Nicolas Wu

Incremental updates for efficient bidirectional transformations

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. The key to handling large data objects that are subject to relatively small modifications is to process the updates incrementally. Incrementality has been explored in the semi-structured settings of relational databases and graph transformations; this flexibility in structure makes it relatively easy to divide the data into separate parts that can be transformed and updated independently. The same is not true if the data is to be encoded with more general-purpose algebraic datatypes, with transformations defined as functions: dividing data into well-typed separate parts is tricky, and recursions typically create interdependencies. In this paper, we study transformations that support incremental updates, and devise a constructive process to achieve this incrementality.

Folding domain-specific languages: deep and shallow embeddings (functional Pearl)

A domain-specific language can be implemented by embedding within a general-purpose host language. This embedding may be deep or shallow, depending on whether terms in the language construct syntactic or semantic representations. The deep and shallow styles are closely related, and intimately connected to folds; in this paper, we explore that connection.

Effect handlers in scope

Algebraic effect handlers are a powerful means for describing effectful computations. They provide a lightweight and orthogonal technique to define and compose the syntax and semantics of different effects. The semantics is captured by handlers, which are functions that transform syntax trees. Unfortunately, the approach does not support syntax for scoping constructs, which arise in a number of scenarios. While handlers can be used to provide a limited form of scope, we demonstrate that this approach constrains the possible interactions of effects and rules out some desired semantics. This paper presents two different ways to capture scoped constructs in syntax, and shows how to achieve different semantics by reordering handlers. The first approach expresses scopes using the existing algebraic handlers framework, but has some limitations. The problem is fully solved in the second approach where we introduce higher-order syntax.

Fusion for Free

Algebraic effect handlers are a recently popular approach for modelling side-effects that separates the syntax and semantics of effectful operations. The shape of syntax is captured by functors, and free monads over these functors denote syntax trees. The semantics is captured by algebras, and effect handlers pass these over the syntax trees to interpret them into a semantic domain.

Heuristics Entwined with Handlers Combined: From Functional Specification to Logic Programming Implementation

A long-standing problem in logic programming is how to cleanly separate logic and control. While solutions exist, they fall short in one of two ways: some are too intrusive, because they require significant changes to Prologs underlying implementation; others are lacking a clean semantic grounding. We resolve both of these issues in this paper. We derive a solution that is both lightweight and principled. We do so by starting from a functional specification of Prolog based on monads, and extend this with the effect handlers approach to capture the dynamic search tree as syntax. Effect handlers then express heuristics in terms of tree transformations. Moreover, we can declaratively express many heuristics as trees themselves that are combined with search problems using a generic entwining handler. Our solution is not restricted to a functional model: we show how to implement this technique as a library in Prolog by means of delimited continuations.

Conjugate Hylomorphisms -- Or: The Mother of All Structured Recursion Schemes

The past decades have witnessed an extensive study of structured recursion schemes. A general scheme is the hylomorphism, which captures the essence of divide-and-conquer: a problem is broken into sub-problems by a coalgebra; sub-problems are solved recursively; the sub-solutions are combined by an algebra to form a solution. In this paper we develop a simple toolbox for assembling recursive coalgebras, which by definition ensure that their hylo equations have unique solutions, whatever the algebra. Our main tool is the conjugate rule, a generic rule parametrized by an adjunction and a conjugate pair of natural transformations. We show that many basic adjunctions induce useful recursion schemes. In fact, almost every structured recursion scheme seems to arise as an instance of the conjugate rule. Further, we adapt our toolbox to the more expressive setting of parametrically recursive coalgebras, where the original input is also passed to the algebra. The formal development is complemented by a series of worked-out examples in Haskell.

Histo- and dynamorphisms revisited

Dynamic programming algorithms embody a widely used programming technique that optimizes recursively defined equations that have repeating subproblems. The standard solution uses arrays to share common results between successive steps, and while effective, this fails to exploit the structural properties present in these problems. Histomorphisms and dynamorphisms have been introduced to expresses such algorithms in terms of structured recursion schemes that leverage this structure. In this paper, we revisit and relate these schemes and show how they can be expressed in terms of recursion schemes from comonads, as well as from recursive coalgebras. Our constructions rely on properties of bialgebras and dicoalgebras, and we are careful to consider optimizations and efficiency concerns. Throughout the paper we illustrate these techniques through several worked-out examples discussed in a tutorial style, and show how a recursive specification can be expressed both as an array-based algorithm as well as one that uses recursion schemes.

Modules over Monads and their Algebras

Modules over monads (or: actions of monads on endofunctors) are structures in which a monad interacts with an endofunctor, composed either on the left or on the right. Although usually not explicitly identified as such, modules appear in many contexts in programming and semantics. In this paper, we investigate the elementary theory of modules. In particular, we identify the monad freely generated by a right module as a generalisation of Moggi’s resumption monad and characterise its algebras, extending previous results by Hyland, Plotkin and Power, and by Filinski and Stovring. Moreover, we discuss a connection between modules and algebraic eects: left modules have a similar feeling to Eilenberg‐Moore algebras, and can be seen as handlers that are natural in the variables, while right modules can be seen as functions that run eectful computations in an appropriate context (such as an initial state for a stateful computation). 1998 ACM Subject Classification F.3.2 Semantics of Programming Languages, F.3.3 Studies of Program Constructs

Relational Algebra by Way of Adjunctions

Bulk types such as sets, bags, and lists are monads, and therefore support a notation for database queries based on comprehensions. This fact is the basis of much work on database query languages. The monadic structure easily explains most of standard relational algebra---specifically, selections and projections---allowing for an elegant mathematical foundation for those aspects of database query language design. Most, but not all: monads do not immediately offer an explanation of relational join or grouping, and hence important foundations for those crucial aspects of relational algebra are missing. The best they can offer is cartesian product followed by selection. Adjunctions come to the rescue: like any monad, bulk types also arise from certain adjunctions; we show that by paying due attention to other important adjunctions, we can elegantly explain the rest of standard relational algebra. In particular, graded monads provide a mathematical foundation for indexing and grouping, which leads directly to an efficient implementation, even of joins.

Towards formally templated relational database representations in z

Many authors have drawn parallels between the relational model of data and the formal description technique Z, yet none of these contributions have managed to be both close to the relational model in terms of providing a practical means of database design and fully formal in terms of providing an appropriate metamodel. We compare these various formalisms, and suggest how the use of the formal template approach of Amalio et al might help to overcome some of the issues faced. We demonstrate the application of this work via a short case study, and suggest further enhancements to the template language.