Homotopic Foundations of the Theory of Computation

What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Homotopic Foundations of the Theory of
Computation
Ruy de Queiroz
(joint work with A. de Oliveira)
Centro de Informática
Universidade Federal de Pernambuco (UFPE)
Recife, Brazil
Colóquio do Depto de Matemática
17/08/2023
Ruy de Queiroz (joint work with A. de Oliveira) Centro de Informática Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Homotopic Foundations of the Theory of Computation

Hilbert’s 10th problem
Solvability
10th problem: Given a Diophantine equation with any number of
unknown quantities and with rational integral numerical coefficients:
To devise a process according to which it can be determined in a
finite number of operations whether the equation is solvable in
rational integers. (Hilbert, D. (1901). ”Mathematische Probleme”.
Archiv der Math. und Physik.)
Entscheidungsproblem: the Entscheidungsproblem (German for
’decision problem’; pronounced is a challenge posed by David Hilbert
and Wilhelm Ackermann in 1928. The problem asks for an algorithm
that considers, as input, a statement and answers ”yes” or ”no”
according to whether the statement is universally valid, i.e., valid in
every structure satisfying the axioms. (D. Hilbert and W. Ackermann
(1928). Grundzüge der theoretischen Logik. Springer-Verlag)

Church λ-calculus
An Unsolvable Problem of Elementary Number Theory
“The purpose of the present paper is to propose a definition of
effective calculability which is thought to correspond satisfactorily to
the somewhat vague intuitive notion in terms of which problems of
this class are often stated, and to show, by means of an example, that
not every problem of this class is solvable.”
(A. Church. 1936. An Unsolvable Problem of Elementary Number
Theory. American Journal of Mathematics, Vol. 58, No. 2. (Apr.,
1936), pp. 345-363)
(A. Church 1932: “There may, indeed, be other applications of the
system [i.e. lambda calculus] than its use as a logic”.)
(A. Church 1933: “in the course of working with these postulates, it
has since become clear that the set as originally given requires some
modification in order to render it free from contradiction.”)

Intuitionistic Type Theory
Formalisation of Russell’s Type Theory
P. Martin-Löf’s A Theory of Types (Feb 1971): “After the recent proof
theoretical investigations of simple type theory, it would seem natural
to make an attempt at set theory as tormalized by Zermelo and
Fraenkel. However, mainly because of the form of the replacement
axiom, it does not seem as if set theory lends itself very well to a
proof theoretical analysis.
Instead, I have formulated an intuitionistic theory of types in which
(the intuitionistic and intentional version of) the simple theory of finite
types imbedded as a subsystem.”
(Martin-Löf’s theory of types: typed λ-calculus + axiom of
selfreference)
(J.Y. Girard’s dissertation (1972) proved inconsistent.)

Impact of Intuitionistic Type Theory
2020 Rolf Schock prize in logic and philosophy

Impact of Intuitionistic Type Theory in Mathematics
Proofs of equality
By introducing (in 1973) a framework in which a formalisation of
the logical notion of equality, via the so-called “identity type”,
Martin-Löf’s Type Theory allows for a surprising connection
between term rewriting and geometric concepts such as path
and homotopy.
The impact in mathematics has been felt more strongly since
the start of Vladimir Voevodsky’s program on the univalent
foundations of mathematics around 2005, joined by Steve
Awodey in building an approach referred to in 2007 as
homotopy type theory.

Vladimir Voevodsky on ‘isomorphism invariance principle’
“One of the keystones of contemporary mathematics is the
isomorphism invariance principle: for any statement P about X
and any isomorphism X
ϕ
≃ X′
, there is a statement Pϕ about X′
such
that P holds iff Pϕ holds
The equality problem in formalizations comes in part from the fact
that when one encodes X and X′
the isomorphism is lost.
There is more to the equality problem than isomorphism invariance:
• equality is a good notion for ‘elements’ – individuals, but fails for
collections.
• isomorphism is a good notion for collections, but fails for collections
of collections.
This leads to a theory of iterated n-equivalences which are the
correct replacements for such “iterated collection” ”
Foundations of Mathematics and Homotopy Theory, IAS (2006)

Working with ‘sameness’
From Michael Shulman’s Homotopy type theory - A high-level
language for invariant mathematics (March 2019):
Homotopy type theory is a high-level abstract frame-
work for working with sameness.
Sameness:
• Two groups are the same if they are isomorphic
• Two topological spaces are the same if they are homeomorphic
• Two categories are the same if they are equivalent
• Two sets are the same if they have the same extension
• Two mappings are homotopically the same if one can be
continuously deformed into the other
one should also add:
• Two λ-terms are the same if they are convertible to one another

Coffee cup and donut: the same?
“An often-repeated mathematical joke is that topologists can’t tell the
difference between a coffee cup and a donut, since a sufficiently
pliable donut could be reshaped to the form of a coffee cup by
creating a dimple and progressively enlarging it, while preserving the
donut hole in the cup’s handle.” (Wikipedia)

