We present a new modular proof method of termination for second-order computation, and report its implementation SOL. The proof method is useful for proving termination of higher-order foundational calculi. To establish the method, we use a variation of semantic labelling translation and Blanqui's General Schema: a syntactic criterion of strong normalisation. As an application, we apply this method to show termination of a variant of call-by-push-value calculus with algebraic effects and effect handlers. We also show that our tool SOL is effective to solve higher-order termination problems.

Abstract By using algebraic structures in a presheaf category over finite sets, following Fiore, Plotkin and Turi, we develop sound and complete models of second-order rewriting systems called second-order computation systems (CSs). Restricting the algebraic structures to those equipped with well-founded relations, we obtain a complete characterisation of terminating CSs. We also extend the characterisation to rewriting on meta-terms using the notion of $\Sigma$ -monoid.

Abstract We present a general methodology of proving the decidability of equational theory of programming language concepts in the framework of second-order algebraic theories. We propose a Haskell-based analysis tool, i.e. Second-Order Laboratory, which assists the proofs of confluence and strong normalisation of computation rules derived from second-order algebraic theories. To cover various examples in programming language theory, we combine and extend both syntactical and semantical results of the second-order computation in a non-trivial manner. We demonstrate how to prove decidability of various algebraic theories in the literature. It includes the equational theories of monad and λ-calculi, Plotkin and Power’s theory of states and bits, and Stark’s theory of π-calculus. We also demonstrate how this methodology can solve the coherence of monoidal categories.

The aim of this paper is to provide mathematical foundations of a graph transformation language, called UnCAL, using categorical semantics of type theory and fixed points. About 20 years ago, Bunemanet al. developed a graph database query language UnQL on the top of a functional meta-language UnCAL for describing and manipulating graphs. Recently, the functional programming community has shown renewed interest in UnCAL, because it provides an efficient graph transformation language which is useful for various applications, such as bidirectional computation. In order to make UnCAL more flexible and fruitful for further extensions and applications, in this paper, we give a more conceptual understanding of UnCAL using categorical semantics. Our general interest of this paper is to clarify what is the algebra of UnCAL. Thus, we give an equational axiomatisation and categorical semantics of UnCAL, both of which are new. We show that the axiomatisation is complete for the original bisimulation semantics of UnCAL. Moreover, we provide a clean characterisation of the computation mechanism of UnCAL called ‘structural recursion on graphs’ using our categorical semantics. We show a concrete model of UnCAL given by the λG-calculus, which shows an interesting connection to lazy functional programming.

We present a general methodology of proving the decidability of equational theory of programming language concepts in the framework of second-order algebraic theories. We propose a Haskell-based analysis tool SOL, Second-Order Laboratory, which assists the proofs of confluence and strong normalisation of computation rules derived from second-order algebraic theories. To cover various examples in programming language theory, we combine and extend both syntactical and semantical results of second-order computation in a non-trivial manner. We demonstrate how to prove decidability of various algebraic theories in the literature. It includes the equational theories of monad and lambda-calculi, Plotkin and Power's theory of states, and Stark's theory of pi-calculus.

We formalise and study the notion of polymorphic algebraic theory, as understood in the mathematical vernacular as a theory presented by equations between polymorphically-typed terms with both type and term variable binding. The prototypical example of a polymorphic algebraic theory is System F, but our framework applies more widely. The extra generality stems from a mathematical analysis that has led to a unified theory of polymorphic algebraic theories with the following ingredients: ; polymorphic signatures that specify arbitrary polymorphic operators (e.g. as in extended λ-calculi and algebraic effects); ; metavariables, both for types and terms, that enable the generic description of meta-theories; ; multiple type universes that allow a notion of translation between theories that is parametric over different type universes; ; polymorphic structures that provide a general notion of algebraic model (including the PL-category semantics of System F); ; a Polymorphic Equational Logic that constitutes a sound and complete logical framework for equational reasoning. Our work is semantically driven, being based on a hierarchical two-levelled algebraic modelling of abstract syntax with variable binding.

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.

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 Turi's 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.

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-Turi's presheaf semantics of variable binding and Lüth-Ghani's 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.