Outline
Motivation
NNOs
Conclusions

Motivation
NNOs
Conclusions



Outline
Motivation
NNOs
Conclusions

Linear Logic perspective into recursion and iteration?

Linear Logic can be seen as magnifying lens to understand
logic



Outline
Motivation
NNOs
Conclusions


logic
Decomposing implication via Girard’s translation
A → B :=!A−◦B gives new insights on computational
phenomena



Outline
Motivation
NNOs
Conclusions


logic
Decomposing implication via Girard’s translation
A → B :=!A−◦B gives new insights on computational
phenomena
Want to use linear perspective for iteration and recursion



Outline
Motivation
NNOs
Conclusions

Linear Recursion and Iteration?

Iteration and recursion in intuitionistic logic done via PCF



Outline
Motivation
NNOs
Conclusions


the mother of all programming languages



Outline
Motivation
NNOs
Conclusions


the mother of all programming languages
and its denotational models



Outline
Motivation
NNOs
Conclusions

Categorical Models of PCF?

Cartesian closed category



Outline
Motivation
NNOs
Conclusions


with booleans, NNO and ﬁxpoints



Outline
Motivation
NNOs
Conclusions


Want to start with Linear PCF and linear natural number
objects...



Outline
Motivation
NNOs
Conclusions


Want to start with Linear PCF and linear natural number
objects...
This talk: linear natural number objects in a speciﬁc model of
Linear Logic, Dialectica categories



Outline
Motivation
NNOs
Conclusions

What are Natural Numbers Objects?

Lawvere’s way of modelling the natural numbers with Peano’s
Induction



Outline
Motivation
NNOs
Conclusions


Induction
CCC with NNO ⇒ all primitive recursive functions.



Outline
Motivation
NNOs
Conclusions


Induction
A Natural Numbers Object (or NNO) is an object in a
category equipped with structure giving it properties similar to
those of the set of natural numbers N in the category Sets.



Outline
Motivation
NNOs
Conclusions


Induction
A Natural Numbers Object (or NNO) is an object in a
category equipped with structure giving it properties similar to
those of the set of natural numbers N in the category Sets.
Want to linearize this setting...



Outline
Motivation
NNOs
Conclusions

A NNO in a CCC C consists of an object N of C together with two
morphisms, zero : 1 → N and a successor mapping succ : N → N.
The triple (N, zero, succ) is required to satisfy the condition that,
given any pair of morphisms f : 1 → B and g : B → B in C, there
exists a unique h : N → B such that the following diagram
commutes.
1

zero E
N
d
d
d
f d
‚
d

h

h

c

B

succ E
N

g

c
E B


Outline
Motivation
NNOs
Conclusions

Natural Numbers Objects in Sets
A NNO in Sets consists of the object N in Sets together with two
morphisms, zero : 1 → N (where we really choose the element 0 of
the natural numbers N, and the successor mapping succ : N → N is
really + 1 : N → N.
1

zero E
N
d
d
d
f d
‚
d

h

h

c

B


succ E
N

g

c
E B


Outline
Motivation
NNOs
Conclusions

Natural Numbers Objects in Sets
Given f : 1 −→ X (or f (∗) = x0 ) and g : X −→ X we have the
map h given by h(n) = g n (f (∗)) makes the following diagram
commute and is the unique map doing so.
1

zero E
N

+1 E
N

d
d
h
d
f d
‚
d c

X


h

g

c
E X


Outline
Motivation
NNOs
Conclusions

Natural Numbers Objects in a SMCC?
Par´ and Rom´n (thinking of Linear Logic) have the following:
e
a
Given a monoidal category C a NNO is an object N of C and
morphisms zero : I −→ N and succ : N −→ N such that for any
object B of C and morphisms b : I −→ B and g : B −→ B there is
a morphism h : N −→ B such that the diagrams below commute:
I

zero E
N
d
d
d
b d
‚
d

succ E
N

h

h

c

B

g

c
E B

Can we calculate with it?


Outline
Motivation
NNOs
Conclusions


Conceived as an internal model of G¨del’s Dialectica
o
Interpretation, turned out to be also a model of Linear Logic.
Objects of the Dialectica category Dial2 (Sets) are triples,
A = (U, X , R), where U and X are sets and R ⊆ U × X is a
relation.
Given elements u in U and x in X , either they are related by R,
R(u, x) = 1 or they are not and R(u, x) = 0, hence the 2 in the
name of the category.



