Dialectica Categories Surprising Application: Cardinalities of the Continuum

Dialectica Categories Surprising Application:Cardinalities of the Continuum

Valeria de Paiva

15th SLALM

June, 2012

Valeria de Paiva (15th SLALM) June, 2012 1 / 46

Outline

Outline

1 Surprising application?

2 Dialectica Categories

3 Category PV

4 Cardinal characteristics

5 Several years later...


Surprising application?

Introduction

I'm a logician, but set theory is de�nitely not my area of expertise.Some times getting out of you comfort zone is a good thing.

I reckon that mathematicians should try some more of it,especially with di�erent kinds of mathematics.

�There are two ways to do great mathematics. The �rst way is to

be smarter than everybody else. The second way is to be stupider

than everybody else � but persistent." � Raoul Bott



A little bit of personal history...

Some twenty years ago I �nished my PhD thesis "The DialecticaCategories" in Cambridge. My supervisor was Dr Martin Hyland.



Dialectica categories came from Gödel's DialecticaInterpretation

The interpretation is named after the Swiss journal Dialectica where itappeared in a special volume dedicated to Paul Bernays 70th birthday in1958.I was originally trying to provide an internal categorical model of theDialectica Interpretation. The categories I came up with proved (also) tobe a model of Linear Logic...



Dialectica categories are models of Linear Logic

Linear Logic was introduced by Girard (1987):�linear logic comes from aproof-theoretic analysis of usual logic.�the dualities of classical logic plus the constructive content of proofs ofintuitionistic logic.Programme to elevate the status of proofs to �rst class logical objects:instead of asking `when is a formula A true', we ask `what is a proof of A?'.Using Frege's distinction between sense and denotation:proofs are the senses of logical formulas,whose denotations might be truth values.



The Challenges of Linear Logic

There is a problem with viewing proofs as logical objects:We do not have direct access to proofs.Only to syntactic representations of them, in the form of derivations insome proof system (e.g. axiomatic, natural deduction or sequent calculus).

Syntactic derivations are �awed means of accessing the underlying proofobjects. The syntax can introduce spurious di�erences between derivationsthat do not correspond to di�erences in the underlying proofs; it can alsomask di�erences that really are there.

Semantics of proofs is what categorical logic is about.Categorical modeling of Linear Logic was a problem.Dialectica Categories are (one of the) good solutions.



Dialectica Interpretation: motivations...

For Gödel (in 1958) the interpretation was a way of proving consistency ofarithmetic. Aim: liberalized version of Hilbert's programme � to justifyclassical systems in terms of notions as intuitively clear as possible.Since Hilbert's �nitist methods are not enough (Gödel's incompleteness theorem and experience with consistency

proofs) must admit some abstract notions. G's approach: computable (or primitive recursive) functionals of �nite

type.

For me (in 1988) the interpretation was an interesting case study for(internal) categorical modelling, that turned out to produce models ofLinear Logic instead of models of Intuitionistic Logic, which were theexpected ones...

Blass (in 1995) realized that the Dialectica categories were a way ofconnecting work of Votjá² in Set Theory with mine and his own work onLinear Logic and cardinalities of the continuum.



Dialectica categories: useful for proving what...?

Blass (1995) Dialectica categories as a tool for proving inequalities betweennearly countable cardinals.Questions and Answers � A Category Arising in Linear Logic, Complexity

Theory, and Set Theory in Advances in Linear Logic (ed. J.-Y. Girard, Y.Lafont, and L. Regnier) London Math. Soc. Lecture Notes 222 (1995).Also quite a few other papers, including the survey Combinatory Cardinal

Characteristics of the Continuum.



Questions and Answers: The short story

Blass realized that my category for modelling Linear Logic was also used byPeter Votjá² for set theory, more speci�cally for proving inequalitiesbetween cardinal invariants and wrote Questions and Answers � A Category

Arising in Linear Logic, Complexity Theory, and Set Theory (1995).(When we discussed the issue in 1994/95 I simply did not read the sections of thepaper on Set Theory. or Computational Complexity.)

Two years ago I learnt from Samuel Gomes da Silva about his and CharlesMorgan's work using Blass/Votjá²' ideas and got interested in understanding theset-theoretical connections. This talk is about trying to understand theseconnections.


