Cofree coalgebras and differential linear logic

0.23 MB·

by  James Clift

Cofree coalgebras and differential linear logic
James Clift, Daniel Murfet
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1 January 6, 2017
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Abstract
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We prove that the semantics of intuitionistic linear logic in vector spaces which

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usescofreecoalgebras tomodeltheexponentialis amodelofdifferential linearlogic.

] Thus, in this semantics, proof denotations have natural derivatives. We give several
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examples of these derivatives.
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.
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c 1 Introduction
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The idea of taking derivatives of programsis anold one[28, §2] with many manifestations,
v
5 including automatic differentiation of algorithms computing real-valued functions [3] and
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incremental computation [29]. However, these approaches are limited to restricted classes
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1 of computations, and it is only recently with the development of the differential λ-calculus
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by Ehrhard-Regnier [8] and its refinement differential linear logic [2, 6] that derivatives
.
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have been defined for general higher-order programs. As with ordinary calculus, the aim
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7 of these theories is to assign to a program P another program ∂P (the derivative) which
1 computes thechange intheoutputof P resulting fromaninfinitesimal change toitsinput.
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v Here we give ourselves arbitrary C-linear combinations of programs (meaning λ-terms or
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X proofs in linear logic) as a starting point so that “small” changes to the input make sense.
r This paper is about the semantics of differential linear logic, following [2]. The aim is
a
to explain how the natural semantics of intuitionistic linear logic in vector spaces [19, 26]
is already a model of differential linear logic. The key point is that tangent vectors and
derivatives appear as soon as we introduce cofree coalgebras to model the exponential,
which shows that the differential structure is intrinsic to the algebra of linear logic.
To see this, let J−K denote semantics in vector spaces and suppose we are given a proof
π in linear logic computing a function from inputs of type A to outputs of type B:
π
.
.
.
!A ⊢ B.
The space of inputs to JπK is JAK, and a small change in the input starting fromP ∈ JAKis
a tangent vector ν at P, viewing JAK as a smooth manifold or a scheme. This is equivalent
1
to the data of a linear map
(C[ε]/ε2)∗ //JAK (1)
where C[ε]/ε2 is the ring of dual numbers (this bijection is reviewed in Appendix A.1).
If J!AK is the universal cocommutative counital coalgebra mapping to JAK then there is a
unique lifting of this linear map to a morphism of coalgebras
(C[ε]/ε2)∗ // J!AK. (2)
SimilarlythelinearmapJπK : J!AK −→ JBKliftstoamorphismofcoalgebrasJ!AK −→ J!BK
which may be composed with (2) to give a morphism of coalgebras
(C[ε]/ε2)∗ // J!AK //J!BK (3)
which, in turn, defines a tangent vector at the point JπK|∅i ∈ JBK, where |∅i is the
P P
point of J!AK corresponding to P. The tangent vector (3) gives the infinitesimal variation
of the output of π on the input P, when the input is varied in the direction of ν.
The formal statement is that for any algebraically closed field k of characteristic zero
the semantics of intuitionistic linear logic in k-vector spaces defined using cofree coalge-
bras is model of differential linear logic (Theorem 2.3). We refer to this as the Sweedler
semantics, since the explicit description of this universal coalgebra is due to him [31, 26].
The proof is elementary and we make no claim here to technical novelty; the link between
the symmetric coalgebra and differential calculus is well-known. Perhaps our main con-
tribution is to give several detailed examples showing how to compute these derivatives.
We do this with the aim of reinforcing the fact that differentiating programs, even higher-
order ones, is a natural thing to do.
We conclude this introduction with a sketch of one such example and a comparison of
our work to other semantics of differential linear logic. To elaborate a little more on the
notation: for any type A of linear logic (which for us has only connectives ⊗,⊸,!) there is
a vector space JAK, and for any proof π of A ⊢ B there is a linear map JπK : JAK −→ JBK.
