A Dialectica Model of Relevant Type Theory

Introduction
Categorical Proof Theory
Back to the future: Linear Logic
Dialectica categories
Relevant Dialectica?
A Dialectica Model of
Relevant Type Theory
Valeria de Paiva
SYSMICS Final Meeting
Jan, 2019
Valeria de Paiva SYSMICS 2019 – Amsterdam

Introduction
Thank you, Yde Venema and
Nick Bezhanishvili, for the invitation!

Introduction
Introduction
I’m a logician, a proof-theorist and a category theorist.
I work in industry, have done so for the last 20 years, applying the
purest of pure mathematics, in surprising ways.
Today I want to tell you about some under-appreciated
mathematics of the 20th century.
Curry-Howard Correspondence
My small part on this
A new application, perhaps...

Introduction
Mathematics is full of surprises...
It often happens that there are similarities between the
solutions to problems. Sometimes, these similarities point
to more general phenomena that simultaneously explain
several diﬀerent pieces of mathematics. These more
general phenomena can be very diﬃcult to discover, but
when they are discovered, they have a very important
simplifying and organizing role, and can lead to the
solutions of further problems, or raise new and
fascinating questions. – T. Gowers, The Importance of Mathematics, 2000

Introduction
Proofs are Programs?
Mathematics got crystallized in the last years of the 19th century,
ﬁrst years of the 20th century. The shock is still being felt.
A Revolution in Mathematics? What Really Happened a Century
Ago and Why It Matters Today Frank Quinn (Notices of the AMS,
Jan 2012)
Today: relationships between Algebra, Proofs and Programs

Introduction
Bourbaki on Algebra
The axiomatization of algebra was begun by Dedekind
and Hilbert, and then vigorously pursued by Steinitz
(1910). It was then completed in the years following
1920 by Artin, Noether and their colleagues at G¨ottingen
(Hasse, Krull, Schreier, van der Waerden). It was
presented to the world in complete form by van der
Waerden’s book (1930).
Bourbaki didn’t say: Category Theory!

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

Introduction
Proofs?
Mathematics in turmoil because of paradoxes
Hilbert’s Program: provide secure foundations for all mathematics.
How? Formalization!
all mathematical statements should be written in a precise formal
language, and manipulated according to well deﬁned rules.
Base all of mathematics in ﬁnitistic methods
Proving the consistency of Arithmetic: the big quest

Introduction
Hilbert’s Program
Mathematics should be
Consistent: no contradiction can be obtained in the formalism.
Complete: all true mathematical statements can be proven.
Consistency proof use only “ﬁnitistic”reasoning about ﬁnite mathematical objects.
Conservative: any result about “real objects”obtained using
reasoning about “ideal objects”(such as uncountable sets) can be
proved without ideal objects.
Decidable: an algorithm for deciding the truth or falsity of any
mathematical statement.

Introduction
Gödel’s Incompleteness Theorems (1931)
Hilbert’s program impossible. BUT:
The development of proof theory itself is an outgrowth of
Hilbert’s program. Gentzen’s development of natural
deduction and the sequent calculus [too]. Gödel obtained
his incompleteness theorems while trying to prove the
consistency of analysis. The tradition of reductive proof
theory of the Gentzen-Schütte school is itself a direct
continuation of Hilbert’s program.
R. Zach, 2005

Introduction
Better Proofs
To prove the consistency of Arithmetic
Gentzen systems of
NATURAL DEDUCTION
(how mathematicians think)
SEQUENT CALCULUS
(how to formalize the thinking to obtain his Hauptsatz. (1934))

Introduction
Church and lambda-calculus
Lambda terms can be used to express every function that could
ever be computed by a machine. Instead of “the function f where
f (x) = t”, simply write λx.t.
Lambda calculus as an universal programming language.
The Curry-Howard correspondence: logicians and computer
scientists developed a cornucopia of new logics/program
constructs based on the correspondence between proofs and
programs.

