In many occasions would one encounter the task of maintaining the consistency of two pieces of structured data that are related by some transform — synchronising bookmarks in different web browsers, the source and the view in an editor, or views in databases, to name a few. This paper proposes a formal model of such tasks, basing on a programming language allowing injective functions only. The programmer designs the transformation as if she is writing a functional program, while the synchronisation behaviour is automatically derived by algebraic reasoning. The main advantage is being able to deal with duplication and structural changes. The result will be integrated to our structure XML editor in the Programmable Structured Document project.
Bidirectional Programming Languages
The need to edit source data through a view arises in a host of applications across many different areas of computing. Unfortunately, few existing systems provide support for updatable views. In practice, when they are needed, updatable views are usually implemented using two separate programs: one that computes the view from the source and another that handles updates. This rudimentary design is tedious for programmers, difficult to reason about, and a nightmare to maintain. This dissertation presents bidirectional programming languages, which provide an elegant and effective mechanism for describing updatable views. Unlike programs written in an ordinary language, which only work in one direction, programs in a bidirectional language can be run both forwards and backwards: from left to right, they describe functions that map sources to views, and from right to left, they describe functions that map updated views back to updated sources. Besides eliminating redundancy, these languages can be designed to ensure correctness, guaranteeing by construction that the two functions work well together. Starting from the foundations, we define a general semantic space of well-behaved bidirectional transformations called lenses. Then, building on this foundation, we describe a particular language for defining lenses on strings with syntax based on the familiar regular operators (union, concatenation, and Kleene star). We present extensions to the basic framework that address the subtle complications that arise when lenses are used to manipulate, data containing unimportant details, ordered data, and confidential data.
A Categorical Theory of Patches
When working with distant collaborators on the same documents, one often uses a version control system, which is a program tracking the history of files and helping importing modifications brought by others as patches. The implementation of such a system requires to handle lots of situations depending on the operations performed by users on files, and it is thus difficult to ensure that all the corner cases have been correctly addressed. Here, instead of verifying the implementation of such a system, we adopt a complementary approach: we introduce a theoretical model, which is defined abstractly by the universal property that it should satisfy, and work out a concrete description of it. We begin by defining a category of files and patches, where the operation of merging the effect of two coinitial patches is defined by pushout. Since two patches can be incompatible, such a pushout does not necessarily exist in the category, which raises the question of which is the correct category to represent and manipulate files in conflicting state. We provide an answer by investigating the free completion of the category of files under finite colimits, and give an explicit description of this category: its objects are finite sets labeled by lines equipped with a transitive relation and morphisms are partial functions respecting labeling and relations.
Combinators for bidirectional tree transformations: A linguistic approach to the view-update problem: ACM Transactions on Programming Languages and Systems: Vol 29, No 3
We propose a novel approach to the view-update problem for tree-structured data: a domain-specific programming language in which all expressions denote bidirectional transformations on trees. In one direction, these transformations---dubbed lenses---map ...
A Conflict-Free Replicated JSON Datatype
Many applications model their data in a general-purpose storage format such as JSON. This data structure is modified by the application as a result of user input. Such modifications are well understood if performed sequentially on a single copy of the data, but if the data is replicated and modified concurrently on multiple devices, it is unclear what the semantics should be. In this paper we present an algorithm and formal semantics for a JSON data structure that automatically resolves concurrent modifications such that no updates are lost, and such that all replicas converge towards the same state (a conflict-free replicated datatype or CRDT). It supports arbitrarily nested list and map types, which can be modified by insertion, deletion and assignment. The algorithm performs all merging client-side and does not depend on ordering guarantees from the network, making it suitable for deployment on mobile devices with poor network connectivity, in peer-to-peer networks, and in messaging systems with end-to-end encryption.
MICROCOSM: An Open Hypermedia Environment for Information Integration.
This report examines open hypermedia systems, and argues that such systems provide users with richer and more diverse ways to access and integrate information from large and dynamic data sets in a distributed, heterogeneous environment. In particular, the enhanced Microcosm model for open hypermedia is examined, and the ways in which it provides such an environment are discussed. The paper continues by investigating the advantages and the short comings of this model and identifies areas in which further work must be completed before such systems can become widely adopted, in particular the granularity of link anchors, editing, and version control. Possible solutions to these problems are presented and discussed.
Edit lenses
A lens is a bidirectional transformation between a pair of connected data structures, capable of translating an edit on one structure into an appropriate edit on the other. Many varieties of lenses have been studied, but none, to date, has offered a satisfactory treatment of how edits are represented. Many foundational accounts only consider edits of the form "overwrite the whole structure," leading to poor behavior in many situations by failing to track the associations between corresponding parts of the structures when elements are inserted and deleted in ordered lists, for example. Other theories of lenses do maintain these associations, either by annotating the structures themselves with change information or using auxiliary data structures, but every extant theory assumes that the entire original source structure is part of the information passed to the lens.We offer a general theory of edit lenses, which work with descriptions of changes to structures, rather than with the structures themselves. We identify a simple notion of "editable structure"--a set of states plus a monoid of edits with a partial monoid action on the states--and construct a semantic space of lenses between such structures, with natural laws governing their behavior. We show how a range of constructions from earlier papers on "state-based" lenses can be carried out in this space, including composition, products, sums, list operations, etc. Further, we show how to construct edit lenses for arbitrary containers in the sense of Abbott, Altenkirch, and Ghani. Finally, we show that edit lenses refine a well-known formulation of state-based lenses, in the sense that every state-based lens gives rise to an edit lens over structures with a simple overwrite-only edit language, and conversely every edit lens on such structures gives rise to a state-based lens.