Associating groups to topological spaces
“In modern mathematics it is common to study a category by
associating to every object of this category a simpler object that
still retains sufficient information about the object of interest.
Homotopy groups are such a way of associating groups to
topological spaces.
That link between topology and groups lets mathematicians
apply insights from group theory to topology. ” (Wikipedia)

Calculation of fundamental groups
“Homotopy groups are used in algebraic topology to classify
topological spaces. The first and simplest homotopy group is
the fundamental group, which records information about loops
in a space. Intuitively, homotopy groups record information
about the basic shape, or holes, of a topological space.
Topological spaces with differing homotopy groups are never
equivalent (homeomorphic), but topological spaces that are not
homeomorphic can have the same homotopy groups.
Calculation of homotopy groups is in general much
more difficult than some of the other homotopy invariants
learned in algebraic topology. ” (Wikipedia)

Calculation of fundamental groups of surfaces

Calculation of fundamental groups
Using a natural deduction system in the Curry–Howard tradition, we
demonstrate how to formalise the concept of computational paths
(sequences of rewrites) as equalities between two terms of the same
type. The aim is to formulate a term rewriting system in order to
illustrate how one can perform computations within these
computational paths, establishing equalities between equalities. We
shall proceed to use the Labelled Natural Deduction – LND – based
on the concept of computational paths (which is a system of rewrites)
as a tool for obtaining the results on the fundamental group of the
circle, the torus and the real projective plane.
A Topological Application of Labelled Natural Deduction. Tiago M.
L.Veras, Arthur F. Ramos, R. J. G. B. de Q., Anjolina G. de Oliveira
(2019) https://arxiv.org/abs/1906.09105 (to appear in South Amer J of
Logic)

Computational paths: weak groupoid
We use a labelled deduction system (LNDED−TRS) to formalise the
concept of computational paths (sequences of rewrites) as equalities
between two terms of the same type. This has allowed us to carry out
a formal counterpart to equality between paths which is dealt with in
homotopy theory, but this time with an approach using the device of
term-rewriting paths. Using such formal calculus dealing with paths,
we construct the fundamental groupoid of a path-connected X type
and we define the concept of isomorphism between types. Next, we
show that the computational paths determine a weak category, which
will be called Cpaths. Finally, we show that the weak category Cpaths
determines a weak groupoid.
Computational Paths - A Weak Groupoid. Tiago M. L.Veras, Arthur F.
Ramos, R. J. G. B. de Q., T.D.O. Silva, Anjolina G. de Oliveira (2020)
https://arxiv.org/abs/2007.07769 (submitted for publication)

Computational Paths as part of the syntax
Definition (The circle S1)
The circle is the type generated by:
(i) A base point - x0 : S1
(ii) A base computational path - x0 =
loop
x0 : S1.
The first thing one should notice is that this definition does not use
only the points of the type S1
, but also a base computational path
called loop between those points. That is why it is called a higher
inductive type . Our approach differs from the one developed in
the HoTT book on the fact that we do not need to simulate the
path-space between those points, since we add
computational paths to the syntax of the theory.

Building Bridges! Opening (new) Paths!
Computation– (Algebraic) Topology–Logic–(Higher) Categories–(Higher) Algebra
A single concept may serve as a bridging bond: path
Computation: convertibility between λ-terms
(Algebraic) Topology: homotopy theory
Logic: proofs of equality
(Higher) Categories: polycategories
(Higher) Algebra: ∞-groupoids
Paths as structure-preserving maps.
Path equivalences: homotopy theory

Equality in λ-Calculus: convertibility
Proofs of equality as (reversible) sequences of contractions, i.e. paths
Church’s (1936) original λ-calculus paper:
NB: equality as the reflexive, symmetric and transitive closure of
1-step contraction: symmetric closure of rewriting paths.

Equality in λ-Calculus: composition of definitional
contractions
Proofs of equality: paths
Definition (Hindley & Seldin 2008)
P is βη-equal or βη-convertible to Q (notation P =βη Q) iff Q is
obtained from P by a finite (perhaps empty) series of
β-contractions, η-contractions, reversed β-contractions,
reversed η-contractions, or changes of bound variables. That is,
P =β Q iff there exist P0, . . . , Pn (n ≥ 0) such that
P0 ≡ P, Pn ≡ Q,
(∀i ≤ n − 1) (Pi ▷1β Pi+1 or Pi+1 ▷1β Pi
or Pi ▷1η Pi+1 or Pi+1 ▷1η Pi
or Pi ≡α Pi+1).

