Makoto Hamana

Bidirectionalization transformation based on automatic derivation of view complement functions

Bidirectional transformation is a pair of transformations: a view function and a backward transformation. A view function maps one data structure called source onto another called view. The corresponding backward transformation reflects changes in the view to the source. Its practically useful applications include replicated data synchronization, presentation-oriented editor development, tracing software development, and view updating in the database community. However, developing a bidirectional transformation is hard, because one has to give two mappings that satisfy the bidirectional properties for system consistency. In this paper, we propose a new framework for bidirectionalization that can automatically generate a useful backward transformation from a view function while guaranteeing that the two transformations satisfy the bidirectional properties. Our framework is based on two known approaches to bidirectionalization, namely the constant complement approach from the database community and the combinator approach from the programming language community, but it has three new features: (1) unlike the constant complement approach, it can deal with transformations between algebraic data structures rather than just tables; (2) unlike the combinator approach, in which primitive bidirectional transformations have to be explicitly given, it can derive them automatically; (3) it generates a view update checker to validate updates on views, which has not been well addressed so far. The new framework has been implemented and the experimental results show that our framework has promise.

Free Σ-Monoids: A Higher-Order Syntax with Metavariables

The notion of Σ-monoids is proposed by Fiore, Plotkin and Turi, to give abstract algebraic model of languages with variable binding and substitutions. In this paper, we give a free construction of Σ-monoids. The free Σ-monoid over a given presheaf serves a well-structured term language involving binding and substitutions. Moreover, the free Σ-monoid naturally contains interesting syntactic objects which can be viewed as “metavariables” and “environments”. We analyse the term language of the free Σ-monoid by relating it with several concrete systems, especially the λ-calculus extended with contexts.

Universal algebra for termination of higher-order rewriting

We show that the structures of binding algebras and Σ-monoids by Fiore, Plotkin and Turi are sound and complete models of Klops Combinatory Reduction Systems (CRSs). These algebraic structures play the same role of universal algebra for term rewriting systems. Restricting the algebraic structures to the ones equipped with well-founded relations, we obtain a complete characterisation of terminating CRSs. We can also naturally extend the characterisation to rewriting on meta-terms by using the notion of Σ-monoids.

A Logic Programming Language Based on Binding Algebras

We give a logic programming language based on Fiore, Plotkin and Turis binding algebras.In this language, we can use not only first-order terms but also terms involving variable binding.The aim of this language is similar to Nadathur and Millers ?Prolog, which can also deal with binding structure by introducing ?-terms in higher-order logic. But the notion of binding used here is finer in a sense than the usual ?-binding. We explicitly manage names used for binding and treat ?-conversion with respect to them. Also an important difference is the form of application related to s-conversion, i.e. we only allow the form (M x), where x is a (object) variable, instead of usual application (M N). This notion of binding comes from the semantics of binding by the category of presheaves. We firstly give a type theory which reflects this categorical semantics. Then we proceed along the line of first-order logic programming language, namely, we give a logic of this language, an operational semantics by SLD-resolution and unification algorithm for binding terms.

Term rewriting with variable binding: an initial algebra approach

We present an extension of first-order term rewriting systems, which involves variable binding in the term language. We develop the systems called binding term rewriting systems (BTRSs) in a stepwise manner; firstly we present the term language, then formulate equational logic, and finally define rewrite systems by two styles: rewrite logic and pattern matching styles. The novelty of this development is that we follow an initial algebra approach in an extended notion of Σ-algebras in various functor categories. These are based on Fiore-Plotkin-Turis presheaf semantics of variable binding and Lüth-Ghanis monadic semantics of term rewriting systems. We characterise the terms, equational logic and rewrite systems for BTRSs are initial algebras in suitable categories. Finally, we discuss our design choice of BTRSs from a semantic perspective.

Higher-order semantic labelling for inductive datatype systems

We give a novel transformation for proving termination of higher-order rewrite systems in the format of Inductive Data Type Systems (IDTSs) by Blanqui, Jouannaud and Okada. The transformation called higher-order semantic labelling attaches algebraic semantics of the arguments to each function symbol. We systematically define the labelling and show that labelled systems give termination models in the frame-work of Fiore, Plotkin and Turis binding algebras. As applications, we give simple proofs of termination of the explicit substitution system λ X and currying transformation via higher-order semantic labelling. Moreover, we prove a new result of modularity of termination of IDTSs by introducing the notion of solid IDTSs. We prove that termination is preserved under the disjoint union of an IDTS and a higher-order program scheme.

A foundation for GADTs and inductive families: dependent polynomial functor approach

Every Algebraic Datatype (ADT) is characterised as the initial algebra of a polynomial functor on sets. This paper extends the characterisation to the case of more advanced datatypes: Generalised Algebraic Datatypes (GADTs) and Inductive Families. Specifically, we show that GADTs and Inductive Families are characterised as initial algebras of dependent polynomial functors. The theoretical tool we use throughout is an abstract notion of polynomial between sets together with its associated general form of polynomial functor between categories of indexed sets introduced by Gambino and Hyland. In the context of ADTs, this fundamental result is the basis for various generic functional programming techniques. To establish the usefulness of our approach for such developments in the broader context of inductively defined dependent types, we apply the theory to construct zippers for Inductive Families.

Explicit substitutions and higher-order syntax

Recently there has been a great deal of interest in higher-order syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding. The canonical example studied in the literature is that of the untyped λ-calculus which is handled as an instance of the general theory of binding algebras, cf. Fiore et al. [13].Another important syntactic construction is that of explicit substitutions which are used to model local definitions and to implement reduction in the λ-calculus. The syntax of a language with explicit substitutions does not form a binding algebra as an explicit substitution may bind an arbitrary number of variables. Thus explicit substitutions are a natural test case for the further development of the theory and applications of syntax with variable binding.This paper shows that a language containing explicit substitutions and a first-order signature Σ is naturally modelled as the initial algebra of the Id + FΣ∘_ +_ ∘ _ endofunctor. We derive a similar formula for adding explicit substitutions to the untyped λ-calculus and then show these initial algebras provide useful datatypes for manipulating abstract syntax by implementing two reduction machines. We also comment on the apparent lack of modularity in syntax with variable binding as compared to first-order languages.

Semantic labelling for proving termination of combinatory reduction systems

We give a novel transformation method for proving termination of higher-order rewrite rules in Klops format called Combinatory Reduction System (CRS). The format CRS essentially covers the usual pure higher-order functional programs such as Haskell. Our method called higher-order semantic labelling is an extension of a method known in the theory of term rewriting. This attaches semantics of the arguments to each function symbol. We systematically define the labelling by using the complete algebraic semantics of CRS, Σ-monoids. We also examine the power of higher-order semantic labelling by several examples. This includes an interesting example from the viewpoint of functional programming.

Polymorphic abstract syntax via Grothendieck construction

Abstract syntax with variable binding is known to be characterised as an initial algebra in a presheaf category. This paper extends it to the case of polymorphic typed abstract syntax with binding. We consider two variations, secondorder and higher-order polymorphic syntax. The central idea is to apply Fiores initial algebra characterisation of typed abstract syntax with binding repeatedly, i.e. first to the type structure and secondly to the term structure of polymorphic system. In this process, we use the Grothendieck construction to combine differently staged categories of polymorphic contexts