Dialectica Categories

Dialectica Categories as Models of Linear Logic

In Linear Logic formulas denote resources.These resources are premises, assumptions and conclusions, as they areused in logical proofs. For example:

$1 −◦ latteIf I have a dollar, I can get a Latte

$1 −◦ cappuccinoIf I have a dollar, I can get a Cappuccino

$1I have a dollar

Can conclude:� Either: latte� Or: cappuccino� But not: latte⊗ cappuccinoI can't get Cappuccino and Latte with $1



Linear Implication and (Multiplicative) Conjunction

Traditional implication: A,A→ B ` BA,A→ B ` A ∧B Re-use A

Linear implication: A,A −◦ B ` BA,A −◦ B 6` A⊗B Cannot re-use A

Traditional conjunction: A ∧B ` A Discard B

Linear conjunction: A⊗B 6` A Cannot discard B

Of course: !A ` A⊗!A Re-use

!(A)⊗B ` B Discard



The challenges of modeling Linear Logic

Traditional categorical modeling of intuitionistic logic:

formula A object A of appropriate categoryA ∧B A×B (cartesian product)A→ B BA (set of functions from A to B)

But these are real products, so we have projections (A×B → A)and diagonals (A→ A×A), which correspond to deletion and duplicationof resources.NOT linear!!!

Need to use tensor products and internal homs in Category Theory.

But ...hard to de�ne the "make-everything-usual" operator "!".



Original Dialectica Categories

My thesis has four chapters, four main de�nitions and four main theorems.

The �rst two chapters are about the �original" dialectica categories, modelsof the Dialectica Interpretation.

The second two chapters are about the Girard categories, a simpli�cation ofthe original dialectica categories, that are a better model for Linear Logic.These categories ended up being more used than the original ones andhence also called Dialectica categories. For the purposes of this talk weconcentrate on those.



Original Dialectica Categories

Given the category Sets and usual functions, we de�ne:The category Dial2(Sets) has as objects triples A = (U,X, α), where U,Xare sets and α is an ordinary relation between U and X. (so either u and xare α related, α(u, x) = 1 or not.)

A map from A = (U,X, α) to B = (V, Y, β) is a pair of functions (f, F ),where f : U → V and F : X → Y such that

U α X

⇓ ∀u ∈ U,∀y ∈ Y α(u, Fy) implies β(fu, y)

V

f

?β Y

F

6

or α(u, F (y)) ≤ β(f(u), y)

An object A is not symmetric: think of (U,X, α) as ∃u.∀x.α(u, x), aproposition in the image of the Dialectica interpretation.Valeria de Paiva (15th SLALM) June, 2012 15 / 46



Theorem 3 of my thesis (construction over any cartesian closed category,here we only use it for Sets) gives us:

Theorem (Model of Linear Logic without exponentials, 1988)

The category Dial2(Sets) has products, coproducts, tensor products, a par

connective, units for the four monoidal structures and a linear function

space, satisfying the appropriate categorical properties. The category

Dial2(Sets) is symmetric monoidal closed with an involution∗ that makes it

a model of classical linear logic without exponentials.

We need to describe the whole structure mentioned in the theorem, butthis is a faithful model of the Logic. We do not discard resources, nor dowe duplicate them.

How to get modalities/exponentials? Need to de�ne two special comonadsand lots of work to prove the desired theorem...



Dialectica Categories: Exponentials

!A must satisfy !A→!A⊗!A, !A⊗B → I ⊗B, !A→ A and !A→!!A,together with several equations relating them.The point is to de�ne a comonad such that its coalgebras are commutativecomonoids and the coalgebra and the comonoid structure interact nicely.

Theorem (Model of Linear Logic with exponentials, 1988)

We can de�ne comonads T and S on Dial2(Sets) such that the Kleisli

category of their composite ! = T ;S, Dial2(Sets)! is cartesian closed.

De�ne T by saying A = (U,X, α) goes to (U,X∗, α∗) where X∗ is the freecommutative monoid on X and α∗ is the multiset version of α.De�ne S by saying A = (U,X, α) goes to (U,XU , αU ) where XU is theset of functions from U into X. Then compose T and S.Loads of calculations prove that the linear logic modalitiy ! is well-de�nedand we obtain a model of classical linear logic. Construction generalized inmany ways, cf. dePaiva, TAC, 2006.


Category PV

Questions and Answers: Dial2(Sets) and PVIs this simply a coincidence?

