type references and with either call/cc or no control operator. In game semantics, differences between the contexts can be captured by the absence or presence of the O-visibility and O-bracketing conditions.The proposed technique, which we call Kripke normal-form bisimulations, combines insights from normal-form bisimulation and Kripke logical relations with game semantics. In particular, the role of the heap and the name history is abstracted away using Kripke-style world transition systems. The differences between the four kinds of contexts manifest themselves through simple local conditions that can be shown to correspond to O-visibility and O-bracketing, as applicable.The technique is sound and complete by virtue of correspondence with operational game semantics. Moreover, it sheds a new light on other related developments, such as backtracking and private transitions in Kripke logical relations, which can be related to specific phenomena in game models.

show abstract

“…It should be observed that such an equivalence check is less problematic for first-order programs than it is for higher-order one (the authors of [18] are able to actually check observational equivalence through an SMT solver). On the other hand, various notions of equivalence which are included in contextual equivalence and sometimes coincide with it (e.g., applicative bisimilarity, denotational semantics, or logical relations themselves) have been developed for higher-order languages, and this starts to give rise to actual automatic tools for deciding contextual equivalence [38]. We give in Figure 5 the typing rule for conditionals.…”

Section: Typing Conditionalsmentioning

confidence: 99%

On the Versatility of Open Logical Relations

Barthe

¹

,

Crubillé

²

,

Lago

³

et al. 2020

Lecture Notes in Computer Science

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Logical relations are one among the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on terms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called open logical relation, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed λ-calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any collection of real-valued firstorder functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.

show abstract

SyTeCi: automating contextual equivalence for higher-order programs with references

Cited by 17 publications

References 39 publications

On the Versatility of Open Logical Relations: Continuity, Automatic Differentiation, and a Containment Theorem

On the Versatility of Open Logical Relations: Continuity, Automatic Differentiation, and a Containment Theorem

Compositional relational reasoning via operational game semantics

On the Versatility of Open Logical Relations

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