Dialectica Categories Surprising Application:                        Cardinalities of the Continuum                       ...
OutlineOutline1   Surprising application?2   Dialectica Categories3   Category    PV4   Cardinal characteristics5   Severa...
Surprising application?IntroductionIm a logician, but set theory is denitely not my area of expertise.Some times getting o...
Surprising application?A little bit of personal history...Some twenty years ago I nished my PhD thesis The DialecticaCateg...
Surprising application?Dialectica categories came from Gödels DialecticaInterpretationThe interpretation is named after th...
Surprising application?Dialectica categories are models of Linear LogicLinear Logic was introduced by Girard (1987):linear...
Surprising application?The Challenges of Linear LogicThere is a problem with viewing proofs as logical objects:We do not h...
Surprising application?Dialectica Interpretation: motivations...For Gödel (in 1958) the interpretation was a way of provin...
Surprising application?Dialectica categories: useful for proving what...?Blass (1995) Dialectica categories as a tool for ...
Surprising application?Questions and Answers: The short storyBlass realized that my category for modelling Linear Logic wa...
Dialectica CategoriesDialectica Categories as Models of Linear LogicIn Linear Logic formulas denote resources.These resour...
Dialectica CategoriesLinear Implication and (Multiplicative) ConjunctionTraditional implication:         A, A → B         ...
Dialectica CategoriesThe challenges of modeling Linear LogicTraditional categorical modeling of intuitionistic logic:formu...
Dialectica CategoriesOriginal Dialectica CategoriesMy thesis has four chapters, four main denitions and four main theorems...
Dialectica CategoriesOriginal Dialectica CategoriesGiven the category            Sets   and usual functions, we dene:The c...
Dialectica CategoriesDialectica CategoriesTheorem 3 of my thesis (construction over any cartesian closed category,here we ...
Dialectica CategoriesDialectica Categories: Exponentials!A   must satisfy    !A →!A⊗!A, !A ⊗ B → I ⊗ B , !A → A           ...
Category   PVQuestions and Answers: Dial2                                  (Sets)       and        PVIs this simply a coin...
Category   PVQuestions and Answers: The category                    PVIn Category Theory morphisms are more important than...
Category   PVCan we give some intuition for these morphisms?Blass makes the case for thinking of problems in computational...
Category   PVWhich problems?A   decision problem is given by a set of instances of the problem, togetherwith a set of posi...
Category   PVComputational Complexity 101A   search problem is given by a set of instances of the problem, a set ofwitness...
Category   PVComputational Complexity 101A   many-one reduction from one search problem to another is a mapsending instanc...
Category   PVQuestions and AnswersObjects in    PV    are only certain objects of Dial2 (Sets), as they have tosatisfy the...
Category   PVQuestions and AnswersSimilarly the coproduct of        A   and     B    in Dial2 (Sets) is the object(U + V, ...
Category   PVThe structure of            PVNote that neither       T   or   0   are objects in      PV ,as they dont satis...
Category   PVThe structure of            PVBack to morphisms of            PV ,   the use that is made of the category in ...
Category   PVExamples of objects in                PV1. The object      (N, N, =)    where   n                   m        ...
Category   PVWhat about Dial2                (Sets)?Dial2 (Sets) is a category for categorical logic, we have:            ...
Cardinal characteristicsWhat have Set Theorists done with                                    PV ?Cardinal Characteristics ...
Cardinal characteristicsCardinals from AnalysisRecall the main denitions from Blass in Questions and Answers and hismain t...
Cardinal characteristicsCardinals from Analysis   - If   f   and   g   are functions       N → N, we say that f dominates ...
Cardinal characteristicsConnecting Cardinals to the CategoryBlass theorems : We have the following inequalities           ...
Cardinal characteristicsThe structure of the category                              PVGiven an object    A = (U, X, α) of P...
Cardinal characteristicsThe structure of            PVProposition[Rangel] The object (R, R, =) is maximal amongst objects ...
Cardinal characteristicsThe structure of             PVAxiom of Choice is essentialGiven cardinality fn        ϕ : U → R, ...
