An algebraic approach to higher-order theories and rewriting€¦ · Introduction Here: A...

Introduction Rewrite systems as 2-signatures 2-categorical semantics Cartesian closed case Conclusion

Laboratoire de MathematiquesUniversite de Savoie UMR 5127

An algebraic approach to higher-order theoriesand rewriting

Tom Hirschowitz

Brno, August 2010

Hirschowitz Algebraic higher-order rewriting 1/34


Original motivation

Understood from a talk by Fiore in Lyon:I will to explain variable binding by a λ-calculus limited to 2nd

order;I complex categorical picture.

Alternative point of view:I the λ-calculus perfectly explains variable binding;I simple categorical picture: cartesian closed categories.



Introduction

Here:

A categorical semantics forI higher-order rewriting (Klop, Nipkow, Wolfram), andI permutation equivalence (Bruggink).

In cartesian closed 2-categories.

Corollary: a semantics for variable binding in cartesian closedcategories.



Related work

Cartesian closed sketches (Kinoshita, Power, Takeyama andWells).

Bruggink’s permutation equivalence, Hilken’s 2-λ-calculus,Jacobs’ book.

Capriotti’s semantics for flat permutation equivalence insesqui-categories.

By extension, Fiore et al., my father and Maggesi.



Plan

1 Introduction.

2 The known case:I rewrite systems as fp 2-signatures,I their fp 2-categorical semantics.

3 Sketch of the cartesian closed case.



Types

Given a set X , let L0(X ) be the set of formulas, as generated bythe grammar:

A,B ::= 1 | x | A× B.

Proposition

This extends to a monad on Set with substitution as multiplication.

Definition

Let S0(X ) = L0(X )∗ × L0(X ) be the set of sequents over X .



1-Signatures

Definition

A 1-signature Σ = (X0,X1) consists of:

a set X0 of sorts,

a map X1 → S0(X0) of operations.

The latter amounts to a set indexed by sequents in S0(X0).



Example: combinatory logic

One sort X0 = {t}.Three operations X1 = {K ,S ,A}:

I K ,S : 1→ t,

I A : t, t → t,i.e.,

I K ,S 7→ ([], t),

I A 7→ ([t, t], t).



Non-example: the λ-calculus

Arity of λ: “tt → t”.

Not expressible by our arities.

Workarounds do exist.



A category of 1-signatures

Consider 1-signatures Σ = (X0,X1) and Σ′ = (Y0,Y1).A morphism Σ→ Σ′ consists of:

a map f0 : X0 → Y0,

a map f1 : X1 → Y1 making

X1 Y1

S0(X0) S0(Y0)

f1

S0(f0)

commute.




Proposition

This yields a category Sig1 of 1-signatures.



The term monad

Given a 1-signature Σ = (X0,X1), let L1(Σ) be the 1-signaturewith:

sorts X0,

operations A1, . . . ,An → A the termsx1 : A1, . . . , xn : An ` M : A in the language generated by Σ.

Proposition

L1 extends to a monad on Sig1.



Parallel operations

For a 1-signature Σ = (X0,X1), let Σ|| be the pullback:

Σ|| X1

X1 S0(X0).

Example: L1(Σ)|| is the set of parallel terms over Σ.

Proposition

This extends to a functor (−)|| : Sig1 → Set.



Rewrite systems = 2-signatures

Definition

A rewrite system, or a 2-signature, X consists of:

a 1-signature Σ = (X0,X1),

a map X2 → L1(Σ)|| of rewrite rules.

Definition

The rewrite relation generated by X is the smallest congruencegenerated by the rewrite rules.

A bit more general than the original definition.



Example: combinatory logic

Remember K , S ,A.

Two rewrite rules (with usual shorthand):

Kxy → x Sxyz → xy(xz).




A morphism X → Y of 2-signatures consists of

a morphism (f0, f1) : (X0,X1)→ (Y0,Y1) of 1-signatures,

a map f2 : X2 → Y2 making

X2 Y2

L1(X )|| L1(Y )||

f2

L1(f1)||

commute.




Proposition

This yields a category Sig.



Reductions and permutation equivalence

We now define a monad L on Sig.

We then use L to construct adjunctions

Sig ⊥ L-Alg ⊥ 2FPCat.