In particular every proof ξ of type A has a denotation JξK ∈ JAK, and the promotion of ξ
has for its denotation a vector |∅i ∈ J!AK, see [26, §5.3].
JξK
For any binary sequence S ∈ {0,1}∗ there is an encoding of S as a proof S of type
bint = !(A ⊸ A) ⊸ !(A ⊸ A) ⊸ (A ⊸ A) .
A
(cid:0) (cid:1)
Repetition of sequences can be encoded as a proof
repeat
.
.
.
!bint ⊢ bint .
A A
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The denotation is a linear map J!bint K −→ Jbint K sending |∅i to JSSK. The deriva-
A A JSK
tive of repeat according to the theory of differential linear logic is another a proof
∂repeat
.
.
.
!bint ,bint ⊢ bint
A A A
which can be derived from repeat by new deduction rules called codereliction, cocontrac-
tion and coweakening (see Section 2.1). We prove in Section 3.2 that the denotation of
this derivative in the Sweedler semantics is the linear map
J∂repeatK : J!bint K⊗Jbint K −→ Jbint K,
A A A
|∅i ⊗JTK 7−→ JSTK+JTSK
JSK
whose value on the tensor |∅i ⊗JTK we interpret as the derivative of the repeat program
JSK
at the sequence S in the direction of the sequence T. This can be justified informally by
the following calculation using an infinitesimal ε
(S +εT)(S +εT) = SS +ε(ST +TS)+ε2TT,
which says that varying the sequence infinitesimally from S in the direction of T causes
a variation of the repetition in the direction of ST +TS.
The Sweedler semantics is far from the first semantics of differential linear logic: basic
examples include the categories of sets and relations [2, §2.5.1] and suplattices [2, §2.5.2].
The motivating examples using topological spaces and differentiable functions are the
Ko¨theandfinitenessspacesemanticsofEhrhard[4,5]andthesemanticsofBlute-Ehrhard-
Tasson [1] based on the theory of convenient vector spaces [12]. These papers explain that
the geometric “avatar” of the exponential connective of linear logic is the functor sending
a space X to the space of distributions on X (for a precise statement, see Remark A.2).
This remarkable analogybetween logicandgeometry deserves further study. Oneobstacle
is that it seems difficult to compute examples of denotations and their derivatives in the
convenient vector space setting of [1]. For example the coproduct [1, p.12] is defined by
extension to a Mackey closure, and is rather implicit.
Conceptually the Sweedler semantics is similar to these examples in that the exponen-
tial is modelled by a space of distributions (with finite support) but it is purely algebraic
and there are simple explicit formulas for all the structure maps. Moreover in the alge-
braicapproachthedifferential structureemerges naturallyfromtheexponential structure,
rather than being “baked in”. The downside is that the smoothness of proof denotations
in our semantics is obscured; in particular, in the case k = C some extra work is required
to see the relationbetween our differential structure andthe derivatives in theusual sense.
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For background material on linear logic and its semantics see [13, 17, 24]. The formal
theory of coalgebras is simpler over algebraically closed fields, which explains why we use
k = C in our examples, but this is not really important: one could work over k = R by
taking C-points into account in the explicit description of the cofree coalgebra.
Acknowledgements. Thanks to Kazushige Terui, who stimulated this project by asking
if the cofree coalgebra gave a model of differential linear logic.
2 Main Theorem
Let k be an algebraically closed field of characteristic zero and V the category of k-vector
spaces. This is a model of intuitionistic linear logic, as observed in [19, p.5] and developed
in detail in [26]. Our notation is as in [25] and [26, §5], see Appendix A for a review.
For a vector space V we denote by !V the universal cocommutative counital coalgebra
mapping to V, with d : !V −→ V the universal map. The map V 7→ !V extends to a
comonad on V with natural transformation δ : ! −→ !!. The coproduct and counit are
respectively denoted ∆ : !V −→ !V ⊗!V and w : !V −→ k. Given vector spaces V,W we
write σ : V ⊗W −→ W ⊗V for the swap map σ (x⊗y) = y ⊗x.
