Reduction monads and their signatures

Abstract

In this work, we study reduction monads, which are essentially the same as monads relative to the free functor from sets into multigraphs. Reduction monads account for two aspects of the lambda calculus: on the one hand, in the monadic viewpoint, the lambda calculus is an object equipped with a well-behaved substitution; on the other hand, in the graphical viewpoint, it is an oriented multigraph whose vertices are terms and whose edges witness the reductions between two terms.We study presentations of reduction monads. To this end, we propose a notion of reduction signature. As usual, such a signature plays the role of a virtual presentation, and specifies arities for generating operationspossibly subject to equations-together with arities for generating reduction rules. For each such signature, we define a category of models; any model is, in particular, a reduction monad. If the initial object of this category of models exists, we call it the reduction monad presented (or specified) by the given reduction signature.Our main result identifies a class of reduction signatures which specify a reduction monad in the above sense. We show in the examples that our approach covers several standard variants of the lambda calculus.

A Cellular Howe Theorem

“…We first introduce transition Σ 0 -monoids, which are intuitively transition systems whose vertices are equipped with Σ 0 -monoid structure. Then, adapting ideas from Fiore [17] and Ahrens et al [4,20] to the cellular approach, we consider transition rules specified by a syntactically free endofunctor Σ 1 on transition Σ 0 -monoids, models being given by a special kind of algebras, called vertical. Under finitarity hypotheses, as our second contribution, we characterise the syntactic transition system as the initial vertical algebra (Theorem 4.20).…”

“…As far as we know, these approaches do not cover higherorder languages like the -calculus, which was one of the motivations for our work. Among more recent work, quite some inspiration was drawn from Ahrens et al [4,20], notably in the use of vertical algebras. However, a difference is that we do not insist that transitions be stable under substitution.…”

“…The relevant models are jusť Σ 1 -algebras whose structure map is sent to the identity by the forgetful functor Σ 0 -Mon → Σ 0 -mon. Following Ahrens et al [4,20], we deem such algebras vertical. As we will prove below (Theorem 4.20), free vertical algebras may be characterised inductively, which in the present case means that the free vertical algebra over any transition Σ 0 -monoid is inductively defined by the modified rules (5), augmented with an axiom…”

Modules over monads and operational semantics (expanded version)

“…In recent work, following previous work on modules over monads for syntax with binding [HM07, AHLM19] (see also [Ahr16]), Ahrens et al [AHLM20] introduce reduction monads, and show how they cover several standard variants of the 𝜆-calculus. Furthermore, as expected in similar contexts, they propose a mechanism for specifying reduction monads by suitable signatures.…”

A categorical framework for congruence of applicative bisimilarity in higher-order languages

“…This was one of the main motivations for our work. Among more recent work, quite some inspiration was drawn from Ahrens et al [5,24], notably in the use of vertical algebras. However, a difference is that we do not insist that transitions be stable under substitution.…”