Cardinal characteristicsThe structure of            PVProposition[Rangel] The object (R, R, =) is minimal amongst objects ...
Cardinal characteristicsThe structure of               PVAxiom of Choice is essentialGiven cardinality fn        ϕ : X → R...
Cardinal characteristicsMore structure of Dial2                    (Sets)Given objects     A   and   B      of Dial2 (Sets...
Cardinal characteristicsTensor Products in Dial2                      (Sets)Given objects         A   and   B   of Dial2 (...
Cardinal characteristicsMore structure in Dial2                    (Sets)By reverse engineering from the desired adjunctio...
Cardinal characteristicsAn easy theorem of Dial2                        (Sets)/PV ...Because its fun, let us calculate tha...
Several years later...Back to Dialectica categoriesIt is pretty: produces models of Intuitionistic and classical linear lo...
Several years later...ConclusionsIntroduced you to dialectica categories Dial2 (Sets)/PV .Hinted at Blass and Votjá² use o...
Several years later...More Conclusions...Working in interdisciplinary areas is hard, but rewarding.The frontier between lo...
Several years later...ReferencesCategorical Semantics of Linear Logic for All, Manuscript.Dialectica and Chu constructions...
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Dialectica Categories Surprising Application: Cardinalities of the Continuum

  1. 1. Dialectica Categories Surprising Application: Cardinalities of the Continuum Valeria de Paiva 15th SLALM June, 2012Valeria de Paiva (15th SLALM) June, 2012 1 / 46
  2. 2. OutlineOutline1 Surprising application?2 Dialectica Categories3 Category PV4 Cardinal characteristics5 Several years later...Valeria de Paiva (15th SLALM) June, 2012 2 / 46
  3. 3. Surprising application?IntroductionIm a logician, but set theory is denitely 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 dierent 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 BottValeria de Paiva (15th SLALM) June, 2012 3 / 46
  4. 4. Surprising application?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.Valeria de Paiva (15th SLALM) June, 2012 4 / 46
  5. 5. Surprising application?Dialectica categories came from Gödels DialecticaInterpretationThe 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...Valeria de Paiva (15th SLALM) June, 2012 5 / 46
  6. 6. Surprising application?Dialectica categories are models of Linear LogicLinear 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 Freges distinction between sense and denotation:proofs are the senses of logical formulas,whose denotations might be truth values.Valeria de Paiva (15th SLALM) June, 2012 6 / 46
  7. 7. Surprising application?The Challenges of Linear LogicThere 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 dierences between derivationsthat do not correspond to dierences in the underlying proofs; it can alsomask dierences 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.Valeria de Paiva (15th SLALM) June, 2012 7 / 46
  8. 8. Surprising application?Dialectica Interpretation: motivations...For Gödel (in 1958) the interpretation was a way of proving consistency ofarithmetic. Aim: liberalized version of Hilberts programme to justifyclassical systems in terms of notions as intuitively clear as possible.Since Hilberts nitist methods are not enough (Gödels incompleteness theorem and experience with consistencyproofs) must admit some abstract notions. Gs approach: computable (or primitive recursive) functionals of nitetype.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.Valeria de Paiva (15th SLALM) June, 2012 8 / 46
  9. 9. Surprising application?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, ComplexityTheory, 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 CardinalCharacteristics of the Continuum.Valeria de Paiva (15th SLALM) June, 2012 9 / 46
  10. 10. Surprising application?Questions and Answers: The short storyBlass realized that my category for modelling Linear Logic was also used byPeter Votjá² for set theory, more specically for proving inequalitiesbetween cardinal invariants and wrote Questions and Answers A Category When we discussed the issue in 1994/95 I simply did not read the sections of theArising in Linear Logic, Complexity Theory, and Set Theory (1995).paper on Set Theory. or Computational Complexity.)(Two years ago I learnt from Samuel Gomes da Silva about his and CharlesMorgans work using Blass/Votjá² ideas and got interested in understanding theset-theoretical connections. This talk is about trying to understand theseconnections.Valeria de Paiva (15th SLALM) June, 2012 10 / 46