V,W V,W
By [2, Proposition 2.6] to equip V with the coalgebra modality (!,δ,d,∆,w) as a
differential category, we need to define a deriving transformation [2, Definition 2.5].
Definition 2.1.A deriving transformation for (V,!,δ,d,∆,w) is a family of morphisms
D : !V ⊗V −→ !V
V
natural in V, satisfying the following properties for all V:
(D.1) w◦D = 0, that is,
V ⊗!V D //!V w // k = 0. (4)
(D.2) ∆◦D = (1⊗D)◦(∆⊗1)+(D ⊗1)◦(1⊗σ)◦(∆⊗1), that is,
!V ⊗V D //!V ∆ //!V ⊗!V (5)
is equal to the sum
!V ⊗V ∆⊗1 // !V ⊗!V ⊗V 1⊗D // !V ⊗!V +
!V ⊗V ∆⊗1 // !V ⊗!V ⊗V 1⊗σ!V,V //!V ⊗V ⊗!V D⊗1 // !V ⊗!V
∼=
(D.3) d◦D = a◦(w⊗1), that is,
!V ⊗V D // !V d //V = !V ⊗V w⊗1 //k ⊗V a //V
∼=
(6)
where a(λ⊗x) = λx.
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(D.4) δ ◦D = D ◦(δ ⊗D)◦(∆⊗1), that is,
!V
!V ⊗V D // !V δ //!!V (7)
is equal to
!V ⊗V ∆⊗1 // !V ⊗!V ⊗V δ⊗D //!!V ⊗!V D!V //!!V . (8)
We refer to [2, §2.2] for an explanation of these axioms. Briefly, (D.1) says the deriva-
tive of constant mapsis zero, (D.2)is theproduct rule, (D.3)says the derivative of a linear
map is constant, and (D.4) is the chain rule. Clearly the rules specify how to commute D
past the structural maps δ,d,∆,w. Here d stands for the dereliction rule in linear logic, ∆
for contraction and w for weakening. The map δ stands for promotion, since for a linear
map φ : !V −→ W the unique lifting to a morphism of coalgebras Φ : !V −→ !W can be
obtained as the composite
!V δ // !!V !φ // !W . (9)
Definition 2.2.We define the k-linear map D : !V ⊗V −→ V by
V
D |ν ,...,ν i ⊗ν = |ν,ν ,...,ν i . (10)
V 1 s P 1 s P
(cid:0) (cid:1)
See Remark A.4 for a justification of this definition from the point of view of [25].
Theorem 2.3. D is a deriving transformation for any vector space V.
V
We split the proof into a series of lemmas. We prefer to give the proofs without first
choosing a basis of V, but if one is willing to do so, then the connection between these
identities andtheusual rulesofcalculusfollowsfromwriting theformulaforthecoproduct
∆ as a kind of Taylor expansion; see for example [30, (B.65)].
Lemma 2.4. (D.1) holds for V.
Proof. This is clear, since the counit w : !V −→ V vanishes on |ω ,...,ω i if t > 0.
1 t P
Lemma 2.5. (D.2) holds for V.
Proof. Setting ν = ν we have
0
∆D |ν ,...,ν i ⊗ν = ∆|ν,ν ,...,ν i
1 s P 1 s P
(cid:0) (cid:1)
= |ν i ⊗|ν i
I P Ic P
X
I⊆{0,1,...,s}
= |ν i ⊗|ν i + |ν i ⊗|ν i
I P Ic P I P Ic P
X X
0∈I 0∈/I
= |ν,ν i ⊗|ν i + |ν i ⊗|ν,ν i
J P Jc P J P Jc P
X X
J⊆{1,...,s} J⊆{1,...,s}
= D |ν i ⊗ν ⊗|ν i +|ν i ⊗D |ν i ⊗ν
J P Jc P J P Jc P
J⊆X{1,...,s}n (cid:0) (cid:1) (cid:0) (cid:1)o
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as claimed, where for I ⊆ {0,1,...,s} we write Ic for {0,...,s}\I and for J ⊆ {1,...,s},
we write Jc for {1,...,s}\J.