Introduction
Curry-Howard Enablers

Introduction
Curry-Howard for Implication
Natural 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

Introduction
Categorical 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

Introduction

Introduction
Model derivations/proofs, not whether theorems are true or
not
Proofs deﬁnitely ﬁrst-class citizens
How? Uses extended Curry-Howard correspondence
Why is it good? Modeling derivations useful where you need
proofs themselves, in linguistics, functional programming,
compilers, etc.
Why is it important? Widespread use of logic/algebra in CS
means new important problems to solve with our favorite
tools.
Why so little impact on maths, CS or logic?

Introduction
How many Curry-Howard Correspondences?
Many!!!
Intuitionistic Propositional Logic, System F, Dependent Type
Theory (Martin-L¨of), Linear Logic, Constructive Modal Logics,
various versions of Classical Logic since the early 90’s.
The programs corresponding to these logical systems are
‘futuristic’ programs.
The logics inform the design of new type systems, that can be used
in new applications.

Introduction
Dialectica Interpretation
If we cannot do Hilbert’s program with finitistic means, can we do
it some other way?
Can we, at least, prove consistency of arithmetic?
Try: liberalized version of Hilbert’s programme – justify classical
systems in terms of notions as intuitively clear as possible.
Gödel’s approach: computable (or primitive recursive) functionals
of finite type (System T), using the Dialectica Interpretation
(named after the Swiss journal Dialectica, special volume
dedicated to Paul Bernays 70th birthday) in 1958.

Introduction
Dialectica Categories
Hyland suggested that to provide a categorical model of the
Dialectica Interpretation, one should look at the functionals
corresponding 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 of
proofs of intuitionistic logic.

Introduction
Linear Logic: a tool for semantics
A proof theoretic logic described by Girard in 1986.
Basic idea: assumptions cannot be discarded or duplicated.
They must be used exactly once – just like dollar bills (except
when they’re marked by a modality !)
Other approaches to accounting for logical resources before.
Great win of Linear Logic:
Account for resources when you want to, otherwise fall back
to traditional logic
A → B iﬀ !A −◦ B

Introduction
Linear Implication and (Multiplicative) Conjunction
Traditional implication: A, A → B B
A, A → B A ∧ B Re-use A
Linear implication: A, A −◦ B B
A, A −◦ B A ⊗ B Cannot re-use A
Traditional conjunction: A ∧ B A Discard B
Linear conjunction: A ⊗ B A Cannot discard B
Of course: !A !A⊗!A Re-use
!(A) ⊗ B B Discard

Introduction
The challenges of modeling Linear Logic
Traditional modeling of intuitionistic logic:
formula A object A of appropriate category
A ∧ B A × B (real product)
A → B BA (set of functions from A to B) and
they relate via adjunction
A ∧ B → C ⇐⇒ A → Hom(B, C) = CB
These are real products, have projections (A × B → A)
and diagonals (A → A × A) corresponding to deletion and
duplication of resources.
Not linear!!!
Need to use tensor products and internal homs.
Hard to deﬁne the “make-everything-usual”operator ”!”.

Introduction
Why Dialectica Categories
Based on Gödel’s Dialectica Interpretation (1958):
Result: an interpretation of intuitionistic arithmetic HA in a
quantifier-free theory of functionals of finite type T.
Idea: translate every formula A of HA to AD = ∃u∀x.AD, where
AD is quantifier-free.
Use: If HA proves A then T proves AD(t, y) where y is string of
variables for functionals of finite type, t a suitable sequence of
terms not containing y
Goal: to be as constructive as possible while being able to interpret
all of classical arithmetic

Introduction
Motivation and interpretation. . .
For G¨odel (in 1958) the Dialectica interpretation was a way of
proving consistency of arithmetic.
For me (in 1988) an internal way of modelling Dialectica turned
out to produce models of Linear Logic
For Hyland (from the beginning) Proof Theory in the Abstract
(2004), see recent work of von Glehn and Moss

Introduction
Objects of DDial2(Sets) are triples
A = (U, X, α)
U and X are sets and α ⊆ U × X is a relation
A morphism from A to B = (V , Y , β) consists of a pair of
functions f : U → V and F : U × Y → X such that
uαF(u, y) → fuβy.
U α X
⇓
V
f
?
β Y
6
F
(Note dependency of F in U and direction!)

