edwardian proofs as futuristic programs for personal assistants

IntroductionEdwardian Proofs

Futuristic ProgramsCategorical Proof Theory

Back to the future: Linear Logic

Edwardian Proofs as Futuristic Programsfor Personal Assistants

Valeria de PaivaNuance Communications, CA

May, 2014

Valeria de Paiva ASL 2014 – Boulder, CO




Thanks!...





Introduction

I’m a logician, a proof-theorist and a category theorist.I work in industry, have done so for the last 15 years, applying thepurest of pure mathematics, in surprising ways.Today I want to show you what I think is a most under-appreciatedpiece of mathematics on the 20th century.

The Curry-Howard Correspondence

Categorical Proof Theory

(as much as time permits) my small part on that...





Mathematics is full of surprises...

It often happens that there are similarities between thesolutions to problems. Sometimes, these similarities pointto more general phenomena that simultaneously explainseveral different pieces of mathematics. These moregeneral phenomena can be very difficult to discover, butwhen they are discovered, they have a very importantsimplifying and organizing role, and can lead to thesolutions of further problems, or raise new andfascinating questions. – T. Gowers, The Importance of Mathematics, 2000





Proofs are Programs?

The bulk of mathematics today got crystallized in the last years ofthe 19th century, first years of the 20th century.

The shock is still being felt. A Revolution in Mathematics? WhatReally Happened a Century Ago and Why It Matters Today FrankQuinn (Notices of the AMS, Jan 2012)Today: the relationship between Algebra, Proofs and Programs





Birth of Algebra

[...] a fundamental shift occurred in mathematics fromabout 1880 to 1940–the consideration of a wide varietyof mathematical ”structures,”defined axiomatically andstudied both individually and as the classes of structuressatisfying those axioms. This approach is so commonnow that it is almost superfluous to mention it explicitly,but it represented a major conceptual shift in answeringthe question: What is mathematics?

The axiomatization of Linear Algebra, Moore, Historia Mathematica, 1995.





Edwardian Algebra

Bourbaki on Abstract Algebra

The axiomatization of algebra was begun by Dedekindand Hilbert, and then vigorously pursued by Steinitz(1910). It was then completed in the years following1920 by Artin, Noether and their colleagues at Gottingen(Hasse, Krull, Schreier, van der Waerden). It waspresented to the world in complete form by van derWaerden’s book (1930).





Bourbaki didn’t say: Algebra became Category Theory...

Category Theory: there’s an underlying unity of mathematicalconcepts/theories.More important than the mathematical concepts themselves is howthey relate to each other.Topological spaces come with continuous maps, while vectorspaces come with linear transformations.Morphisms: how structures transform into others in the (mostreasonable) way to organize the mathematical edifice.Abstract Nonsense...





Edwardian Proofs

Frege: one of the founders of modern symbolic logic put forwardthe view that mathematics is reducible to logic.Begriffsschrift, 1879Was the first to write proofs using a collection of abstract symbols:instead of B → A and B hence A





Why Proofs?

Mathematics in turmoil in the turn of the century because ofparadoxes e.g. Russell’s ParadoxHilbert’s Program: Base all of mathematics in finitistic methodsProving the consistency of Arithmetic: the big questRead the graphic novel!!





Edwardian Turmoil...

Hilbert’s program:provide secure foundations for all mathematics.How? Formalization all mathematical statements should be writtenin a precise formal language, and manipulated according to welldefined rules.There is no ignorabimus in mathematics.. .Sounds good, doesn’t it?





Hilbert’s Program

Consistent: no contradiction can be obtained in the formalism ofmathematics.Complete: all true mathematical statements can be proven in theformalism. Consistency proof use only “finitistic”reasoning about finite mathematical objects.

Conservative: any result about “real objects”obtained usingreasoning about “ideal objects”(such as uncountable sets) can beproved without ideal objects.Decidable: an algorithm for deciding the truth or falsity of anymathematical statement.