Lemma 2.6. (D.3) holds for V.
Proof. We have
dD |ν ,...,ν i ⊗ν = d|ν,ν ,...,ν i = δ ν
1 s P 1 s P s=0
(cid:0) (cid:1)
while
a(w ⊗1) |ν ,...,ν i ⊗ν = w|ν ,...,ν i ·ν = δ ν.
1 s P 1 s P s=0
(cid:0) (cid:1)
Lemma 2.7. (D.4) holds for V.
Proof. The trivial case is, with Q = |∅i ,
P
δD(|∅i ⊗ν) = δ|νi = |νi
P P P Q
(cid:12) (cid:11)
and on the other side (cid:12)
D (δ ⊗D)(∆⊗1)(|∅i ⊗ν) = D (δ ⊗D)(|∅i ⊗|∅i ⊗ν)
!V P !V P P
= D (|∅i ⊗|νi )
!V Q P
= |νi .
P Q
(cid:12) (cid:11)
Now we consider the case s > 0. Putting ν =(cid:12)ν and writing P for the set of partitions
0 T
of T we have
δD |ν ,...,ν i ⊗ν = ⊗ |ν i
1 s P x∈X x P
(cid:0) (cid:1) X∈PX{0,1,...,s}(cid:12)(cid:12) EQ
(cid:12)
where for a partition X = {x ,...,x } the notation means
1 t
⊗ |ν i = |ν i ,...,|ν i .
x∈X x P x1 P xt P
(cid:12) EQ (cid:12) EQ
(cid:12) (cid:12)
See Appendix A for the (cid:12)definition of |ν i w(cid:12)hen x is a set. There is a surjective function
x
θ : P −→ P
{0,1,...,s} {1,...,s}
θ(X) = x\{0}|x ∈ X and x 6= {0}
(cid:8) (cid:9)
and given a partition X = {x ,...,x } of {1,...,s},
1 t
θ−1(X) = {x ∪{0},x ,...,x },
1 2 t
n
{x ,x ∪{0},...,x },
1 2 t
...,
{x ,x ,...,x ,x ∪{0}}
1 2 t−1 t
{x ,x ,...,x ,{0}} .
1 2 t
o
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With this in mind we have, writing ⊗ |ν i for the list of |ν i as x′ ranges over
x′6=x x′ P x′ P
elements of X \{x}, that δD |ν ,...,ν i ⊗ν is equal to
1 s P
(cid:0) (cid:1)
|ν,ν i ,⊗ |ν i + |νi ,⊗ |ν i . (11)
x P x′6=x x′ P P x∈X x P
X nX(cid:12) EQ (cid:12) EQo
X∈P{1,...,s} x∈X(cid:12) (cid:12)
(cid:12) (cid:12)
Note that when X = {1,...,s} the summand is
(cid:8) (cid:9)
|ν,ν ,...,ν i + |νi ,|ν ,...,ν i .