Introduction
Theorem: You have to ﬁnd the right structure. . .
(deP1989) The category DDial2(Sets) has a symmetric monoidal
closed structure, which makes it a model of (exponential-free) in-
tuitionistic multiplicative linear logic.
Tensor products and internal homs relate appropriately
A ⊗ B → C ⇐⇒ A → Hom(B, C)
Theorem(Hard): You also want usual logic. . .
(deP1989) There is a comonad ! in DDial2(Sets) which models ex-
ponentials/modalities (using free commutative monoids in the base)
and recovers Intuitionistic (and Classical) Logic from LL.
The comonad is !A = (U, X∗, α∗), where X∗ is the free
commutative monoid on X, α∗ the natural relation

Introduction
Two Kinds of Dialectica Categories
Girard’s sugestion in Boulder 1987:
Dialectica category Dial2(Sets)
Same objects as before.
BUT morphism from A to B = (V , Y , β) is a pair of functions
f : U → V and F : Y → X such that uαFy → fuβy.
U α X
⇓
V
f
?
β Y
6
F
(Simpliﬁed maps, no dependency on U!)

Introduction
Girard-style Dialectica Categories
Theorem: You just have to ﬁnd the right structure. . .
(deP1989) The category Dial2(Sets) has a symmetric monoidal clo-
sed structure, and involution which makes it a model of (exponential-
free) classical multiplicative linear logic.
Internal homs are similar to previous DDial. Tensor products are
more complicated, but the cat is simpler.
Theorem (Even Harder): You still want usual logic. . .
There is a comonad ! which models exponentials/modalities, hence
recovers Intuitionistic Logic.
The comonad is composition of two others, plus a distributive law.
TA = (U, XU, αU), SA = (U, X∗, α∗) composing
!A = (U, (X∗
)U
, (α∗
)U
)

Introduction
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)
(U, X, α)
can be seen as representing a problem.
The elements of U are instances of the problem, while the
elements of X are possible answers to the problem instances.
The relation α says whether the answer is correct for that instance
of the problem or not.

Introduction
Can we give some intuition for these morphisms?
Better perhaps to think of objects as propositions that the left side
is trying to prove correct and the right-side is trying to prove false.
Intuitively in an object of Dial2(Sets)
(U, X, α)
each u can be seen as a witness and each x as a counterexample
for the proposition ∃u.∀xA(u, x)
The elements of U are trying to prove the theorem/solve the
problem, while the elements of X are possible counterexamples, so
they’re trying to prove the theorem false.

Introduction
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 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, =) with usual equality or
inequality.

Introduction
The Right Structure?
To “internalize”the notion of map between objects we need to
consider the collection of all maps from U to V , V U, the collection
of all maps from Y to X, XY and we need to make sure that a
pair f : U → V and F : Y → X in that set, satisﬁes our dialectica
condition:
∀u ∈ U, y ∈ Y , uαFy → fuβy
This give us an object (V U × XY , U × Y , eval) where
eval: V U × XY × (U × Y ) → 2 is the ‘relation’ that evaluates the
pair (f , F) on the pair (u, y) and checks the dialectica implication
between relations.