Godel’s Incompleteness Theorems (1931)

Hilbert’s program impossible, if interpreted in the most obviousway. BUT:

The development of proof theory itself is an outgrowth ofHilbert’s program. Gentzen’s development of naturaldeduction and the sequent calculus [too]. Godel obtainedhis incompleteness theorems while trying to prove theconsistency of analysis. The tradition of reductive prooftheory of the Gentzen-Schutte school is itself a directcontinuation of Hilbert’s program.

R. Zach, 2005





Proof theory: poor sister or cinderella?

Logic traditionally divided into:Model Theory,Proof Theory,Set Theory andRecursion Theory.What about Complexity Theory?





20th Century Proofs

To prove the consistency of Arithmetic Gentzen invented hissystems ofNATURAL DEDUCTION(how mathematicians think)SEQUENT CALCULUS(how he could formalize the thinking to obtain the main result heneeded, his Hauptsatz. (1934))





Church and lambda-calculus

Alonzo Church: the lambda calculus (1932)Church realized that lambda terms could be used to express everyfunction that could ever be computed by a machine.Instead of “the function f where f (x) = t”, he simply wrote λx .t.

The lambda calculus is an universal programming language.

The Curry-Howard correspondence: logicians and computerscientists developed a cornucopia of new logics/program constructsbased on the correspondence between proofs and programs.





Curry-Howard for ImplicationNatural deduction rules for implication (without λ-terms)

A→ B A

B

[A]····π

B

A→ B

Natural deduction rules for implication (with λ-terms)

M : A→ B N : A

M(N) : B

[x : A]····π

M : B

λx .M : A→ B

function application abstraction





Proofs are Programs!

Types are formulae/objects in appropriate category,Terms/programs are proofs/morphisms in the category,Logical constructors are appropriate categorical constructions.Most important: Reduction is proof normalization (Tait)Outcome: Transfer results/tools from logic to CT to CSci





Proof Theory using Categories...

Category: a collection of objects and of morphisms, satisfyingobvious lawsFunctors: the natural notion of morphism between categoriesNatural transformations: the natural notion of morphisms betweenfunctorsConstructors: products, sums, limits, duals....Adjunctions: an abstract version of equalityHow does this relate to logic?Where’s the theorem?A long time coming:Curry, Schoenfinkel, Howard (1969, published in 1980)





Categorical Proof Theory

Model derivations/proofs, not whether theorems are true or notProofs definitely first-class citizensHow? Uses extended Curry-Howard correspondenceWhy is it good? Modeling derivations useful in linguistics,functional programming, compilers..Why is it important? Widespread use of logic/algebra in CS meansnew important problems to solve with our favorite tools.Why so little impact on logic itself?





How many Curry-Howard Correspondences?

Easier to count, if thinking about the logics:Intuitionistic Propositional Logic, System F, Dependent TypeTheory (Martin-Lof), Linear Logic, Constructive Modal Logics,various versions of Classical Logic since the early 90’s.The programs corresponding to these logical systems are futuristicprograms.The logics inform the design of new type systems, that can be usedin new applications.





Dialectica Interpretation

If we cannot do Hilbert’s program with finitistic means, can we doit some other way?

Can we, at least, prove consistency of arithmetic?

Try: liberalized version of Hilbert’s programme – justify classicalsystems in terms of notions as intuitively clear as possible.

Godel’s approach: computable (or primitive recursive) functionalsof finite type (System T ), using the Dialectica Interpretation(named after the Swiss journal Dialectica, special volumededicated to Paul Bernays 70th birthday) in 1958.





Dialectica Categories

Hyland suggested that to provide a categorical model of theDialectica Interpretation, one should look at the functionalscorresponding to the interpretation of logical implication.

The categories in my thesis proved to be a model of Linear Logic...

Linear Logic introduced by Girard (1987) as a proof-theoretic tool:the symmetries of classical logic plus the constructive content ofproofs of intuitionistic logic.

Linear Logic: a tool for semantics of Computing.





Linear Logic

A proof theoretic logic described by Jean-Yves Girard in 1986.

Basic idea: assumptions cannot be discarded or duplicated. Theymust be used exactly once – just like dollar bills...

Other approaches to accounting for logical resources before.