The dialectica category Dial2(Sets) is the dual of GT the Galois-Tukeyconnections category in Votjá² work.Blass calls this category PV for Paiva and Votjá².The objects of PV are triples A = (U,X, α) where U,X are sets andα ⊆ U ×X is a binary relation, which we usually write as uαx or α(u, x).Blass writes it as (A−, A

+, A) but I get confused by the plus and minus signs.

Two conditions on objects in PV (not the case in Dial2(Sets)):1. U,X are sets of cardinality at most |R|.2. The condition in Moore, Hrusák and Dzamonja (MHD) holds:

∀u ∈ U∃x ∈ X such that α(u, x)

and∀x ∈ X∃u ∈ U such that ¬α(u, x)


Category PV

Questions and Answers: The category PVIn Category Theory morphisms are more important than objects.

Given objects A = (U,X, α) and B = (V, Y, β)a map from A to B in GSets is a pair of functions f : U → V andF : Y → X such that α(u, Fy) implies β(fu, y).

As before write maps as

U α X

⇓ ∀u ∈ U,∀y ∈ Y α(u, Fy) implies β(fu, y)

V

f

?β Y

F

6

But a map in PV satis�es the dual condition that is β(fu, y)→ α(u, Fy).Trust set-theorists to create morphisms A→ B where the relations go inthe opposite direction!...Valeria de Paiva (15th SLALM) June, 2012 19 / 46

Category PV

Can we give some intuition for these morphisms?

Blass makes the case for thinking of problems in computational complexity.Intuitively an object of Dial2(Sets) or PV

(U,X, α)

can be seen as representing a problem.The elements of U are instances of the problem, while the elements of Xare possible answers to the problem instances.The relation α say whether the answer is correct for that instance of theproblem or not.


Category PV

Which problems?

A decision problem is given by a set of instances of the problem, togetherwith a set of positive instances.The problem is to determine, given an instance whether it is a positive oneor not.Examples:1. Instances are graphs and positive instances are 3-colorable graphs;2. Instances are Boolean formulas and positive instances are satis�ableformulas.A many-one reduction from one decision problem to another is a mapsending instances of the former to instances of the latter, in such a waythat an instance of the former is positive if and only its image is positive.An algorithm computing a reduction plus an algorithm solving the latterdecision problem can be combined in an algorithm solving the former.


Category PV

Computational Complexity 101

A search problem is given by a set of instances of the problem, a set ofwitnesses and a binary relation between them.The problem is to �nd, given an instance, some witness related to it.The 3-way colorability decision problem (given a graph is it 3-waycolorable?) can be transformed into the 3-way coloring search problem:Given a graph �nd me one 3-way coloring of it.Examples:1. Instances are graphs, witnesses are 3-valued functions on the vertices ofthe graph and the binary relation relates a graph to its proper 3-waycolorings2. Instances are Boolean formulas, witnesses are truth assignments and thebinary relation is the satis�ability relation.Note that we can think of an object of PV as a search problem, the set ofinstances is the set U , the set of witnesses the set X.


Category PV

Computational Complexity 101

A many-one reduction from one search problem to another is a mapsending instances of the former to instances of the latter, in such a waythat an instance of the former is positive if and only its image is positive.An algorithm computing a reduction plus an algorithm solving the latterdecision problem can be combined in an algorithm solving the former.

We can think of morphisms in PV as reductions of problems.If this intuition is useful, great!!!If not, carry on thinking simply of sets and relations and a complicatednotion of how to map triples to other triples...


Category PV

Questions and Answers

Objects in PV are only certain objects of Dial2(Sets), as they have tosatisfy the two extra conditions. What happens to the structure of thecategory when we restrict ourselves to this subcategory?

Fact: Dial2(Sets) has products and coproducts, as well as a terminal andan initial object.

Given objects of Dial2(Sets), A = (U,X, α) and B = (V, Y, β)the product A×B in GSets is the object (U × V,X + Y, choice)where choice : U × V × (X + Y )→ 2 is the relation sending(u, v, (x, 0)) to α(u, x) and (u, v, (y, 1)) to β(v, y).

The terminal object is T = (1, 0, e) where e is the empty relation,e : 1× 0→ 2 on the empty set.


Category PV

Questions and Answers

Similarly the coproduct of A and B in Dial2(Sets) is the object(U + V,X × Y, choice)where choice : U + V × (X × Y )→ 2 is the relation sending((u, 0), x, y) to α(u, x) and ((v, 1), x, y) to β(v, y)

The initial object is 0 = (0, 1, e) where e : 0× 1→ 2 is the empty relation.

