A bidirectional transformation consists of a function get that takes a source (document or value) to a view and a function put that takes an updated view and the original source back to an updated source, governed by certain consistency conditions relating the two functions. Both the database and programming language communities have studied techniques that essentially allow a user to specify only one of get and put and have the other inferred automatically. All approaches so far to this bidirectionalization task have been syntactic in nature, either proposing a domain-specific language with limited expressiveness but built-in (and composable) backward components, or restricting get to a simple syntactic form from which some algorithm can synthesize an appropriate definition for put.Here we present a semantic approach instead. The idea is to take a general-purpose language, Haskell, and write a higher-order function that takes (polymorphic) get-functions as arguments and returns appropriate put-functions. All this on the level of semantic values, without being willing, or even able, to inspect the definition of get, and thus liberated from syntactic restraints. Our solution is inspired by relational parametricity and uses free theorems for proving the consistency conditions. It works beautifully.

Abstract-We provide a syntactic analysis of contextual preorder and equivalence for a polymorphic programming language with effects. Our approach applies uniformly across a range of algebraic effects, and incorporates, as instances: errors, input/output, global state, nondeterminism, probabilistic choice, and combinations thereof. Our approach is to extend Plotkin and Power's structural operational semantics for algebraic effects (FoSSaCS 2001) with a primitive "basic preorder" on ground type computation trees. The basic preorder is used to derive notions of contextual preorder and equivalence on program terms. Under mild assumptions on this relation, we prove fundamental properties of contextual preorder (hence equivalence) including extensionality properties and a characterisation via applicative contexts, and we provide machinery for reasoning about polymorphism using relational parametricity.

Parametric polymorphism constrains the behavior of pure functional programs in a way that allows the derivation of interesting theorems about them solely from their types, i.e., virtually for free. Unfortunately, the standard parametricity theorem fails for nonstrict languages supporting a polymorphic strict evaluation primitive like Haskell's seq. Contrary to the folklore surrounding seq and parametricity, we show that not even quantifying only over strict and bottom-reflecting relations in the ∀-clause of the underlying logical relation -and thus restricting the choice of functions with which such relations are instantiated to obtain free theorems to strict and total ones -is sufficient to recover from this failure. By addressing the subtle issues that arise when propagating up the type hierarchy restrictions imposed on a logical relation in order to accommodate the strictness primitive, we provide a parametricity theorem for the subset of Haskell corresponding to a Girard-Reynolds-style calculus with fixpoints, algebraic datatypes, and seq. A crucial ingredient of our approach is the use of an asymmetric logical relation, which leads to "inequational" versions of free theorems enriched by preconditions guaranteeing their validity in the described setting. Besides the potential to obtain corresponding preconditions for standard equational free theorems by combining some new inequational ones, the latter also have value in their own right, as is exemplified with a careful analysis of seq's impact on familiar program transformations.

Matsuda et al. [2007, ICFP] and Voigtländer [2009, POPL] introduced two techniques that given a source-to-view function provide an update propagation function mapping an original source and an updated view back to an updated source, subject to standard consistency conditions. Being fundamentally different in approach, both techniques have their respective strengths and weaknesses. Here we develop a synthesis of the two techniques to good effect. On the intersection of their applicability domains we achieve more than what a simple union of applying the techniques side by side delivers.

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Parametric polymorphism constrains the behavior of pure functional programs in a way that allows the derivation of interesting theorems about them solely from their types, i.e., virtually for free. The formal background of such 'free theorems' is well developed for extensions of the Girard-Reynolds polymorphic lambda calculus by algebraic datatypes and general recursion, provided the resulting calculus is endowed with either a purely strict or a purely nonstrict semantics. But modern functional languages like Clean and Haskell, while using nonstrict evaluation by default, also provide means to enforce strict evaluation of subcomputations at will. The resulting selective strictness gives the advanced programmer explicit control over evaluation order, but is not without semantic consequences: it breaks standard parametricity results. This paper develops an operational semantics for a core calculus supporting all the language features emphasized above. Its main achievement is the characterization of observational approximation with respect to this operational semantics via a carefully constructed logical relation. This establishes the formal basis for new parametricity results, as illustrated by several example applications, including the first complete correctness proof for short cut fusion in the presence of selective strictness. The focus on observational approximation, rather than equivalence, allows a finer-grained analysis of computational behavior in the presence of selective strictness than would be possible with observational equivalence alone.

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