  11. 11. Dialectica CategoriesDialectica Categories as Models of Linear LogicIn Linear Logic formulas denote resources.These resources are premises, assumptions and conclusions, as they areused in logical proofs. For example: $1 −◦ latte If I have a dollar, I can get a Latte $1 −◦ cappuccino If I have a dollar, I can get a Cappuccino $1 I have a dollarCan conclude: Either: latte Or: cappuccino But not: latte ⊗ cappuccino I cant get Cappuccino and Latte with $1Valeria de Paiva (15th SLALM) June, 2012 11 / 46
  12. 12. Dialectica CategoriesLinear Implication and (Multiplicative) ConjunctionTraditional implication: A, A → B B A, A → B A∧B Re-use ALinear implication: A, A −◦ B B A, A −◦ B A⊗B Cannot re-use ATraditional conjunction: A∧B A Discard BLinear conjunction: A⊗B A Cannot discard BOf course: !A A⊗!A Re-use !(A) ⊗ B B DiscardValeria de Paiva (15th SLALM) June, 2012 12 / 46
  13. 13. Dialectica CategoriesThe challenges of modeling Linear LogicTraditional categorical modeling of intuitionistic logic:formula A object A of appropriate categoryA∧B A × B (cartesian product)A→B B A (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 dene the make-everything-usual operator !.Valeria de Paiva (15th SLALM) June, 2012 13 / 46
  14. 14. Dialectica CategoriesOriginal Dialectica CategoriesMy thesis has four chapters, four main denitions 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 simplication 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.Valeria de Paiva (15th SLALM) June, 2012 14 / 46
  15. 15. Dialectica CategoriesOriginal Dialectica CategoriesGiven the category Sets and usual functions, we dene: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 6 f ⇓ F ∀u ∈ U, ∀y ∈ Y α(u, F y) implies β(f u, y) ? V β Y 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
  16. 16. Dialectica CategoriesDialectica CategoriesTheorem 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 parconnective, units for the four monoidal structures and a linear functionspace, satisfying the appropriate categorical properties. The category ∗Dial2 (Sets) is symmetric monoidal closed with an involution that makes ita 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 dene two special comonadsand lots of work to prove the desired theorem...Valeria de Paiva (15th SLALM) June, 2012 16 / 46
  17. 17. Dialectica CategoriesDialectica 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 dene 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 dene comonads T and S on Dial2 (Sets) such that the Kleislicategory of their composite ! = T ; S , Dial2 (Sets)! is cartesian closed.Dene T A = (U, X, α) goes to (U, X ∗ , α∗ ) where X ∗ is the free by saying ∗commutative monoid on X and α is the multiset version of α. U U UDene S by saying A = (U, X, α) goes to (U, X , α ) where X is theset of functions from U into X . Then compose T and S .Loads of calculations prove that the linear logic modalitiy ! is well-denedand we obtain a model of classical linear logic. Construction generalized inmany ways, cf. dePaiva, TAC, 2006.Valeria de Paiva (15th SLALM) June, 2012 17 / 46
  18. 18. Category PVQuestions 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) Valeria de Paiva (15th SLALM) June, 2012 18 / 46
  19. 19. Category PVQuestions and Answers: The category PVIn Category Theory morphisms are more important than objects. A = (U, X, α) and B = (V, Y, β)Given objectsa map fromA to B in GSets is a pair of functions f : U → V andF : Y → X such that α(u, F y) implies β(f u, y).As before write maps as U α X 6 f ⇓ F ∀u ∈ U, ∀y ∈ Y α(u, F y) implies β(f u, y) ? V β YBut a map in PV satises the dual condition that is β(f u, y) → α(u, F y).Trust set-theorists to create morphisms A→B where the relations go inthe opposite direction!...Valeria de Paiva (15th SLALM) June, 2012 19 / 46
  20. 20. Category PVCan 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.Valeria de Paiva (15th SLALM) June, 2012 20 / 46