L

U

F

V

Reminiscent of multigraphs a monoidal categories throughmulticategories.

Though missing the language for representable L-algebras.



Reductions and permutation equivalence

L preserves underlying 1-signature.

The rewrite rules of L(X ) are the reductions, modulopermutation equivalence, which we now define.

Along the way, we sketch:I how L(X ) yields an fp 2-category,I how any fp 2-category yields an L-algebra.



Reductions

First, for (r ∈ X (G ` M,N : t)), we have the formation rule:

. . . Γ ` Pi : Mi → Ni : Gi . . .

Γ ` r(P1, . . . ,Pn) : M[M1, . . . ,Mn]→ N[N1, . . . ,Nn] : A·

2-categorically:

∏Γ

∏G A,

(M1, . . . ,Mn)

(N1, . . . ,Nn)

M

N

rP with P = (P1, . . . ,Pn).



Reductions

There is a rule for sequential composition:

Γ ` P : M1 → M2 : A Γ ` Q : M2 → M3 : A

Γ ` P ;M2 Q : M1 → M3 : A·

2-categorically:

∏G A.

M1

M3

P

Q



Reductions

Plus a few other rules:

for empty reductions, i.e., identities,

for reductions in context, i.e., making reductions into acongruence.



Permutation equivalence

We then quotient out reductions by permutation equivalence.

Also inductively defined.

Permutation actually happens in the rules equating

C B A,r with C B A

C B Ar

and the other way around. Namely:

r(P ;N2 Q) ≡ M1[P] ;M1[N2] r(Q).



The monad

Theorem

This defines a functor L : Sig→ Sig, which is a monad withsubstitution as multiplication.



The adjunction

We use L to construct an adjunction

Sig ⊥ 2FPCat

H

W

as the composite

Sig ⊥ L-Alg ⊥ 2FPCat.

L

U

F

V



Soundness and completeness

Proposition

Given a 2-signature X , form the fp 2-category X ∗ with:

objects: tuples of sorts,

morphisms: tuples of terms,

preordered hom-categories:I X ∗(M,N) = 1 iff M →∗ N,I X ∗(M,N) = ∅ otherwise.




Theorem

For all sorts t1, . . . , tn, t, there is a functor

H(X )(Πti , t)→ X ∗(Πti , t),

which is full and identity-on-objects (to a preorder).

Hence:

If P : M → N in H(X ) then M →∗ N in the standard sense.

Conversely.



Cartesian closedness

Up to now: first-order rewriting.

2-categorical semantics known, or at least folklore.

New presentation as far as I know.

Now, sketch for higher-order rewriting.



2-signatures

Types: A,B ::= 1 | x | A× B | BA.

L1(Σ): simply-typed λ-calculus on Σ.

2-signature: set indexed by pairs of parallel λ-terms inβ-normal, η-long form.



Example

Signature Σ: ` : tt → t a : t × t → t.

Pure λ-terms with variables in n in bijection with

L1(Σ)(t, . . . , t︸︷︷︸n times

→ t).

One rule β : a(`(Mtt),Nt)→ MN, i.e.,

t × t

tt × t t.

`× t a

ev

β



Reductions and permutations

As before, plus formation rules

Γ, x : A ` P : M → N : B

Γ ` λx : A.P : λx : A.M → λx : A.N : BA

Γ ` P : M → M ′ : BA Γ ` Q : N → N ′ : A

Γ ` PQ : MN → M ′N ′ : B

with equations for βη.




From any 2-signature X , higher-order rewriting yields a reductionrelation between terms on the underlying 1-signature.

Proposition

M →∗ N iff there exists a reduction P : M → N.



Corollary

Cartesian closed categories are locally discrete cartesian closed

2-categories: 2CCCat ⊥ CCCat.

Composing with the above: Sig ⊥ CCCat.

K

Fiore and Hur’s syntactic category on X is a full subcategoryof K(X ).

More objects in K(X ), e.g., ttt.



Uncaught

The π-calculus (and all calculi with a structural congruence).

Would need to incorporate equations between reductions.

Even with that, no efficient presentation of π yet.


An algebraic approach to higher-order theories and rewriting€¦ · Introduction Here: A...

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