Introduction
The Right Structure!
Do the “reverse engineering” necessary to obtain a model of Linear
Logic...
A ⊗ B → C if and only if A → [B −◦ C]
U × V (α ⊗ β)XV
× Y U
U α X
⇓ ⇓
W
f
?
γ Z
6
(g1, g2)
W V
× Y Z
?
(β −◦ γ)V × Z
6

Introduction
Newest Old Application

Introduction
Relevant Logic
Philosophers claim that implication in logic is counterintuitive.
The following fail to be valid if we interpret arrow as our natural
notion of logical implication
Relevantists want to reject theses and arguments that commit
‘fallacies of relevance’:
antecedents and consequents could be on completely diﬀerent
topics, unrelated

Introduction
Relevance Logic
Precursors: Orlov(1928, Dosen review), Ackerman (1958)
Mostly here: Anderson and Belnap I (1975) and II (1992) and M.
Dunn
Also Avron (1984-2014).
Connections with paraconsistency important
Agree with Restall
Hilbert systems, with many axioms and few rules, are not
so suited to a project of understanding the internal
structure of a family of logical systems. (Handbook of
the History of Logic (2006))

Introduction
My Relevance Logic Masterplan
1. Instead of Relevance logic proper, use ILL+!c
2. Then add modality !w to get IL
3. Build dual context system DIRL where contexts are relevant and
intuitionistic, following DILL (Plotkin/Barber, Barber PhD)
4. Provide Dialectica Models of ILL+!c, appendix Mer´e (1993)
5. Provide models of DIRL
6. Provide models where modality !w disappears, but we keep two
implications, relevant and intuitionist, like ILT (Maietti et al, 2000)

Introduction
Failure of Masterplan
Hoped to do 1-6 today.
Can only do 1, 2 and 4.
But 3 done by Maraist and Bierman, independently.
This talk is a promissory note for more work on Relevant logic,
from the perspective of Linear Logic.
Conjecture
The category DialR has a symmetric monoidal closed structure.
There is a comonad !w which recovers Intuitionistic Logic over rele-
vant logic.

Introduction
(Weak) evidence for conjecture
DialR similar to DDial, but objects satisfy A −◦ A ⊗ A plus
coherence conditions.
Discussed in page 39 of ”The Dialectica Categories”,
(www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf)
where we have a “tricky map”, (∆, D) : A → A ⊗ A.
Map deﬁned by easy ∆=diagonal on U, BUT D(u, x, x ) = x , if
uαx, and x otherwise.
This is a new(?) example of Dosen/Petric’s “relevant category”,
as described in the nLab, I believe.

Introduction
Some Dialectica Categories Applications
Petri nets (more than 2 phds), non-commutative version for
Lambek calculus (linguistics), a model of state (Correa et al)
Generic models of Linear Logic (Schalk04), syntax-semantics
interface for Natural Language, the Glue Approach
(Dalrymple, Lamping and Gupta).
Biering’s ‘Copenhagen Interpretation’ (ﬁrst ﬁbrational
version), P. Hofstra ”The dialectica monad and its cousins”.
”The Compiler Forest”Budiu, Galenson and Plotkin (2012).
Pradic and Riba and (indep) Hyvernat
Von Glehn and Moss ”polynomials”/containers (2015),
(2017). Pedrot (2015) Krivine machine interpretation.
Shulman on 2-Chu-Dialectica construction arxiv:1806.06082.

Introduction
Conclusions
Introduced/reacquainted you with the Curry-Howard
correspondence.
Hinted at its importance for interdisciplinarity:
Described two examples of Dialectica categories hinted at 3rd one
(only behavior involv modalities change)
Much more explaining needed... But pictures help.

Introduction
Thank you!

Introduction
Some References
(see https://github.com/vcvpaiva/DialecticaCategories)
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 and
Computer Science, Springer, (1989) 341–356.
de Paiva, The Dialectica Categories, In Proc of Categories in Computer
Science and Logic, Boulder, CO, 1987. Contemporary Mathematics, vol
92, American Mathematical Society, 1989 (eds. J. Gray and A. Scedrov)

A Dialectica Model of Relevant Type Theory

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