Great win of Linear Logic: Account for resources when you wantto, otherwise fall back on traditional logic, A→ B iff !A −◦ B





Dialectica Categories as Models of Linear LogicIn Linear Logic formulas denote resources. Resources are premises,assumptions and conclusions, as they are used 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 using my dollar and one of the premisses above, say

$1 −◦ latte gives me a latte but the dollar is gone— Usual logic doesn’t pay attention to uses of premisses, A implies B

and A gives me B but I still have A...





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 (real 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 andduplication of resources.Not linear!!!Need to use tensor products and internal homs in Category Theory.Hard to decide how to define the“make-everything-as-usual”operator ”!”.





My version of Curry-Howard: Dialectica Categories

Based on Godel’s Dialectica Interpretation (1958):Result: an interpretation of intuitionistic arithmetic HA in aquantifier-free theory of functionals of finite type T .

Idea: translate every formula A of HA to AD = ∃u∀x .AD , whereAD is quantifier-free.

Use: If HA proves A then T proves AD(t, y) where y is string ofvariables for functionals of finite type, t a suitable sequence ofterms not containing y

Goal: to be as constructive as possible while being able to interpretall of classical arithmetic





Motivations and interpretations. . .

For Godel (in 1958) the Dialectica interpretation was a way ofproving consistency of arithmetic.

For me (in 1988) an internal way of modelling Dialectica turnedout to produce models of Linear Logic instead of models ofIntuitionistic Logic, which were expected...

For Blass (in 1995) a way of connecting work of Votjas in SetTheory with mine and also his own work on Linear Logic andcardinalities of the continuum.





Dialectica CategoriesObjects of the Dialectica category DDial2(Sets) are triples, ageneric object is A = (U,X ,R), where U and X are sets andR ⊆ U × X is an usual set-theoretic relation. A morphism from Ato B = (V ,Y ,S) is a pair of functions f : U → V andF : U × Y → X such that uRF (u, y)→ fuSy . (Note direction!)

Theorem: You have to find the right structure. . .

(de Paiva 1987) The category DDial2(Sets) has a symmetric monoi-dal closed structure, which makes it a model of (exponential-free)intuitionistic multiplicative linear logic.

Theorem(Hard part): You also want usual logic. . .

There is a comonad ! which models exponentials/modalities andrecovers Intuitionistic and Classical Logic.





Two Kinds of Dialectica CategoriesGirard’s sugestion in Boulder: Dialectica category Dial2(Sets)objects are triples, a generic object is A = (U,X ,R), where U andX are sets and R ⊆ U × X is a set-theoretic relation. A morphismfrom A to B = (V ,Y ,S) is a pair of functions f : U → V andF : Y → X such that uRFy → fuSy . (Simplified maps!)

Theorem: You just have to find the right structure. . .

(de Paiva 1989) The category Dial2(Sets) has a symmetric mo-noidal closed structure, and involution which makes it a model of(exponential-free) classical multiplicative linear logic.

Theorem (Even Harder part): You still want usual logic. . .

There is a comonad ! which models exponentials/modalities, hencerecovers Intuitionistic and Classical Logic.





Can we give some intuition for these morphisms?

Blass makes the case for thinking of problems in computationalcomplexity. Intuitively an object of Dial2(Sets)

(U,X ,R)

can be seen as representing a problem.The elements of U are instances of the problem, while theelements of X are possible answers to the problem instances.The relation R says whether the answer is correct for that instanceof the problem or not.





Examples of objects in Dial2(Sets)

1. The object (N,N,=) where n is related to m iff n = m.

2. The object (NN,N,R) where f is R-related to n iff f (n) = n.

3. The object (R,R,≤) where r1 and r2 are related iff r1 ≤ r2

4. The objects (2, 2,=) and (2, 2, 6=) with usualequality/inequality.





The Right Structure?