Now if the basic sets all U, V,X, Y have cardinality up to |R| then(co-)products will do the same.But the MHD condition is a di�erent story.


Category PV

The structure of PV

Note that neither T or 0 are objects in PV,as they don't satisfy the MHD condition.

∀u ∈ U,∃x ∈ X such that α(u, x) and ∀x ∈ X∃u ∈ U such that ¬α(u, x)

To satisfy the MHD condition neither U nor X can be empty.

Also the object I = (1, 1, id) is not an object of PV, as it satis�es the �rsthalf of the MHD condition, but not the second. And the object⊥ = (1, 1,¬id) satis�es neither of the halves.The constants of Linear Logic do not fare well in PV.


Category PV

The structure of PV

Back to morphisms of PV, the use that is made of the category in theset-theoretic applications is simply of the pre-order induced by themorphisms.

It is somewhat perverse that here, in contrast to usual categorical logic,

A ≤ B ⇐⇒ There is a morphism from B to A


Category PV

Examples of objects in PV

1. The object (N,N,=) where n is related to m i� n = m.To show MHD is satis�ed we need to know that ∀n ∈ N∃m ∈ N(n = m),can take m = n. But also that ∀m ∈ N∃k ∈ N such that ¬(m = k). Herewe can take k = suc(m).

2. The object (N,N,≤) where n is related to m i� n ≤ m.3. The object (R,R,=) where r1 and r2 are related i� r1 = r2, sameargument as 1 but equality of reals is logically much more complicated.4. The objects (2, 2,=) and (2, 2, 6=) where = is usual equality in 2.


Category PV

What about Dial2(Sets)?

Dial2(Sets) is a category for categorical logic, we have:

A ≤ B ⇐⇒ There is a morphism from A to B

Examples of objects in Dial2(Sets):�Truth-value" constants of Linear Logic as discussed T , 0, ⊥ and I.

All the PV objects are in Dial2(Sets). Components of objects such asU,X are not bound above by the cardinality of R.

Also the object 2 of Sets plays an important role in Dial2(Sets), as ourrelations α are maps into 2, but the objects of the form (2, 2, α) haveplayed no major role in Dial2(Sets).


Cardinal characteristics

What have Set Theorists done with PV?Cardinal Characteristics of the Continuum

Blass: �One of Set Theory's �rst and most important contributions tomathematics was the distinction between di�erent in�nite cardinalities,especially countable in�nity and non-countable one."Write N for the natural numbers and ω for the cardinality of N.Similarly R for the reals and 2ω for its cardinality.All the cardinal characteristics considered will be smaller or equal to thecardinality of the reals.They are of little interest if the Continuum Hypothesis holds, as then thereare no cardinalities between the cardinality of the integers ω and thecardinality of the reals 2ω.But if the continuum hypothesis does NOT hold there are many interestingconnections (in general in the form of inequalities) between variouscharacteristics that Votjá² discusses.



Cardinals from Analysis

Recall the main de�nitions from Blass in Questions and Answers and hismain �theorem":

- If X and Y are two subsets of N we say that X splits Y if bothX ∩ Y and Y \X are in�nite.

- The splitting number s is the smallest cardinality of any family S ofsubsets of N such that every in�nite subset of N is split by someelement of S.

Recall the Bolzano-Weierstrass Theorem: Any bounded sequence ofreal numbers (xn)n∈N has a convergent subsequence, (xn)n∈A.One can extend the theorem to say:For any countably bounded many sequences of real numbersxk = (xkn)n∈N there is a single in�nite set A ⊆ N such that thesubsequences indexed by A, (xkn)n∈A all converge.If one tries to extend the result for uncountably many sequences s above isthe �rst cardinal for which the analogous result fail.Valeria de Paiva (15th SLALM) June, 2012 31 / 46


Cardinals from Analysis

- If f and g are functions N→ N, we say that f dominates g if for allexcept �nitely many n's in N, f(n) ≤ g(n).

- The dominating number d is the smallest cardinality of any family Dcontained in NN such that every g in NN is dominated by some f in D.

- The bounding number b is the smallest cardinality of any familyB ⊆ NN such that no single g dominates all the members of B.



Connecting Cardinals to the Category

Blass �theorems� : We have the following inequalities

ω ≤ s ≤ d ≤ 2ω

ω ≤ b ≤ r ≤ rσ ≤ 2ω

b ≤ d

The proofs of these inequalities use the category of Galois-Tukeyconnections and the idea of a "norm" of an object of PV.(and a tiny bit of structure of the category...)