Outline
Motivation
NNOs
Conclusions

A morphism from A to B = (V , Y , S) is a pair of functions
f : U −→ V and F : Y −→ X such that uRF (y ) =⇒ f (u)Sy .
R
u∈U '

X
T

⇓

f
c

V '


S

F
Y

y


Outline
Motivation
NNOs
Conclusions

Tensor in Dialectica Categories
Let A = (U, X , R) and B = (V , Y , S) be objects in Dial2 (Sets).
The tensor product of A and B is given by
A ⊗ B = (U × V , X V × Y U , R ⊗ S)
where the relation R ⊗ S is given by (u, v ) R ⊗ S (f , g ) iﬀ uRf (v )
and vSg (u).
The unit for this tensor product is the object IDial := (1, 1, =),
where 1 = {∗} is a singleton set and = is the identity relation on
the singleton set.



Outline
Motivation
NNOs
Conclusions

Symmetric Monoidal Closed Dialectica Categories
The internal-hom is given by
[A, B] = (V U × X Y , U × Y , [R, S])
where (f , F )[R, S](u, x) iﬀ uRF (y ) implies f (u)Sy .
The tensor product is adjoint to the internal-hom, as usual
HomDial (A ⊗ B, C ) ∼ HomDial (A, [B, C ])
=



Outline
Motivation
NNOs
Conclusions

Other structure of Dialectica Categories
There is an auxiliary tensor product structure given by
A ◦ B = (U × V , X × Y , R ◦ S)
where (u, v )R ◦ S(x, y ) iﬀ uRx and vSy .
This simpler tensor structure is not the adjoint of the internal-hom.
It’s necessary to prove existence of appropriate modality !.
The unit for this tensor product is also IDial .



Outline
Motivation
NNOs
Conclusions

Cartesian structure of Dialectica Categories

The cartesian product is given by A × B = (U × V , X + Y , ch)
where X + Y = X × 0 ∪ Y × 1 and the relation ch (short for
‘choose’) is given by (u, v )ch(x, 0) if uRx and (u, v )ch(y , 1) if vSy .
The unit for this product is (1, ∅, ∅), the terminal object of
Dial2 (Sets). (there are also cartesian coproducts.)



Outline
Motivation
NNOs
Conclusions

NNOs in Dialectica Categories?

To investigate iteration and recursion in linear categories we want
to deﬁne a natural numbers object in Dial2 (Sets).
We can use either the cartesian structure of Dial2 (Sets) or any one
of its tensor structures.
The ﬁrst candidate monoidal structure is the cartesian product in
Dial2 (Sets).
This requires a map corresponding to zero from the terminal object
(1, ∅, ∅) in Dial2 (Sets) to our natural numbers object candidate,
say a generic object like (N, M, E )



Outline
Motivation
NNOs
Conclusions

NNOs in Dialectica Categories?
Reading the deﬁnition, (N, M, E ) is a NNO with respect to the
cartesian structure of Dial2 (Sets)if there are maps
(z, Z ) : (1, ∅, ∅) −→ (N, M, E ) and
(s, S) : (N, M, E ) −→ (N, M, E ) such that for any object
(X , Y , R) and any pair of morphisms
(f , F ) : (1, ∅, ∅) −→ (X , Y , R) and
(g , G ) : (X , Y , R) −→ (X , Y , R) there exists some (unique)
(h, H) : (N, M, E ) −→ (X , Y , R) such that a big diagram
commutes.
Which big diagram?



Outline
Motivation
NNOs
Conclusions

NNO using cartesian structure
∅
∅

1

z

Z

d
sF
d
E

d
d
d
f d
d
‚
d

S

M

'
E

H

T

E

N

s

E

Y

G

'

R
c
X


H

N
h

d
d
h

M

'
T

Y

R
c
g

E

X


Outline
Motivation
NNOs
Conclusions

Proposition 1. The category Dial2 (Sets) has a (trivial) NNO with
respect to its cartesian structure, given by (N, ∅, ∅).
Proof sketch: any possible NNO for Dial2 (Sets) is of the form
N = (N, M, E ) for some set M and some relation E ⊆ N × M,
where N is the usual natural numbers object in Sets, with the
usual zero constant and the usual successor functions.
There must exist a morphism in Dial2 (Sets) zero = (z, Z ) : 1 → N
with two components, z : 1 → N (as in Sets) and Z : M → 0. But
since the only map into the empty set in the category of Sets is
the empty map, we conclude that M is empty and so is E as this is
a relation in the product N × ∅.


Outline
Motivation
NNOs
Conclusions

This trivial NNO works, because given any object B of Dial2 (Sets)
with maps f : 1 → B and g : B → B, we can ﬁnd a unique map
h : N → B making all the necessary diagrams commute: In the
ﬁrst coordinate h is given by the map that exists for N as a NNO
in Sets and in the second coordinate this is simply the empty map.
Not very exciting...
This triviality result is expected, since the ‘main’ structure of the
category Dial2 (Sets) is the tensor product that makes it a
symmetric monoidal closed category, not its cartesian structure.