To “internalize”the notion of map between problems, we need toconsider the collection of all maps from U to V , V U , the collectionof all maps from Y to X , XY and we need to make sure that apair f : U → V and F : Y → X in that set, satisfies our dialecticacondition:

∀u ∈ U, y ∈ Y , uRFy → fuSy

This give us an object (V U × XY ,U × Y , eval) whereeval : V U × XY × (U × Y )→ 2 is the ‘relation’ that evaluates thepair (f ,F ) on the pair (u, y) and checks the dialectica implicationbetween relations.





The Right Structure!

Because it’s fun, let us calculate the “reverse engineering”necessary for a model of Linear Logic..

A⊗ B → C if and only if A→ [B −◦ C ]

U × V (R ⊗ S)XV × Y U U R X

⇓ ⇓

W

f

?T T

6

(g1, g2)

W V × Y Z?

(S −◦ T )V × Z

6





Dialectica Categories ApplicationsIn CS: models of Petri nets (more than 2 phds), non-commutativeversion for Lambek calculus (linguistics), it has been used as amodel of state (Correa et al) and even of quantum groups.

Generic models of Linear Logic (with Schalk04) and for LinguisticsAnalysis of the syntax-semantics interface for NaturalLanguage, the Glue Approach (Dalrymple, Lamping and Gupta).

Recently: Bodil Biering ‘Copenhagen Interpretation’ (firstfibrational version), P. Hofstra. ”The dialectica monad and itscousins”. Also ”The Compiler Forest”Budiu, Galenson and Plotkin(2012) and P. Hyvernat. “A linear category of polynomialdiagrams”.Most recent:Tamara Von Glehn ”polynomials”/containers (2014?).Piedrot (2014) Krivine machine interpretation...





My ‘newest’ Application

Blass (1995) Dialectica categories, or rather category PV as a toolfor proving inequalities between cardinalities of the continuum.

Blass realized that my model of Linear Logic was also used byPeter Votjas for set theory, proving inequalities between cardinalinvariants and wrote Questions and Answers A Category Arising inLinear Logic, Complexity Theory, and Set Theory (1995).

Four years ago I learnt from Samuel Gomes da Silva about his andCharles Morgan’s work using Blass/Votjas’ ideas and we startedworking together.





Goal

Blass (1995)

It is an empirical fact that proofs between cardinal characteristicsof the continuum usually proceed by representing the characteristicsas norms of objects in PV and then exhibiting explicit morphismsbetween those objects.

Why?so far only tiny calculation of natural numbers object in Dialecticacategories. (de Paiva, Morgan and da Silva, Natural NumberObjects in Dialectica Categories, LFSA 2013, to appear in ENTCS)





Conclusions

Introduced you to the under-appreciated Curry-Howardcorrespondence.

Hinted at its importance for interdisciplinarity:Categorical Proof Theory

Described one example: Dialectica categories Dial2(Sets),Illustrated one easy, but essential, theorem in categorical logic.

Hinted at Blass and Votjas use for mapping cardinal invariants.Much more explaining needed...





Take Home

Working in interdisciplinary areas is hard, but rewarding.

The frontier between logic, computing, linguistics and categories isa fun place to be.

Mathematics teaches you a way of thinking, more than specifictheorems.

Barriers: over-specialization, lack of open access and unwillingnessto ‘waste time’ on formalizationsEnablers: international scientific communities, open access,growing interaction between fields?...

Handsome payoff expectedFall in love with your ideas and enjoy talking to many about them..





Thank you!





Some References

A.Blass, Questions and Answers: A Category Arising in Linear Logic,Complexity Theory, and Set Theory, Advances in Linear Logic (ed. J.-Y.Girard, Y. Lafont, and L. Regnier) London Math. Soc. Lecture Notes 222(1995).

de Paiva, A dialectica-like model of linear logic, Category Theory andComputer Science, Springer, (1989) 341–356.

de Paiva, The Dialectica Categories, In Proc of Categories in ComputerScience and Logic, Boulder, CO, 1987. Contemporary Mathematics, vol92, American Mathematical Society, 1989 (eds. J. Gray and A. Scedrov)

P. Vojtas, Generalized Galois-Tukey-connections between explicit relationson classical objects of real analysis. In: Set theory of the reals (RamatGan, 1991), Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan(1993), 619–643.


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