The structure of the category PV

Given an object A = (U,X, α) of PV its "norm" ||A|| is the smallestcardinality of any set Z ⊆ X su�cient to contain at least one correctanswer for every question in U .

Blass again: �It is an empirical fact that proofs between cardinalcharacteristics of the continuum usually proceed by representing thecharacteristics as norms of objects in PV and then exhibiting explicitmorphisms between those objects.�

One of the aims of a proposed collaboration with Gomes da Silva andMorgan is to explain this empirical fact, preferably using categorical tools.

Haven't done it, yet...



The structure of PVProposition[Rangel] The object (R,R,=) is maximal amongst objects ofPV.Given any object A = (U,X, α) of PV we know both U and X havecardinality small than |R|. In particular this means that there is an injectivefunction ϕ : U → R. (let ψ be its left inverse, i.e ψ(ϕu) = u)Since α is a relation α ⊆ U ×X over non-empty sets, if one accepts theAxiom of Choice, then for each such α there is a map f : U → X suchthat for all u in U , uαf(u).Need a map Φ: R→ X such that

U α X

⇑ ∀u ∈ U,∀r ∈ R (ϕu = r)→ α(u,Φr)

R

ϕ

?= R

Φ

6



The structure of PVAxiom of Choice is essential

Given cardinality fn ϕ : U → R, let ψ be its left inverse, ψ(ϕu) = u, for allu ∈ U . Need a map Φ: R→ X such that

U α X

⇑ ∀u ∈ U,∀r ∈ R (ϕu = r)→ α(u,Φr)

R

ϕ

?= R

Φ

6

Let u and r be such that ϕu = r and de�ne Φ as ψ ◦ f .Since ϕu = r can apply Φ to both sides to obtain Φ(ϕ(u)) = Φ(r).Substituting Φ's de�nition get f(ψ(ϕu))) = Φ(r). As ψ is left inverse of ϕ(ψ(ϕu) = u) get fu = Φ(r).Now the de�nition of f says for all u in U uαfu, which is uαΦr holds, asdesired. (Note that we did not need the cardinality function for X.)



The structure of PV

Proposition[Rangel] The object (R,R, 6=) is minimal amongst objects ofPV.This time we use the cardinality function ϕ : X → R for X. We want amap in PV of the shape:

R 6= R

⇑ ∀r ∈ R,∀x ∈ X α(Φr, x)→ (r 6= ϕx)

U

Φ

?α X

ϕ

6

Now using Choice again, given the relation α ⊆ U ×X we can �x afunction g : X → U such that for any x in X g(x) is such that ¬g(x)αx.



The structure of PVAxiom of Choice is essential

Given cardinality fn ϕ : X → R, let ψ : R→ X be its left inverse,ψ(ϕx) = x, for all x ∈ X. Need a map Φ: R→ U such that

R 6= R

⇑ ∀r ∈ R,∀x ∈ X α(Φr, x)→ ¬(ϕx = r)

U

Φ

?α X

ϕ

6

Exactly the same argument goes through. Let r and x be such thatϕx = r and de�ne Φ as ψ ◦ g.Since ϕx = r can apply Φ to both sides to obtain Φ(ϕ(x)) = Φ(r).Substituting Φ's de�nition get g(ψ(ϕx))) = Φ(r). As ψ is left inverse of ϕ,(ψ(ϕx) = x) get gx = Φ(r).But the de�nition of g says for all x in X ¬gxαx, which is ¬α(Φr, x)holds. Not quite the desired, unless you're happy with RAA.Valeria de Paiva (15th SLALM) June, 2012 38 / 46


More structure of Dial2(Sets)

Given objects A and B of Dial2(Sets) we can consider their tensorproducts. Actually two di�erent notions of tensor products were consideredin Dial2(Sets), but only one has an associated internal-hom.(Blass considered also a mixture of the two tensor products of Dial2(Sets),that he calls a sequential tensor product.)Having a tensor product with associated internal hom means that we havean equation like:

A⊗B → C ⇐⇒ A→ (B → C)

Can we do the same for PV? Would it be useful?The point is to check that the extra conditions on PV objects are satis�ed.