1 s P P 1 s P
(cid:12) EQ (cid:12) EQ
(cid:12) (cid:12)
(cid:12) (cid:12)
On the other hand, the right hand side (8) of the (D.4) identity is
D (δ ⊗D)(∆⊗1) |ν ,...,ν i ⊗ν
!V 1 s P
(cid:0) (cid:1)
= D (δ ⊗D) |ν i ⊗|ν i ⊗ν
!V I P Ic P
X (cid:16) (cid:17)
I⊆{1,...,s}
= D δ|ν i ⊗|ν,ν i
!V I P Ic P
X (cid:16) (cid:17)
I⊆{1,...,s}
= D δ|∅i ⊗|ν,ν ,...,ν i
!V P 1 s P
(cid:16) (cid:17)
+ D ⊗ |ν i ⊗|ν,ν i
!V y∈Y y P Q Ic P
X∅⊂I YX∈PI (cid:16)(cid:12)(cid:12) (cid:11) (cid:17)
= |ν,ν ,...,ν i + |ν,ν i ,⊗ |ν i
1 s P Ic P y∈Y y P
(cid:12) EQ X X (cid:12) EQ
(cid:12) ∅⊂I Y∈PI(cid:12)
(cid:12) (cid:12)
= |ν,ν ,...,ν i + |νi ,⊗ |ν i
1 s P P y∈Y y P
(cid:12) EQ X (cid:12) EQ
(cid:12) Y∈P{1,...,s}(cid:12)
(cid:12) (cid:12)
+ |ν,ν i ,⊗ |ν i
Ic P y∈Y y P
X X (cid:12) EQ
∅⊂I⊂{1,...,s}Y∈PI(cid:12)
(cid:12)
which matches (11) since the last sum can be rewritten as
|ν,ν i ,⊗ |ν i .
x P x′6=x x′ P
X X(cid:12) EQ
X∈P{1,...,s}x∈X(cid:12)
with|X|>1 (cid:12)
Corollary 2.8. V is a differential category.
Proof. This follows from [2, Proposition 2.6].
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2.1 Codereliction, cocontraction, coweakening
An alternative formulation of the differential structure in differential linear logic is in
terms of codereliction, cocontraction and coweakening maps; see [10] and [1, §5.1]. This
has the advantage of providing an appealing symmetry to the formulation of the syntax.
In this section we briefly sketch the definition of these maps in the Sweedler semantics.
Throughout linear logic means intuitionistic linear logic with the connectives !,⊗,⊸.
First we recall the canonical commutative Hopf structure on !V of [31, §6.4]. Given
vector spaces V ,V then (see [31, Remark 2.19]) there is an isomorphism of coalgebras
1 2
Ψ : !V ⊗!V −→ !(V ⊕V ),
1 2 1 2
|ν ,...,ν i ⊗|ω ,...,ω i 7−→ |ν ,...,ν ,ω ,...,ω i .
1 s P 1 t Q 1 s 1 t (P,Q)
Using this and the definitions in [31], it is easy to check that the product ∇ is
∇ : !V ⊗!V −→ !V ,
|ν ,...,ν i ⊗|ω ,...,ω i 7−→ |ν ,...,ν ,ω ,...,ω i ,
1 s P 1 t Q 1 s 1 t P+Q
while the antipode S is
S : !V −→ !V ,
|ν ,...,ν i 7−→ |−ν ,...,−ν i
1 s P 1 s −P
and the unit u : k −→ !V is u(1) = |∅i . By [31, Theorem 6.4.8] these maps make !V into
0
a commutative (and cocommutative) Hopf algebra. In the terminology of [6] the map ∇
is the cocontraction map and u is the coweakening map (the antipode seems not to have
a formal role in differential linear logic). Finally,
Definition 2.9.The codereliction d¯is the composite
V ∼= V ⊗k 1⊗u // V ⊗!V D //!V
which is given by ν 7→ |νi .
0
Note that we can recover D as
!V ⊗V 1⊗d¯ //!V ⊗!V ∇ // !V
|ν ,...,ν i ⊗ν 7→ |ν ,...,ν i ⊗|νi 7→ |ν,ν ,...,ν i .
1 s P 1 s P 0 1 s P
It seems more convenient to model differentiation syntactically using the codereliction,
cocontraction and coweakening maps, rather than the deriving transformation D itself.
We briefly sketch how this works, following [6]. In the sequent calculus for linear logic one
introduces three new deduction rules “dual” to dereliction, contraction and weakening:
Γ,!A,∆ ⊢ B
(Codereliction): coder
Γ,A,∆ ⊢ B
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Γ,!A,∆ ⊢ B
(Cocontraction): coctr
Γ,!A,!A,∆ ⊢ B
Γ,!A,∆ ⊢ B
(Coweakening): coweak
Γ,∆ ⊢ B
together with new cut-elimination rules [6, §1.4.3].