Relational foundations for functorial data migration | Proceedings of the 15th Symposium on Database Programming Languages
In this paper we present a simple database definition language: that of categories and functors. A database schema is a small category and an instance is a set-valued functor on it. We show that morphisms of schemas induce three ''data migration ...
Unison | A friendly, statically-typed, functional programming language from the future · Unison programming language
A friendly programming language from the future.
Functorial data migration
In this paper we present a simple database definition language: that of categories and functors. A database schema is a small category and an instance is a set-valued functor on it. We show that morphisms of schemas induce three “data migration functors”, which translate instances from one schema to the other in canonical ways. These functors parameterize projections, unions, and joins over all tables simultaneously and can be used in place of conjunctive and disjunctive queries. We also show how to connect a database and a functional programming language by introducing a functorial connection between the schema and the category of types for that language. We begin the paper with a multitude of examples to motivate the definitions, and near the end we provide a dictionary whereby one can translate database concepts into category-theoretic concepts and vice versa.
Algebraic data integration
In this paper, we develop an algebraic approach to data integration by combining techniques from functional programming, category theory, and database theory. In our formalism, database schemas and instances are algebraic (multi-sorted equational) theories of a certain form. Schemas denote categories, and instances denote their initial (term) algebras. The instances on a schema S form a category, S–Inst, and a morphism of schemas F : S → T induces three adjoint data migration functors: ΣF : S–Inst → T–Inst, defined by substitution along F, which has a right adjoint ΔF : T–Inst → S–Inst, which in turn has a right adjoint ΠF : S–Inst → T–Inst. We present a query language based on for/where/return syntax where each query denotes a sequence of data migration functors; a pushout-based design pattern for performing data integration using our formalism; and describe the implementation of our formalism in a tool we call AQL (Algebraic Query Language).
Specification of graph translators with triple graph grammars
Data integration is a key issue for any integrated set of software tools. A typical CASE environment, for instance, offers tools for the manipulation of requirements and software design documents, and it provides more or less sophisticated assistance for keeping these documents in a consistent state. Up to now, almost all data consistency observing or preserving integration tools are hand-crafted due to the lack of generic implementation frameworks and the absence of adequate specification formalisms. Triple graph grammars are intended to fill this gap and to support the specification of interdependencies between graph-like data structures on a very high level. Furthermore, they are the fundamentals of a new machinery for the production of batch-oriented as well as incrementally working data integration tools.
An Algorithm for Layout Preservation in Refactoring Transformations
Transformations and semantic analysis for source-to-source transformations such as refactorings are most effectively implemented using an abstract representation of the source code. An intrinsic limitation of transformation techniques based on abstract syntax trees is the loss of layout, i.e. comments and whitespace. This is especially relevant in the context of refactorings, which produce source code for human consumption. In this paper, we present an algorithm for fully automatic source code reconstruction for source-to-source transformations. The algorithm preserves the layout and comments of the unaffected parts and reconstructs the indentation of the affected parts, using a set of clearly defined heuristic rules to handle comments.
Composing schema mappings: Second-order dependencies to the rescue: ACM Transactions on Database Systems: Vol 30, No 4
A schema mapping is a specification that describes how data structured under one schema (the source schema) is to be transformed into data structured under a different schema (the target schema). A fundamental problem is composing schema mappings: given ...
A Categorical Unification for Multi-Model Data: Part I Categorical Model and Normal Forms
Modern database systems face a significant challenge in effectively handling the Variety of data. The primary objective of this paper is to establish a unified data model and theoretical framework for multi-model data management. To achieve this, we present a categorical framework to unify three types of structured or semi-structured data: relation, XML, and graph-structured data. Utilizing the language of category theory, our framework offers a sound formal abstraction for representing these diverse data types. We extend the Entity-Relationship (ER) diagram with enriched semantic constraints, incorporating categorical ingredients such as pullback, pushout and limit. Furthermore, we develop a categorical normal form theory which is applied to category data to reduce redundancy and facilitate data maintenance. Those normal forms are applicable to relation, XML and graph data simultaneously, thereby eliminating the need for ad-hoc, model-specific definitions as found in separated normal form theories before. Finally, we discuss the connections between this new normal form framework and Boyce-Codd normal form, fourth normal form, and XML normal form.
Boomerang | Proceedings of the 35th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Bidirectional programming languages are a practical approach to the view update problem. Programs in these languages, called lenses, define both a view and an update policy - i.e., every program can be read as a function mapping sources to views as well ...
Bidirectionalization for free! (Pearl) | Proceedings of the 36th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
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 ...