Tensor Products in Dial2(Sets)

Given objects A and B of Dial2(Sets) we can consider a preliminary tensorproduct, which simply takes products in each coordinate. Write this asA�B = (U × V,X × Y, α× β) This is an intuitive construction toperform, but it does not provide us with an adjunction.To "internalize" the notion of map between problems, we need to considerthe collection of all maps from U to V , V U , the collection of all maps fromY to X, XY and we need to make sure that a pair f : U → V andF : Y → X in that set, satis�es our dialectica (or co-dialectica) condition:

∀u ∈ U, y ∈ Y, α(u, Fy) ≤ β(fu, y) (respectively ≥)

This give us an object (V U ×XY , U × Y, eval) whereeval : V U ×XY × (U × Y )→ 2 is the map that evaluates f, F on the pairu, y and checks the implication between relations.



More structure in Dial2(Sets)

By reverse engineering from the desired adjunction, we obtain the `right'tensor product in the category.The tensor product of A and B is the object (U × V, Y U ×XV , prod),where prod : (U × V )× (Y U ×XV )→ 2 is the relation that �rst evaluatesa pair (ϕ,ψ) in Y U ×XV on pairs (u, v) and then checks the(co)-dialectica condition.Blass discusses a mixture of the two tensor products, which hasn't showedup in the work on Linear Logic, but which was apparently useful in SetTheory.



An easy theorem of Dial2(Sets)/PV ...

Because it's fun, let us calculate that the reverse engineering worked...

A⊗B → C if and only if A→ [B → C]

U × V α⊗ βXV × Y U U α X

⇓ ⇓

W

f

?γ Z

(g1, g2)

6

W V × Y Z?

β → γ V × Z

6


Several years later...

Back to Dialectica categories

It is pretty: produces models of Intuitionistic and classical linear logic andspecial connectives that allow us to get back to usual logic.

Extended it in a number of directions:a robust proof can be pushed in many ways...

used in CS as a model of Petri nets (more than 3 phds),it has a non-commutative version for Lambek calculus (linguistics),it has been used as a model of state (with Correa and Hausler, Reddy ind.)Also in Categorical Logic: generic models (with Schalk04) of Linear Logic,Dialectica interp of Linear Logic and Games (Shirahata and Oliva et al)�brational versions (B. Biering and P. Hofstra).Most recently and exciting:Formalization of partial compilers: correctness of MapReduce in MS .netframeworks via DryadLINQ, �The Compiler Forest", M. Budiu, J. Galenson,G. Plotkin 2011.



Conclusions

Introduced you to dialectica categories Dial2(Sets)/PV.Hinted at Blass and Votjá² use of them for mapping cardinal invariants.(did not discuss parametrized diamond principles MHD's work, nor versionsusing lineale=[0,1] interval...)

Showed one easy, but essential, theorem in categorical logic, for fun...Uses for Categorical Proof Theory very di�erent from uses in Set Theoryfor cardinal invariants.

But I believe it is not a simple coincidence that dialectica categories areuseful in these disparate areas.We're starting our collaboration, so hopefully real "new" theorems willcome up.



More Conclusions...

Working in interdisciplinary areas is hard, but rewarding.The frontier between logic, computing, linguistics and categories is a funplace to be.Mathematics teaches you a way of thinking, more than speci�c theorems.Fall in love with your ideas and enjoy talking to many about them...

Many thanks to Andreas Blass, Charles Morgan and Samuel Gomes daSilva for mentioning my work in connection to their work on set theory...and thanks Alexander Berenstein and the organizers of SLALM for invitingme here.



References

Categorical Semantics of Linear Logic for All, Manuscript.Dialectica and Chu constructions: Cousins?Theory and Applications ofCategories, Vol. 17, 2006, No. 7, pp 127-152.A Dialectica Model of State. (with Correa and Haeusler). In CATS'96,Melbourne, Australia, Jan 1996.Full Intuitionistic Linear Logic (extended abstract). (with Martin Hyland).Annals of Pure and Applied Logic, 64(3), pp.273-291, 1993.Thesis TR: The Dialectica Categorieshttp://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 389The Dialectica Categories. In Proc of Categories in Computer Science andLogic, Boulder, CO, 1987. Contemporary Mathematics, vol 92, AmericanMathematical Society, 1989all available from http://www.cs.bham.ac.uk/~vdp/publications/papers.html


http://www.cs.bham.ac.uk/~vdp/publications/papers.html

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Dialectica Categories Surprising Application: Cardinalities of the Continuum