Definition 2.10.Given a proof π of !A ⊢ B in linear logic, the derivative ∂π is the proof
π
.
.
.
!A ⊢ B (12)
coctr
!A,!A ⊢ B
coder
!A,A ⊢ B
whose denotation is, by our earlier remark, the composite
!JAK⊗JAK D // !JAK JπK //JBK. (13)
Remark 2.11.Given π as above we have the function [26, Definition 5.10]
JπK : JAK −→ JBK, P 7−→ JπK|∅i , (14)
nl P
and for P,ν ∈ JAK we interpret the vector
JπKD(|∅i ⊗ν) = JπK|νi ∈ JBK (15)
P P
as the derivative of JπK at the point P in the direction of ν. Here we implicitly identify
nl
JAK with the tangent space T JAK and JBK with the tangent space T JBK. This
P JπKnl(P)
interpretation is justified by the following elaboration of the remarks in the Introduction.
Let prom(π) denote the proofwhich is the promotionof π, which has for its denotation
the unique morphism of coalgebras Jprom(π)K : !JAK −→ !JBK with d◦Jprom(π)K = JπK.
Let Ψ : (k[ε]/ε2)∗ −→ !JAK be the morphism of coalgebras as in (27) corresponding to the
tangent vector ν at a point P ∈ JAK. Then the morphism of coalgebras
Jprom(π)K◦Ψ : (k[ε]/ε2)∗ −→ !JBK (16)
has the following values, writing Q = JπK (P), we have by [25, Theorem 2.22]
nl
Jprom(π)KΨ(1) = Jprom(π)K|∅i = |∅i ,
P Q
Jprom(π)KΨ(ε∗) = Jprom(π)K|νi = JπK|νi .
P P
(cid:12) EQ
(cid:12)
(cid:12)
Under the bijection of Section A.1 the morphism of coalgebras (16) therefore corresponds
to the tangent vector JπK|νi ∈ JBK at Q.
P
9
It is easy using the formulas for ∇,D to check that the ∇-rule of [2, §4.3] is satisfied:
Lemma 2.12. The diagram
V ⊗!V ⊗!V 1⊗∇ //V ⊗!V (17)
D⊗1 D
(cid:15)(cid:15) (cid:15)(cid:15)
!V ⊗!V //!V
∇
commutes.
This, together with [2, Theorem 4.12], shows that V with the comonad ! and deriving
transformation D is a model of the differential calculus in the sense of [2, Definition 4.11].
Corollary 2.13. V is a categorical model of the differential calculus.
3 Examples
In this section we give various examples of proofs π and the derivatives JπK◦D of their
denotations, according to Definition 2.10. We recall briefly the definition of the semantics
J−K in the category V of k-vector spaces from [19] and [26, §5.1, §5.3]. For a propositional
variable x the denotation JxK is any finite-dimensional vector space, and
JA ⊸ BK = Hom (JAK,JBK),
k
JA⊗BK = JAK⊗JBK,
J!AK = !JAK,
where !V denotes the universal cocommutative counital coalgebra mapping to V.
The encoding of integers and binary sequences in linear logic is based on the following
encoding of the composition rule.
Definition 3.1.For any formula A let C1 denote the proof
A
A ⊢ A A ⊢ A
A,A ⊸ A ⊢ A ⊸L
We define recursively for n > 1 a proof Cn of A,(A ⊸ A)n ⊢ A, where (A ⊸ A)n denotes
A
a sequence of n copies of A ⊸ A, to be
Cn−1
A
.
.
.
A ⊢ A A,(A ⊸ A)n−1 ⊢ A
A,(A ⊸ A)n ⊢ A ⊸L
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