Outline
Motivation
NNOs
Conclusions

NNO using monoidal structure
[Burroni] Peano-Lawvere axiom’s says that for any object X in a
zX E
sX E
category E , there is a diagram X
NX
NX
with the universal property that for any diagram of the form
f E
g E
X
Y
Y there exists h : NX → Y such that
the following diagram commutes
zX E
X
NX

sX E
NX

d
d
h
d
f d
‚
d c

Y

h

g

c
E Y


Outline
Motivation
NNOs
Conclusions

If a category satisﬁes this axiom, we say the category is a
Peano-Lawvere (PL) category.
Dial2 (Sets) is a PL-category if given any object (A, B, C ) of
Dial2 (Sets) there is an object (N, M, E ) and maps
(z, Z ) : (A, B, C ) −→ (N, M, E ) and
(s, S) : (N, M, E ) −→ (N, M, E ) such that for any object of
Dial2 (Sets) (X , Y , R) together with a pair of morphisms
(f , F ) : (A, B, C ) −→ (X , Y , R) and
(g , G ) : (X , Y , R) −→ (X , Y , R) there exists some (unique) map
in Dial2 (Sets) (h, H) : (N, M, E ) −→ (X , Y , R) making the big
diagram commute.


Outline
Motivation
NNOs
Conclusions

B
C

A

z

Z

d
sF
d
E

d
d
d
f d
d
‚
d

S

M

'
E

H

T

E

N

s

E

Y


H

G

'

R
c
X

N
h

d
d
h

M

'
T

Y

R
c
g

E

X


Outline
Motivation
NNOs
Conclusions

Simplifying this picture for the case where the unit (1, 1, =) is
used. If N = (N, M, E ) is a proposed NNO in Dial2 (Sets) then
there must exist morphisms zero = (z, Z ) : (1, 1, =) → (N, M, E )
and succ = (s, S) : (N, M, E ) → (N, M, E ) such that:
1
=

1

z

Z

d
sF
d
E

d
d
d
f d
d
‚
d

S

M

'
E

H

T

E

N

s

E

Y

X

H

G

'

R
c


N
h

d
d
h

M

'
T

Y

R
c
X
g

E

Outline
Motivation
NNOs
Conclusions

Proposition 2. The category Dial2 (Sets) has a (trivial) weak
NNO with respect to its monoidal closed structure described, given
by (N, 1, N × 1).
As before, any possible NNO for Dial2 (Sets) is of the form
N = (N, M, E ) for some set M and some relation E ⊆ N × M,
where N is the usual natural numbers object in Sets, with the usual
zero constant and the usual successor function on natural numbers.
The morphism zero has two components, z : 1 → N (as in Sets)
and Z : M → 1. The map Z has to be the unique map !M : M → 1
sending all m’s in M to the singleton set ∗



Outline
Motivation
NNOs
Conclusions

The morphism succ : N → N has two components (s, S), where
s : N → N is the usual successor function in N, and S : M → M is
to be determined, satisfying some equations.
Fact 1. If there is a map (f , F ) : I → B in Dial2 (Sets) for a
generic object B of the form (X , Y , R) then there exists x0 in X
such for all y in Y we have x0 Ry .
By deﬁnition of maps in Dial2 (Sets), we must have
=
1'
f
c
X '

1
T
F =!

R


Y

Outline
Motivation
NNOs
Conclusions


Fact 2. If there is a NNO in Dial2 (Sets) of the form (N, M, E ),
where M is the singleton set 1, then S : 1 → 1 is the identity on 1
and E relates every n in N to ∗.
If N = (N, 1, E ) is a NNO in Dial2 (Sets) then the map
zero = (z, Z ) : I → N has to be the zero map in N together with
the terminal map in 1 and the succ = (s, S) : N → N consists of
the usual successor function on the integers and S : 1 → 1 has to
be the identity on 1.



Outline
Motivation
NNOs
Conclusions

The fact that (s, S) is map of Dial2 (Sets) gives us the diagram
E
N'

1
T

S

s
c

N'

E

1

and the condition on morphisms says for all n in N and for all ∗ in
1, if nES∗ then n + 1 = s(n)E ∗. But S is the identity on 1, ie
S∗ = ∗, so if nES∗ ⇒ n + 1E ∗, which is just what we need to
prove that E relates every n in N to ∗.


Outline
Motivation
NNOs
Conclusions

Back to the proposition 2: The object of Dial2 (Sets) of the form
(N, 1, E ), where E relates every n in N to ∗, together with
morphisms zero = (0, id1 ) : I → N and succ = (+1, id1 ) : N → N
is a weak NNO in Dial2 (Sets).
Proof. Let B be an object (X , Y , R) of Dial2 (Sets) such that
there are maps (f , F ) : I → B and (g , G ) : B → B. To prove
N = (N, 1, E ), where nE ∗ for all n in N is a weak NNO, we must
define a map (h, H) : N → B such that the main NNO diagram
commutes. It is clear that h : N → X can be defined using the fact
that N is a NNO in Sets. It is clear that we must take H : Y → 1
as the terminal map on Y . We need to check that all the required
conditions are satisfied.