Curry-Howard Interpretation: Intuitionistic Type Theory
‘Formulae-as-Types’, ‘Proofs-as-Programs’
(1934) H. Curry came up with an early version of type inference
for the combinators of Combinatory Logic. The types of
combinators could be seen as axioms of implicational logic:
‘α → β’ could be read as
(1) ‘the type of functions from type α to type β’ ;
(2) ‘the formula “α implies β” ’.
Axioms of Implicational Logic:
α → α
α → β → α
(α → β → γ) → (α → β) → α → γ
Types of Combinators:
I : α → α
K : α → β → α
S : (α → β → γ) → (α → β) → α → γ

Curry-Howard Interpretation: Intuitionistic Type Theory
‘Formulae-as-Types’, ‘Proofs-as-Programs’
(1969) W. Howard came up with an extension of Curry’s
functionality interpretation to full intuitionistic predicate logic:
⊥ as ∅ (empty type)
α ∧ β as α × β (product)
α ∨ β as α + β (sum)
∀xγ as Πxγ (dependent product)
∃xγ as Σxγ (dependent sum)
(1972) P. Martin-Löf came up with Type Theory extending
Howard’s Formulae-as-Types with Natural Numbers and
Universes.
(1973) P. Martin-Löf came up with Intuitionistic Type Theory
extending Type Theory with Identity Types.

Identity Types
Identity Types - Topological and Categorical Structure
Workshop, Uppsala, November 13–14, 2006:
“The identity type, the type of proof objects for the fundamental
propositional equality, is one of the most intriguing constructions of
intensional dependent type theory (also known as Martin-Löf type
theory). Its complexity became apparent with the Hofmann–Streicher
groupoid model of type theory. This model also hinted at some
possible connections between type theory and homotopy theory and
higher categories. Exploration of this connection is intended to be the
main theme of the workshop.”
Michael Shulman’s (2017) ‘Homotopy type theory: the logic of
space’: “For many years, the most mysterious part of Martin-Löf’s
type theory was the identity types “x = y”.”

Type Theory and Derivations-as-Terms
Howard on the so-called Curry-Howard ‘Formulae-as-Types’
“ [de Bruijn] discovered the idea of derivations as
terms, and the accompanying idea of formulae-as-
types, on his own. (...)
Martin-Löf suggested that the derivations-as-terms
idea would work particularly well in connection with
Prawitz’s theory of natural deduction.”
(W.Howard, Wadler’s Blog, 2014)
1-step contraction originating from logic: contractions in
redundant proofs correspond to contractions in redundant
terms.
Curry-Howard ‘Formulae-as-Types’:
Proof equivalence ←→ Term equivalence

Type Theory
Hausdorff Trimester: Types, Sets and Constructions, May 2 – Aug 24, 2018, Unn Bonn
“Type theory, originally conceived as a bulwark against the paradoxes
of naive set theory, has languished for a long time in the shadow of
axiomatic set theory which became the mainstream foundation of
mathematics. The first renaissance of type theory occurred with the
advent of computer science and Bishop’s development of a
practice-oriented constructive mathematics. It was followed by a
second quite recent one that not only champions type theory as a
central framework for achieving the goal of fully formalized
mathematics amenable to verification by computer-based proof
assistants, but also finds deep and unexpected connections between
type theory and homotopy theory. ”

Type Theory
Hausdorff Trimester: Types, Sets and Constructions, May 2 – Aug 24, 2018, Unn Bonn
“Constructive set theory and mathematics distinguishes itself from its
traditional counterpart, classical set theory and mathematics based
on it, by insisting that proofs of existential theorems must afford
means for constructing an instance. Constructive reasoning emerges
naturally in core areas of mathematics and in the theory of
computation. The aim of the Hausdorff Trimester is to create a forum
for research on and dissemination of exciting recent developments,
which are of central importance to modern foundations of
mathematics.”

Homotopy Type Theory
Open Source Book
Institute for Advanced Study, Princeton, 2013 (approx. 600p)
Open-source book: The Univalent Foundations Program
27 main participants. 58 contributors
Available on GitHub. Latest version April 15, 2020
version marker: first-edition-1257-gdc4966e

Open Source Book
“Homotopy type theory is a new branch of mathematics that
combines aspects of several different fields in a surprising way. It is
based on a recently discovered connection between homotopy theory
and type theory. Homotopy theory is an outgrowth of algebraic
topology and homological algebra, with relationships to higher
category theory; while type theory is a branch of mathematical logic
and theoretical computer science. Although the connections between
the two are currently the focus of intense investigation, it is
increasingly clear that they are just the beginning of a subject that will
take more time and more hard work to fully understand. It touches on
topics as seemingly distant as the homotopy groups of spheres, the
algorithms for type checking, and the definition of weak ∞-groupoids.”

Origins of Research Programme
Vladimir Voevodsky (IAS, Princeton) (b. 4 June 1966 – d. 30
September 2017) (IAS, Princeton) (Fields Medal 2002)
1st
(?) use of term ‘homotopy λ-calculus’: tech report Notes on
homotopy λ-calculus (Started Jan 18, Feb 11, 2006): “In this paper
we suggest a new approach to the foundations of mathematics. (...)
A key development (totally unnoticed by the mathematical
community) occurred in the 70-ies when the typed λ-calculus
was enriched by the concept of dependent types.”

