Presentation at POPL'20 2020-01-24 New Orleans, USA Authors: Benedikt Ahrens André Hirschowitz Ambroise Lafont Marco Maggesi

1. Reduction Monads
and Their Signatures
Benedikt Ahrens, André Hirschowitz, Ambroise Lafont, Marco Maggesi
POPL’20
January 24, 2020, New Orleans, USA

2. Overview

3. Abstract point-of-view on programming languages
To specify a (functional untyped) programming language we need:
‣ a syntax for programs (terms, sometimes modulo equations)
‣ a well-behaved substitution
‣ a syntax for reductions
‣ models (semantics)
Our work: a theoretical framework for specifying and reasoning about all the
above aspects of an abstract programming language.
Our paradigmatic example will be the -calculus with -reduction.λ β

4. Initial Semantics

5. Quick introduction on Initial Semantics
The essential steps are:
1. Establish a notion of signature (abstract description of a language)
2. To each signature associate a category of models
3. Prove that this category has an initial object
Then:
‣ the initial object is the language
‣ the initial morphisms are the semantics of this language.

6. Toy example: language of natural numbers
• and are constructors of arities 0 and 1 (signature: (0, 1)).
• Models are diagrams in of the form .
• The diagram is the initial model.
• The initial morphism
encodes the classical recursion principle.
Z S
𝖲𝖾𝗍 ( ∙
ZX
⟶ X
SX
⟶ X)
( ∙
Z
⟶ ℕ
S
⟶ ℕ)
( ∙
Z
⟶ ℕ
S
⟶ ℕ) ⟶ (∙
ZX
⟶ X
SX
⟶ X)
ℕ ::= Z : ℕ
| S : ℕ → ℕ

7. Beyond the syntax of terms
• It is well-known that Initial Semantics can describe the shape of terms of a
language.
• But we need more than just terms: substitution, reduction, …
• We use Initial Semantics to do this all at once.
• In other words: generate semantic morphisms that are well-behaved with
respect to:
- substitution
- binding constructions and free/bound variables
- equations
- reductions

8. Monads and
Modules over monads

9. Example: Monad of -termsλ
Monads (and modules over a monad) occur in our context as an axiomatic
theory of substitution.
Example: Monad of (untyped) -terms modulo -conversion:
‣
set of -terms with free variables in
‣ construction of variables
‣ parallel capture-avoiding
substitution
‣ properties of substitution (e.g. associativity) = monad laws
λ α
Λ(X) := { M | 𝖿𝗏(M) ⊆ X }
λ X
ηΛ : X ⟶ Λ(X)
𝖻𝗂𝗇𝖽Λ : (X → Λ(Y)) ⟶ (Λ(X) → Λ(Y))

10. A module over a monad is an endofunctor endowed with an action of
:
‣
‣
‣ module laws (compatibility laws with ).
NB: Every monad is trivially a module over itself.
M R
R
M : 𝖲𝖾𝗍 ⟶ 𝖲𝖾𝗍
𝗆𝖻𝗂𝗇𝖽 : (X → R(Y)) ⟶ (M(X) → M(Y))
R
R
Modules over a monad

11. Crucial examples of modules
Product: Pairs of terms of a monad :
‣
‣
Derived modules: Terms with bound variables ( set of cardinality )
‣
‣
R
R × R : X ↦ R(X) × R(X)
𝗆𝖻𝗂𝗇𝖽R×R : (X → R(Y)) ⟶ (R(X) × R(X) → R(Y) × R(Y))
n n n
R(n)
: X ↦ R(X + n)
𝗆𝖻𝗂𝗇𝖽R(n) : (X → R(Y)) ⟶ (R(X + n) → R(Y + n))

12. Example: -terms (again)λ
Application and abstraction are -modules morphisms (i.e., natural
transformations that are compatible with the substitution).
• Application:
• Abstraction:
Λ
𝖺𝗉𝗉X
: Λ(X) × Λ(X) ⟶ Λ(X)
𝖺𝗉𝗉X
: (M, N) ↦ MN
𝖺𝖻𝗌X : Λ(1)
(X) ⟶ Λ(X)
𝖺𝖻𝗌X : M[ ] ↦ λx . M[x]

13. -reduction as a pair of module morphismsβ
Metavariables:
Module of metavariables:
(T, U)
Λ(1)
× Λ
𝖺𝗉𝗉( 𝖺𝖻𝗌(T), U)
β
⟶ T{⋆ := U}
β-Red
Source morphism:
Λ(1)
× Λ ⟶ Λ
(T, U) ↦ 𝖺𝗉𝗉( 𝖺𝖻𝗌(T), U)
Target morphism:
Λ(1)
× Λ ⟶ Λ
(T, U) ↦ T{⋆ := U}

14. Our achievement

15. Our stack of signatures
Signatures for syntactic constructions [CSL 2018]
Reduction rules [POPL 2020]
Signatures for equations [FSCD 2019]
}
Signatures
for monads
{
Reduction
signatures

16. What you can find in our paper
• An initial semantic approach to abstract programming language, in particular:
‣ A notion of reduction monad (mathematical model of programming language)
‣ A general notion of reduction signature that can specify reduction monads
‣ The associated notion of category of models
‣ A theorem that ensures (under suitable hypotheses) the existence of an initial
model
• Several examples of -calculi presented with our framework, including:
‣ several kind of rules and reduction strategies ( -red, hnf, whnf, parallel- ,…)
‣ extensions of the of -calculus (fixpoint operator, explicit substitution)
λ
βη β
λ

17. Thanks!