Outline
Motivation
NNOs
Conclusions

The
1.
2.
3.

required conditions amount to showing that
the map (h, H) is a map of Dial2 (Sets);
the triangles commute, and
the squares commute in the diagram below.
1

'

=

1

z =0

Z = id1

1

d
sF =!Y E
d

E N

d
d
d
f d
d
‚
d

H

'
T

S = id1

T

E

s = +1

E

Y


H =!Y

G

'

R
c
X

N
h

d
d
h

1

Y

R
c
g

E

X


Outline
Motivation
NNOs
Conclusions


Items 2 and 3 are boring, but easy.
Need to show that the proposed map (h, H) is indeed a map in
Dial2 (Sets).
The map (h, H) is a map in Dial2 (Sets) if the condition
for all m in N, for all y in Y , if mEH(y ) then h(m)Ry
is satisﬁed.
Since H(y ) = ∗ and we know mE ∗ for all m in N, we need to show
h(m)Ry for all m ∈ N and all y in Y . (vertical square)
By cases, either m = 0 or not.



Outline
Motivation
NNOs
Conclusions

If m = 0 we need to show h(0)Ry for all y ∈ Y . Since N is the
NNO in Sets we know that
1

0

+1

E
d
fd
‚

E

N

N

h

h

c

c

X

g

E

X

commutes, hence h(0) = f (∗) and h(m + 1) = g (h(m)). Since
B = (X , Y , R) is an object that has a map (f , F ) : I → B we know
(Fact 1) that there exists x0 = f (∗) such that f (∗)Ry for all y in
Y and hence h(0) = f (∗)Ry for all y ∈ Y .



Outline
Motivation
NNOs
Conclusions

If m is not zero, then m = n + 1 and h(n + 1) = g (h(n)) by the
deﬁnition of h in Sets. But B is an object of Dial2 (Sets) equipped
with a map (g , G ) : B → B, which means that there exist
g : X → X and G : Y → Y in Sets such that for all x ∈ X and for
all y ∈ Y , if xRG (y ) then g (x)Ry . To show that h(n)Ry , since we
know that h(0)Ry we need to show that if h(n)Ry for all y ∈ Y
then h(n + 1)Ry for all y ∈ Y .
But if h(n)Ry for all y ∈ Y , then in particular h(n)RG (y ) for all
y ’s that happen to be in the range of G , that is if y happens to be
Gy . In this case g (h(n))Ry , that is h(n + 1)Ry .



Outline
Motivation
NNOs
Conclusions

Summing up
We obtained a degenerate weak NNO, where in the first coordinate
we have business as usual in Sets and in the second coordinate we
have simply the singleton set 1 and terminal maps.
We expected to find a NNO in the dialectica categories, with
iteration and recursion as usual in the first coordinate, but
co-recursion/co-iteration in the second coordinate.
It is disappointing to obtain only a ‘degenerate’ NNO as above,
where the second coordinate is trivial. Maybe we have not got the
right level of generality...



Outline
Motivation
NNOs
Conclusions

Summing up
Natural number algebras 1 → N ← N are in bijective
correspondence with F -algebras (for F the endofunctor
F (X ) = 1 + X ) . The initial algebra for this functor in Set is
indeed the usual natural numbers, where we have an isomoprhism
N ∼ 1 + N. Since this is an isomorphism we could also see it as an
=
F -coalgebra, but this is not ﬁnal in the category of sets. As
Plotkin remarks this coalgebra is ﬁnal in the category of sets and
partial functions Pfn.
Can we change our working underlying category of Dial2 (Sets) so
that a non-trivial NNO can be constructed?



Outline
Motivation
NNOs
Conclusions

Some references
Sandra Alves, Maribel Fernandez, Mario Florido, and Ian
Mackie. Linear recursive functions. In Rewriting,
Computation and Proof, pages 182195. Springer, 2007.
Dialectica and Chu constructions: Cousins?Theory and
Applications of Categories, Vol. 17, 2006, No. 7, pp 127-152.
Thesis TR: The Dialectica Categories
http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.html

A Dialectica-like Model of Linear Logic.In Proceedings of
CTCS, Manchester, UK, September 1989. LNCS 389 .
The Dialectica Categories. In Proc of Categories in Computer
Science and Logic, Boulder, CO, 1987. Contemporary
Mathematics, vol 92, American Mathematical Society, 1989


Outline
Motivation
NNOs
Conclusions

Thanks!



Natural Number Objects in Dialectica Categories

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