Vladimir Voevodsky and Motivic Homotopy Theory
Nature, OBITUARY, 06/11/2017, Vladimir Voevodsky (by Daniel R. Grayson)
“[Voevodsky] made major progress towards Grothendieck’s grand
vision, articulated in the 1960s, of a theory of ‘motives’.
Grothendieck’s dream was to produce, for any system of polynomial
equations, the essential nugget that would remain after everything
apart from the shared topological flavour of the system was washed
away. Perhaps borrowing the French musical term for a recurring
theme, Grothendieck dubbed this the motif of the system.
In Voevodsky’s motivic homotopy theory, familiar classical geometry
was replaced by homotopy theory – a branch of topology in which a
line may shrink all the way down to a point. He abandoned the idea
that maps between geometric objects could be defined locally and
then glued together, a concept that Grothendieck considered to be
fundamental. A colleague commented that if mathematics were
music, then Voevodsky would be a musician who invented his own
key to play in.”

Origins of Research Programme
Steve Awodey (Dept Phil, CMU)
1st
(?) use of term ‘homotopy type theory’: Eighty-sixth Peripatetic
Seminar on Sheaves and Logic, Nancy, 8–9 September 2007

HoTT Approach to Spaces: New Proofs
“Homotopy type theory: towards Grothendieck’s dream”, M. Shulman, 2013 Int.
Category Theory conference, Sydney
“Progress in synthetic homotopy theory:
• π1(S1
) = Z (Shulman, Licata)
• πk (Sn
) = 0 for k < n (Brunerie, Licata)
• πn(Sn
) = Z (Licata, Brunerie)
• The long exact sequence of a fibration (Voevodsky)
• The Hopf fibration and π3(S2
) = Z (Lumsdaine, Brunerie)
• The Freudenthal suspension theorem (Lumsdaine)
• The Blakers–Massey theorem (Lumsdaine, Finster, Licata)
• The van Kampen theorem (Shulman)
• Whitehead’s theorem for n-types (Licata)
• Covering space theory (Hou) ”
“Homotopy type theory can also serve as a foundational system
for mathematics whose basic objects are ∞-groupoids rather
than sets.”

Computational Paths: New Proofs
Calculation of Fundamental Groups
Calculation of Fundamental Groups of Surfaces:
circle
cylinder
Möbius band
torus
two-holed torus
real projective plane
Van Kampen theorem
T. M. L. de Veras, A. F. Ramos, R. J. G. B. de Queiroz, A. G. de
Oliveira, On the Calculation of Fundamental Groups in
Homotopy Type Theory by Means of Computational Paths,
arXiv 1804.01413

Geometry and Logic
Alexander Grothendieck
Alexander Grothendieck
b. 28 March 1928, Berlin, Prussia, Germany
d. 13 November 2014 (aged 86), Saint-Girons, Ariège, France

Geometry and Logic
Alexander Grothendieck: The Homotopy Hypothesis
“... the study of n-truncated homotopy types (of
semisimplicial sets, or of topological spaces) [should
be] essentially equivalent to the study of so-called n-
groupoids. . . . This is expected to be achieved by
associating to any space (say) X its “fundamental n-
groupoid” Πn(X).... The obvious idea is that 0-objects
of Πn(X) should be the points of X, 1-objects should
be “homotopies” or paths between points, 2-objects
should be homotopies between 1-objects, etc. ”
(Grothendieck, “Pursuing Stacks” (1983))
homotopy types ←→ ∞-groupoids

∞-Groupoids and polycategories
Voevodsky’s attempt to realize Grothendieck’s dream
“It is known that CW-complexes X such that πi (X) = 0 for
i ≥ 2 can be described by groupoids from the homotopy
point of view. In the unpublished paper “Pursuing stacks”
Grothendieck proposed the idea of a multi-dimensional gen-
eralization of this connection that used polycategories. The
present note is devoted to the realization of this idea.”
(“∞-Groupoids as a model for a homotopy category”, V A
Voevodskii and M M Kapranov, Communications of the Moscow
Mathematical Society, 45:239–240, 1990)

From n-CPO’s to ∞-CPOs
Dana Scott’s D∞ and typed vs untyped λ-Calculus
“This particular paper has, of course, and odd historical role:
in it [October 1969] the author argues against the type-free
calculi of Church and Curry, Kleene and Rosser, and their
later uses by Böhm and Strachey. But then in November of
1969, after writing this report, the author himself saw that
the method of monotone continuous functions (...) could be
applied to posets other than just those generated from the
integers (with bottom) by the very simple type constructors.”
(“A type-theoretical alternative to ISWIN, CUCH, OWHY”, 1993)

Higher-Order Functions, Complete Partial Orders
(“Outline of a mathematical theory of computation”, 1970):
“The process of “completing” spaces is a very general one, and
the full implication of the method are not yet clear. A second
example of the idea concerns function spaces. Let D be given,
and set D0 = D and Dn+1 = (Dn → Dn). The spaces Dn are (a
selection of) the “higher-type” spaces of functions of functions
of functions ... . It turns out there is a way of naturally
embedding each Dn successfully into the next space Dn+1.
These embeddings make it possible to pass to a limit space D∞
which contains the originally given D and is such that
D∞
∼
= (D∞ → D∞).”