  21. 21. Category PVWhich 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 satisableformulas.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.Valeria de Paiva (15th SLALM) June, 2012 21 / 46
  22. 22. Category PVComputational Complexity 101A 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 satisability 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.Valeria de Paiva (15th SLALM) June, 2012 22 / 46
  23. 23. Category PVComputational Complexity 101A 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...Valeria de Paiva (15th SLALM) June, 2012 23 / 46
  24. 24. Category PVQuestions and AnswersObjects 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.Valeria de Paiva (15th SLALM) June, 2012 24 / 46
  25. 25. Category PVQuestions and AnswersSimilarly 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 dierent story.Valeria de Paiva (15th SLALM) June, 2012 25 / 46
  26. 26. Category PVThe structure of PVNote that neither T or 0 are objects in PV ,as they dont 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 satises the rsthalf of the MHD condition, but not the second. And the object⊥ = (1, 1, ¬id) satises neither of the halves.The constants of Linear Logic do not fare well in PV .Valeria de Paiva (15th SLALM) June, 2012 26 / 46
  27. 27. Category PVThe structure of PVBack 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 AValeria de Paiva (15th SLALM) June, 2012 27 / 46
  28. 28. Category PVExamples of objects in PV1. The object (N, N, =) where n m i n = m. is related toTo show MHD is satised 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, =) where = is usual equality in 2.Valeria de Paiva (15th SLALM) June, 2012 28 / 46
  29. 29. Category PVWhat about Dial2 (Sets)?Dial2 (Sets) is a category for categorical logic, we have: A ≤ B ⇐⇒ There is a morphism from A to BExamples 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).Valeria de Paiva (15th SLALM) June, 2012 29 / 46
  30. 30. Cardinal characteristicsWhat have Set Theorists done with PV ?Cardinal Characteristics of the ContinuumBlass: One of Set Theorys rst and most important contributions tomathematics was the distinction between dierent innite cardinalities,especially countable innity 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. Valeria de Paiva (15th SLALM) June, 2012 30 / 46
  31. 31. Cardinal characteristicsCardinals from AnalysisRecall the main denitions from Blass in Questions and Answers and hismain theorem: - IfX and Y are two subsets of N we say that X splits Y if both X ∩ Y and Y X are innite. - The splitting number s is the smallest cardinality of any family S of subsets of N such that every innite subset of N is split by some element 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 innite 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
  32. 32. Cardinal characteristicsCardinals from Analysis - If f and g are functions N → N, we say that f dominates g if for all except nitely many ns in N, f (n) ≤ g(n). - The dominating number d is the smallest cardinality of any family D contained 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 family B ⊆ NN such that no single g dominates all the members of B.Valeria de Paiva (15th SLALM) June, 2012 32 / 46
  33. 33. Cardinal characteristicsConnecting Cardinals to the CategoryBlass theorems : We have the following inequalities ω ≤ s ≤ d ≤ 2ω ω ≤ b ≤ r ≤ rσ ≤ 2ω b≤dThe 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...)Valeria de Paiva (15th SLALM) June, 2012 33 / 46
  34. 34. Cardinal characteristicsThe structure of the category PVGiven an object A = (U, X, α) of PV its norm ||A|| is the smallestcardinality of any set Z ⊆ X sucient 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.Havent done it, yet...Valeria de Paiva (15th SLALM) June, 2012 34 / 46
  35. 35. Cardinal characteristicsThe 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 6 ϕ ⇑ Φ ∀u ∈ U, ∀r ∈ R (ϕu = r) → α(u, Φr) ? R = RValeria de Paiva (15th SLALM) June, 2012 35 / 46
