Substitution in non-wellfounded syntax with variable binding

Abstract

Inspired from the recent developments in theories of non-wellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and non-wellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. The main technical tool is a novel concept of heterogeneous substitution systems.

Generalizing Substitution

“…Future work includes extending the substitution and solution theorems to signatures with binding and explicit substitution operators (joint project of the author with Ralph Matthes, the substitution theorem appears in [15]), proving the free completely iterative monad theorem of [2] for generalized substitution (this will take generalizing their concept of ideal monad), a study of the pragmatics of generalized redecoration, and also a detailed comparison of the concept of substitutioncarrying monad employed here to that of coalgebraic monad by Ghani et al [11].…”

Nominal Coalgebraic Data Types with Applications to Lambda Calculus

“…Roughly speaking, the difference between Nom and Set F is that the latter already comes equipped with a notion of substitution, see [Sta07, Section 7.3] for details. [MU04] further develop substitution for algebraic and coalgebraic datatypes over presheaf-categories and describe the set of infinitary λ-terms as a final coalgebra. [AMV11] furthermore study the so-called rational fixed point, again over Set F , as a semantic universe for solutions of higher-order recursion schemes.…”

Modules over Monads and Linearity

“…[2,11,18]) although some other settings have been considered. Notably in [6] the authors work within a setting roughly based on operads (although they do not write this word down; the definition of operad is on Wikipedia; operads and monads are not too far from each other).…”