Higher-Order Functions, Complete Partial Orders
(∞–Grupoid Generated by an Arbitrary Topological λ-Model.
Daniel Martinez-Rivillas, R.J.G.B. de Q. arXiv:1906.05729
(appeared in Logic Journal of the IGPL 30(3), June 2022)):
“Starting from any topological space that models extensional
λ-calculus, we propose a method to build an ∞-groupoid. This
construction was applied to the particular c.p.o. D∞ with Scott
topology, resulting in a constant cell infinite sequences set,
where each cell sequence is isomorphic to a constant higher
paths infinite matrix.”

Homotopic λ-models
Reflexive Kan complex
(Towards a Homotopy Domain Theory. Daniel Martinez Rivillas,
R.J.G.B. de Q. arXiv:2007.15082 (AML 2022)):
“An appropriate framework is put forward for the construction of
λ-models with ∞-groupoid structure, which we call homotopic
λ-models through the use of an ∞-category with cartesian
closure and enough points. With this, we establish the start of a
project of generalization of Domain Theory and λ-calculus, in
the sense that the concept of proof (path) of equality of λ-terms
is raised to higher proof (homotopy).”

Homotopic λ-models
Reflexive Kan complex
R.J.G.B. de Q. arXiv:2007.15082 (AML, Nov 2022)):
“The initiative to search for λ-models with a ∞-groupoid structure
emerged in a previous work (called homotopic λ-models), which
studied the geometry of any complete partial order (c.p.o) (e.g., D∞),
and found that the topology inherent in these models generated trivial
higher-order groups. From that moment on, the need arose to look for
a type of model that presented a rich geometric structure; where their
higher-order fundamental groups would not collapse. In this sense,
we will gain the semantics of a type-free theory from a version of
HoTT based on computational paths, which distinguishes the
difference between proofs of equality of two λ-terms.”

Homotopic λ-models
Extensional Kan complex
(The Theory of an Arbitrary Higher λ-Model. Daniel Martinez
Rivillas, R.J.G.B. de Q. arXiv:2007.15082 (Bull Sect on Logic,
2023)):
One takes advantage of some basic properties of every
homotopic λ-model (e.g. extensional Kan complex) to explore
the higher βη-conversions, which would correspond to proofs of
equality between terms of a theory of equality of any
extensional Kan complex. Besides, Identity types based on
computational paths are adapted to a type-free theory with
higher λ-terms, whose equality rules would be contained in the
theory of any λ-homotopic model.

The search for λ-models with a ∞-groupoid structure
Homotopy Domain Theory
PhD Th., Dec 2022):
The initiative to search for λ-models with a ∞-groupoid
structure emerged in (Martinez-Rivillas (2020), MSc Diss),
which studied the geometry of any complete partial order
(c.p.o) (e.g., D∞), and found that the topology inherent in these
models generated trivial higher-order groups. From that
moment on, the need arose to look for a type of model that
presented a rich geometric structure; where their higher-order
fundamental groups would not collapse.

Homotopy Domain Theory
PhD Th., Dec 2022):
It is known in the literature that Dana Scott’ Domain-Theory provides
general techniques for obtaining λ-models by solving domain
equations over arbitrary cartesian closed categories. To fulfil the
purpose of getting λ-models with non-trivial ∞-groupoid structure, the
most natural way would be to adapt Dana Scott’s Domain Theory to a
“Homotopy Domain Theory”. Where the cartesian closed categories
(c.c.c) will be replaced by cartesian closed ∞-categories (c.c.i), the
c.p.o’s will be replaced by “c.h.p.o’s (complete homotopy partial
orders)”, the 0-categories by (0, ∞)-categories and the isomorphism
between objects in a cartesian closed 0-categories (at the domain
equation) will be replaced by an equivalence between objects in a
cartesian closed (0, ∞)-category, which we call “homotopy domain
equation”.

Generalisation of Recursive Domain Equations
Towards a Homotopy Domain Theory.
Daniel O. Martı́nez Rivillas, PhD Th., Dec 2022:
Dana Scott’s Domain-Theory Homotopy Domain Theory
Cartesian Closed
Category (c.c.c)
Cartesian Closed ∞-Category (c.c.i)
Complete Partial
Order (c.p.o)
Complete Homotopy Partial Order
(c.h.p.o)
0-category (0, ∞)-category
Domain Equation:
X ∼
= (X ⇒ X)
Homotopy Domain Equation:
X ≃ (X ⇒ X)

Type Theory and Homotopy Theory
Steve Awodey: a calculus to reason about abstract homotopy
“Homotopy type theory is a new field devoted to a recently discovered
connection between Logic and Topology – more specifically, between
constructive type theory, which was originally invented as a
constructive foundation for mathematics and now has many
applications in the theory of programming languages and formal proof
verification, and homotopy theory, a branch of algebraic topology
devoted to the study of continuous deformations of geometric spaces
and mappings. The basis of homotopy type theory is an interpretation
of the system of intensional type theory into abstract homotopy
theory. As a result of this interpretation, one can construct new kinds
of models of constructive logic and study that system semantically,
e.g. proving consistency and independence results. Conversely,
constructive type theory can also be used as a formal calculus to
reason about abstract homotopy.”
(A proposition is the (homotopy) type of its proofs, Jan 2016.)

