Abstract syntax and variable binding

Positive Inductive-Recursive Definitions

“…Among these data types are those with constraints, such as perfect trees [Hin00]; types with variable binding, such as untyped λ-terms [FPT99]; cyclic data structures [GHUV06]; and certain dependent types [MM04].…”

Coproducts of Ideal Monads

“…Monads also model a number of other important structures in computer science, such as (many-sorted) algebraic theories, non-wellfounded syntax [1,8,26], term graphs [9], calculi with variable binders [7], term rewriting systems [18], and, via computational monads [24], state-based computations, exceptions, continuations etc. These applications involve base categories other than Set and the desire for a uniform treatment underpins their monadic axiomatisation.…”

“…One of their first applications was to give a categorical account of universal algebra [17] and they have since remained central to the development of category theory. More recently they have found further significant applications in computer science as they provide (i) an abstract model of syntax covering algebraic theories [21], higher-order abstract syntax [7] and non-wellfounded syntax [1,8,26] etc. where the term algebra construction, substitution and variables are taken as primitive; (ii) a useful abstraction of computational effects permitting a uniform treatment of diverse features of impure programming languages such as stateful computations, exceptions and I/O [24]; and (iii) a semantic framework for modularity where the interaction of different components of a complex system is modelled by the interleaving of the representing monads [20].…”

“…In particular, this paper arose out of research into modularity in (i) non-wellfounded syntax and (ii) higher-order syntax. In the former we want to study coproducts of free completely iterative monads as these model non-wellfounded terms [2,26] while in the latter we study monads which model calculi with variable binding such as the λ-calculus [7] and which typically arise as least fixed points of higher-order functors. The results above do not apply to coproducts of such monads since these monads are not free.…”

From Co-algebraic Specifications to Implementation: The Mihda Toolkit

“…This framework is sufficient to describe and reason about formalisms with name-binding operations. It has been incorporated into various kinds of transition systems that aim at providing syntax-free models of name-passing calculi [5,6,12,15] It is important to emphasise that names of states of HD-automata have local meaning. For instance, assume that A(x, y, z) denotes an agent having three (free) names x, y and z.…”