  36. 36. Cardinal characteristicsThe structure of PVAxiom of Choice is essentialGiven 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 6 ϕ ⇑ Φ ∀u ∈ U, ∀r ∈ R (ϕu = r) → α(u, Φr) ? R = RLet u r be such that ϕu = r and dene Φ as ψ ◦ f . andSince ϕu = r can apply Φ to both sides to obtain Φ(ϕ(u)) = Φ(r).Substituting Φs denition get f (ψ(ϕu))) = Φ(r). As ψ is left inverse of ϕ(ψ(ϕu) = u) get f u = Φ(r).Now the denition of f says for all u in U uαf u, which is uαΦr holds, asdesired. (Note that we did not need the cardinality function for X .) Valeria de Paiva (15th SLALM) June, 2012 36 / 46
  37. 37. Cardinal characteristicsThe structure of PVProposition[Rangel] The object (R, R, =) 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 = R 6 Φ ⇑ ϕ ∀r ∈ R, ∀x ∈ X α(Φr, x) → (r = ϕx) ? U α XNow 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.Valeria de Paiva (15th SLALM) June, 2012 37 / 46
  38. 38. Cardinal characteristicsThe structure of PVAxiom of Choice is essentialGiven 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 = R 6 Φ ⇑ ϕ ∀r ∈ R, ∀x ∈ X α(Φr, x) → ¬(ϕx = r) ? U α XExactly the same argument goes through. Let r and x be such thatϕx = r and dene Φ as ψ ◦ g .Since ϕx = r can apply Φ to both sides to obtain Φ(ϕ(x)) = Φ(r).Substituting Φs denition get g(ψ(ϕx))) = Φ(r). As ψ is left inverse of ϕ,(ψ(ϕx) = x) get gx = Φ(r).But the denition of g says for all x in X ¬gxαx, which is ¬α(Φr, x)holds. Not quite the desired, unless youre happy with RAA. Valeria de Paiva (15th SLALM) June, 2012 38 / 46
  39. 39. Cardinal characteristicsMore structure of Dial2 (Sets)Given objects A and B of Dial2 (Sets) we can consider their tensorproducts. Actually two dierent 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 satised.Valeria de Paiva (15th SLALM) June, 2012 39 / 46
  40. 40. Cardinal characteristicsTensor 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 from YY to X , X and we need to make sure that a pair f: U →V andF: Y →X in that set, satises our dialectica (or co-dialectica) condition: ∀u ∈ U, y ∈ Y, α(u, F y) ≤ β(f u, y) (respectively ≥)This give us an object (V U × X Y , U × Y, eval) where U Yeval : V × X × (U × Y ) → 2 is the map that evaluates f, F on the pairu, y and checks the implication between relations.Valeria de Paiva (15th SLALM) June, 2012 40 / 46
  41. 41. Cardinal characteristicsMore structure in Dial2 (Sets)By reverse engineering from the desired adjunction, we obtain the `righttensor product in the category.The tensor product of A and B is the object (U × V, Y U × X V , prod), Uwhere prod : (U × V ) × (Y × X V ) → 2 is the relation that rst evaluates Ua pair (ϕ, ψ) in Y × X V on pairs (u, v) and then checks the(co)-dialectica condition.Blass discusses a mixture of the two tensor products, which hasnt showedup in the work on Linear Logic, but which was apparently useful in SetTheory.Valeria de Paiva (15th SLALM) June, 2012 41 / 46
  42. 42. Cardinal characteristicsAn easy theorem of Dial2 (Sets)/PV ...Because its fun, let us calculate that the reverse engineering worked... A⊗B →C if and only if A → [B → C] U × V α ⊗ βX V × Y U U α X 6 6 f ⇓ (g1 , g2 ) ⇓ ? ? V W γ Z W × Y Zβ → γ V × ZValeria de Paiva (15th SLALM) June, 2012 42 / 46
  43. 43. Several years later...Back to Dialectica categoriesIt 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.Valeria de Paiva (15th SLALM) June, 2012 43 / 46
  44. 44. Several years later...ConclusionsIntroduced 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 MHDs work, nor versionsusing lineale=[0,1] interval...)Showed one easy, but essential, theorem in categorical logic, for fun...Uses for Categorical Proof Theory very dierent from uses in Set Theoryfor cardinal invariants.But I believe it is not a simple coincidence that dialectica categories areuseful in these disparate areas.Were starting our collaboration, so hopefully real new theorems willcome up.Valeria de Paiva (15th SLALM) June, 2012 44 / 46
  45. 45. Several years later...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 specic 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.Valeria de Paiva (15th SLALM) June, 2012 45 / 46
  46. 46. Several years later...ReferencesCategorical 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 CATS96,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.htmlA 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.htmlValeria de Paiva (15th SLALM) June, 2012 46 / 46

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