The Functional Interpretation of Equality
Sequences of contractions
(λx.(λy.yx)(λw.zw))v ▷1η (λx.(λy.yx)z)v ▷1β (λy.yv)z ▷1β zv
(λx.(λy.yx)(λw.zw))v ▷1β (λx.(λw.zw)x)v ▷1η (λx.zx)v ▷1β zv
(λx.(λy.yx)(λw.zw))v ▷1β (λx.(λw.zw)x)v ▷1β (λw.zw)v ▷1η zv
There is at least one sequence of contractions from the initial term to
the final term. Thus, in the formal theory of λ-calculus, the term
(λx.(λy.yx)(λw.zw))v is declared to be equal to zv.
Now, some natural questions arise:
1 Are the sequences themselves normal?
2 What are the non-normal sequences?
3 How are the latter to be identified and (possibly) normalised?

Brouwer–Heyting–Kolmogorov Interpretation
Proofs rather than truth-values
a proof of the proposition: is given by:
A ∧ B a proof of A and
a proof of B
A ∨ B a proof of A or
a proof of B
A → B a function that turns a proof of A
into a proof of B
∀x.P(x) a function that turns an element a
into a proof of P(a)
∃x.P(x) an element a (witness)
and a proof of P(a)

What is a proof of an equality statement?
t1 = t2 ?
(Perhaps a path from t1 to t2?)
What is the logical status of the symbol “=”?
What would be a canonical/direct proof of t1 = t2?
What is an equality between paths?
What is an equality between homotopies (i.e., paths between
paths)?

Type Theory and Equality
Proposition vs Judgements
In type theory, two main kinds of judgements:
1 x : A
2 x = y : A
Via the so-called Curry-Howard interpretation, “x : A” can be read as
“x is a proof of proposition A”.
Also, “x = y : A” can be read as “x and y are (definitionally) equal
proofs of proposition A”.
What about the judgement of “p is a proof of the statement that x and
y are equal elements of type A”? This is where the so-called Identity
type comes into the picture:
p : IdA(x, y)

Type Theory and Equality
Explicit Terms for Paths
Those paths are not part of the syntax of type theory. This is
clear from an answer given by Vladimir Voevodsky for the
following question in a short interview (22 Oct 2015):
- Martin Escardò: What was your first reaction when
you first saw the type of identity? Did you immediately
connect it with path spaces?
- Vladimir Voevodsky: Not at all. I did not make this
connection until late 2009. All the time before it I was
hypnotized by the mantra that the only inhabitant
of the Id type is reflexivity which made it useless
from my point of view.

What is the formal counterpart of a proof of an equality?
In talking about proofs of an equality statement, two dichotomies
arise:
1 definitional equality versus propositional equality
2 intensional equality versus extensional equality
First step on the formalisation of proofs of equality statements: Per
Martin-Löf’s Intuitionistic Type Theory (Log Coll ’73, published 1975)
with the so-called Identity Type

Identity Types
Type Theory and Homotopy Theory
The groupoid structure exposed in the Hofmann–Streicher (1994)
countermodel to the principle of Uniqueness of Identity Proofs (UIP).
In Hofmann & Streicher’s own words,
“We give a model of intensional Martin-Löf type theory
based on groupoids and fibrations of groupoids in which
identity types may contain two distinct elements which are
not even propositionally equal. This shows that the principle
of uniqueness of identity proofs is not derivable in the syn-
tax”.
(M Hofmann, T Streicher, “The groupoid model refutes uniqueness of
identity proofs”. In Logic in Computer Science, 1994 (LICS’94), pp.
208–212.)

Identity Types
Identity Types as Topological Spaces
“All of this work can be seen as an elaboration of the fol-
lowing basic idea: that in Martin-Löf type theory, a type
A is analogous to a topological space; elements a, b ∈ A
to points of that space; and elements of an identity type
p, q ∈ IdA(a, b) to paths or homotopies p, q : a → b in
A.”.
(B. van den Berg and R. Garner, “Topological and simplicial models of
identity types”, ACM Transactions on Computational Logic, Jan 2012)

Identity Types
Identity Types as Topological Spaces
From the Homotopy type theory collective book (2013):
“In type theory, for every type A there is a (formerly some-
what mysterious) type IdA of identifications of two objects
of A; in homotopy type theory, this is just the path space AI
of all continuous maps I → A from the unit interval. In this
way, a term p : IdA(a, b) represents a path p : a ⇝ b in A.”

