Relational Parametricity and Separation Logic

Separation logic is a recent extension of Hoare logic for reasoning about programs with references to shared mutable data structures. In this paper, we provide a new interpretation of the logic for a programming language with higher types. Our interpretation is based on Reynolds's relational parametricity, and it provides a formal connection between separation logic and data abstraction

Proofs for free - parametricity for dependent types

Reynolds' abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We obtain a similar result for pure type systems: for any PTS used as a programming language, there is a PTS that can be used as a logic for parametricity. Types in the source PTS are translated to relations (expressed as types) in the target. Similarly, values of a given type are translated to proofs that the values satisfy the relational interpretation. We extend the result to inductive families. We also show that the assumption that every term satisfies the parametricity condition generated by its type is consistent with the generated logic

Two for the Price of One: Lifting Separation Logic Assertions

Recently, data abstraction has been studied in the context of separation logic, with noticeable practical successes: the developed logics have enabled clean proofs of tricky challenging programs, such as subject-observer patterns, and they have become the basis of efficient verification tools for Java (jStar), C (VeriFast) and Hoare Type Theory (Ynot). In this paper, we give a new semantic analysis of such logic-based approaches using Reynolds's relational parametricity. The core of the analysis is our lifting theorems, which give a sound and complete condition for when a true implication between assertions in the standard interpretation entails that the same implication holds in a relational interpretation. Using these theorems, we provide an algorithm for identifying abstraction-respecting client-side proofs; the proofs ensure that clients cannot distinguish two appropriately-related module implementations

Parametricity and Local Variables

We propose that the phenomenon of local state may be understood in terms of Strachey\u27s concept of parametric (i.e., uniform) polymorphism. The intuitive basis for our proposal is the following analogy: a non-local procedure is independent of locally-declared variables in the same way that a parametrically polymorphic function is independent of types to which it is instantiated. A connection between parametricity and representational abstraction was first suggested by J. C. Reynolds. Reynolds used logical relations to formalize this connection in languages with type variables and user-defined types. We use relational parametricity to construct a model for an Algol-like language in which interactions between local and non-local entities satisfy certain relational criteria. Reasoning about local variables essentially involves proving properties of polymorphic functions. The new model supports straightforward validations of all the test equivalences that have been proposed in the literature for local-variable semantics, and encompasses standard methods of reasoning about data representations. It is not known whether our techniques yield fully abstract semantics. A model based on partial equivalence relations on the natural numbers is also briefly examined

Step-Indexed Relational Reasoning for Countable Nondeterminism

Programming languages with countable nondeterministic choice are computationally interesting since countable nondeterminism arises when modeling fairness for concurrent systems. Because countable choice introduces non-continuous behaviour, it is well-known that developing semantic models for programming languages with countable nondeterminism is challenging. We present a step-indexed logical relations model of a higher-order functional programming language with countable nondeterminism and demonstrate how it can be used to reason about contextually defined may- and must-equivalence. In earlier step-indexed models, the indices have been drawn from {\omega}. Here the step-indexed relations for must-equivalence are indexed over an ordinal greater than {\omega}

Comprehensive parametric polymorphism : categorical models and type theory

This paper combines reflexive-graph-category structure for relational parametricity with fibrational models of impredicative polymorphism. To achieve this, we modify the definition of fibrational model of impredicative polymorphism by adding one further ingredient to the structure: comprehension in the sense of Lawvere. Our main result is that such comprehensive models, once further endowed with reflexive-graph-category structure, enjoy the expected consequences of parametricity. This is proved using a type-theoretic presentation of the category-theoretic structure, within which the desired consequences of parametricity are derived. The formalisation requires new techniques because equality relations are not available, and standard arguments that exploit equality need to be reworked

Bifibrational functorial semantics of parametric polymorphism

Reynolds' theory of parametric polymorphism captures the invariance of polymorphically typed programs under change of data representation. Semantically, reflexive graph categories and fibrations are both known to give a categorical understanding of parametric polymorphism. This paper contributes further to this categorical perspective by showing the relevance of bifibrations. We develop a bifibrational framework for models of System F that are parametric, in that they verify the Identity Extension Lemma and Reynolds' Abstraction Theorem. We also prove that our models satisfy expected properties, such as the existence of initial algebras and final coalgebras, and that parametricity implies dinaturality

Bidirectionalization for Free with Runtime Recording: Or, a Light-Weight Approach to the View-Update Problem

A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. Over the years, a lot of effort has been made to offer better language support for programming such transformations. In particular, a technique known as bidirectionalization is able to analyze and transform unidirectional programs written in general purpose languages, and "bidirectionalize" them. Among others, a technique termed as semantic bidirectionalization proposed by Voigtländer stands out in term of user-friendliness. The unidirectional program can be written using arbitrary language constructs, as long as the function is polymorphic and the language constructs respect parametricity. The free theorems that follow from the polymorphic type of the program allow a kind of forensic examination of the transformation, determining its effect without examining its implementation. This is convenient, in the sense that the programmer is not restricted to using a particular syntax; but it does require the transformation to be polymorphic. In this paper, we lift this polymorphism requirement to improve the applicability of semantic bidirectionalization. Concretely, we provide a type class PackM γ α μ, which intuitively reads "a concrete datatype γ is abstracted to a type α, and the 'observations' made by a transformation on values of type γ are recorded by a monad μ". With PackM, we turn monomorphic transformations into polymorphic ones, that are ready to be bidirectionalized. We demonstrate our technique with a case study of standard XML queries, which were considered beyond semantic bidirectionalization because of their monomorphic nature

A Computational Interpretation of Parametricity

Reynolds' abstraction theorem has recently been extended to lambda-calculi with dependent types. In this paper, we show how this theorem can be internalized. More precisely, we describe an extension of Pure Type Systems with a special parametricity rule (with computational content), and prove fundamental properties such as Church-Rosser's and strong normalization. All instances of the abstraction theorem can be both expressed and proved in the calculus itself. Moreover, one can apply parametricity to the parametricity rule: parametricity is itself parametric