Identity Types: Iteration
From Propositional to Predicate Logic and Beyond
In the same aforementioned workshop, B. van den Berg in his
contribution “Types as weak omega-categories” draws attention to the
power of the identity type in the iterating types to form a globular set:
“Fix a type X in a context Γ. Define a globular set as
follows: A0 consists of the terms of type X in context
Γ,modulo definitional equality; A1 consists of terms of the
types Id(X; p; q) (in context Γ) for elements p, q in A0, mod-
ulo definitional equality; A2 consists of terms of well-formed
types Id(Id(X; p; q); r; s) (in context Γ) for elements p, q in
A0, r, s in A1, modulo definitional equality; etcetera...”

The homotopy interpretation
Here is how we can see the connections between proofs of equality
and homotopies:
a, b : A
p, q : IdA(a, b)
α, β : IdIdA(a,b)(p, q)
· · · : IdIdId...
(· · · )
Now, consider the following interpretation:
Types ⇝ Spaces
Terms ⇝ Maps
a : A ⇝ Points a : 1 → A
p : IdA(a, b) ⇝ Paths p : a ⇒ b
α : IdIdA(a,b)(p, q) ⇝ Homotopies α : p ⇛ q

The homotopy interpretation (Awodey (2016))
point, path, homotopy, ...

Identity Types
Univalent Foundations of Mathematics
From Vladimir Voevodsky (IAS, Princeton) “Univalent
Foundations: New Foundations of Mathematics”, Mar 26, 2014:
“There were two main problems with the existing foun-
dational systems which made them inadequate.
Firstly, existing foundations of mathematics were based
on the languages of Predicate Logic and languages of
this class are too limited.
Secondly, existing foundations could not be used to di-
rectly express statements about such objects as, for ex-
ample, the ones that my work on 2-theories was about.”

The Homotopy Interpretation
Steve Awodey (2016): a “logic of homotopies”
“The homotopy interpretation was first proposed by the
present author and worked out formally (with a student) in
terms of Quillen model categories – a modern, axiomatic
setting for abstract homotopy theory that encompasses not
only the classical homotopy theory of spaces and their com-
binatorial models like simplicial sets, but also other, more
exotic notions of homotopy (...). These results show that
intensional type theory can in a certain sense be regarded
as a “logic of homotopy”, in that the system can be faithfully
represented homotopically, and then used to reason formally
about spaces, continuous maps, homotopies, and so on. ”
(A proposition is the (homotopy) type of its proofs, Jan 2016.)

Propositional Equality
Proofs of equality as (rewriting) computational paths
What is a proof of an equality statement? In what sense it can be
seen as a homotopy? Motivated by looking at equalities in type
theory as arising from the existence of computational paths between
two formal objects, it may be useful to review the role and the power
of the notion of propositional equality as formalised in the so-called
Curry–Howard functional interpretation.
The main idea, namely, proofs of equality statements as (reversible)
sequences of rewrites, i.e. paths, goes back to a paper entitled
“Equality in labelled deductive systems and the functional
interpretation of propositional equality”, presented in Dec 1993 at the
9th Amsterdam Colloquium, and published in the proceedings in
1994.

BHK for Identity Types
Types and Propositions
(source: Awodey (2016))
types vs propositions:
sum/coproduct vs disjunction,
product vs conjunction,
function space vs implication
dependent sum vs existential quantifier,
dependent product vs universal quantifier
path space (?) vs equality symbol

t1 = t2 ?
(Perhaps a sequence of rewrites
starting from t1 and ending in t2?)
What is the logical status of the symbol “=”?
What would be a canonical/direct proof of t1 = t2?

The Functional Interpretation of Direct Computations
Propositional Equality: The Groupoid Laws
With the formulation of propositional equality that we have just
defined, we can also prove that all elements of an identity type
obey the groupoid laws, namely
1 Associativity
2 Existence of an identity element
3 Existence of inverses
Also, the groupoid operation, i.e. composition of
paths/sequences, is actually, partial, meaning that not all
elements will be connected via a path. (The groupoid
interpretation refutes the Uniqueness of Identity Proofs.)

Propositional Equality: The Uniqueness of Identity Proofs
“We will call UIP (Uniqueness of Identity Proofs) the follow-
ing property. If a1, a2 are objects of type A then for any
proofs p and q of the proposition “a1 equals a2” there is an-
other proof establishing equality of p and q. (...) Notice
that in traditional logical formalism a principle like UIP
cannot even be sensibly expressed as proofs cannot be
referred to by terms of the object language and thus are
not within the scope of propositional equality.”
Martin Hofmann and Thomas Streicher, “The groupoid
interpretation of type theory”, Twenty-five years of constructive
type theory (Venice, 1995), Oxford Logic Guides, vol. 36,
Oxford Univ. Press, New York, 1998, pp. 83–111.

Sequences of contractions
(λx.(λy.yx)(λw.zw))v ▷1η (λx.(λy.yx)z)v ▷1β (λy.yv)z ▷1β zv
(λx.(λy.yx)(λw.zw))v ▷1β (λx.(λw.zw)x)v ▷1η (λx.zx)v ▷1β zv
(λx.(λy.yx)(λw.zw))v ▷1β (λx.(λw.zw)x)v ▷1β (λw.zw)v ▷1η zv
There is at least one sequence of contractions from the initial term to
the final term. (In this case we have given three!) Thus, in the formal
theory of λ-calculus, the term (λx.(λy.yx)(λw.zw))v is declared to be
equal to zv.
Now, some natural questions arise:
1 Are the sequences themselves normal?
2 Are there non-normal sequences?
3 If yes, how are the latter to be identified and (possibly)
normalised?
4 What happens if general rules of equality are involved?

Propositional equality
P is β-equal or β-convertible to Q (notation P =β Q) iff Q is
obtained from P by a finite (perhaps empty) series of
β-contractions and reversed β-contractions and changes of
bound variables. That is, P =β Q iff there exist P0, . . . , Pn
(n ≥ 0) such that
P0 ≡ P, Pn ≡ Q,
(∀i ≤ n − 1)(Pi ▷1β Pi+1 or Pi+1 ▷1β Pi or Pi ≡α Pi+1).
NB: equality with an existential force.
NB: equality as the reflexive, symmetric and transitive closure
of 1-step contraction: arising from rewriting

The Functional Interpretation of Direct Computation
Equality: Existential Force and Rewriting Path
The same happens with λβη-equality:
Definition 7.5 (λβη-equality) (Hindley & Seldin 2008)
The equality-relation determined by the theory λβη is
called =βη; that is, we define
M =βη N ⇔ λβη ⊢ M = N.
Note again that two terms are λβη-equal if there exists a proof
of their equality in the theory of λβη-equality.

Gentzen’s ND for propositional equality
Remark
In setting up a set of Gentzen’s ND-style rules for equality we
need to account for:
1 definitional versus propositional equality;
2 there may be more than one normal proof of a certain
equality statement;
3 given a (possibly non-normal) proof, the process of
bringing it to a normal form should be finite and confluent.

Equality in Type Theory
Martin-Löf’s Intuitionistic Type Theory:
Intensional (1975)
Extensional (1982(?), 1984)
Remark (Definitional vs. Propositional Equality)
definitional, i.e. those equalities that are given as rewrite
rules, orelse originate from general functional principles
(e.g. β, η, ξ, µ, ν, etc.);
propositional, i.e. the equalities that are supported (or
otherwise) by an evidence (a sequence of substitutions
and/or rewrites)

Definitional Equality
(α) λx.M = λy.[y/x]M (y /
∈ FV(M))
(β) (λx.M)N = [N/x]M
(η) (λx.Mx) = M (x /
∈ FV(M))
(ξ)
M = M′
λx.M = λx.M′
(µ)
M = M′
NM = NM′
(ν)
M = M′
MN = M′N
(ρ) M = M
(σ)
M = N
N = M
(τ)
M = N N = P
M = P

Intuitionistic Type Theory
→-intro
[x : A]
f(x) = g(x) : B
λx.f(x) = λx.g(x) : A → B
(ξ)
→-elim
x = y : A g : A → B
gx = gy : B
(µ)
→-elim
x : A g = h : A → B
gx = hx : B
(ν)
→-reduc
a : A
[x : A]
b(x) : B
(λx.b(x))a = b(a/x) : B
(β)
c : A → B
λx.cx = c : A → B
(η)

Lessons from Curry–Howard and Type Theory
Harmonious combination of logic and λ-calculus;
Proof terms as ‘record of deduction steps’, i.e.
‘deductions-as-terms’
Function symbols as first class citizens.
Cp.
∃xF(x)
[F(t)]
C
C
with
p : ∃xF(x)
[t : D, g(t) : F(t)]
h(g, t) : C
? : C
in the term ‘?’ the variable g gets abstracted from, and this enforces a
kind of generality to g, even if this is not brought to the ‘logical’ level.

Equality in Martin-Löf’s Intensional Type Theory
A type a : A b : A
Idint
A (a, b) type
Idint
-formation
a : A
r(a) : Idint
A (a, a)
Idint
-introduction
a = b : A
r(a) : Idint
A (a, b)
Idint
-introduction
a : A b : A c : Idint
A (a, b)
[x:A]
d(x):C(x,x,r(x))
[x:A,y:A,z:Idint
A (x,y)]
C(x,y,z) type
J(c, d) : C(a, b, c)
Idint
-elimination
a : A
[x : A]
d(x) : C(x, x, r(x))
[x : A, y : A, z : Idint
A (x, y)]
C(x, y, z) type
J(r(a), d(x)) = d(a/x) : C(a, a, r(a))
Idint
-equality

Equality in Martin-Löf’s Extensional Type Theory
A type a : A b : A
Idext
A (a, b) type
Idext
-formation
a = b : A
r : Idext
A (a, b)
Idext
-introduction
c : Idext
A (a, b)
a = b : A
Idext
-elimination
c : Idext
A (a, b)
c = r : Idext
A (a, b)
Idext
-equality

The missing entity
Considering the lessons learned from Type Theory, the
judgement of the form:
a = b : A
which says that a and b are equal elements from domain D, let
us add a function symbol:
a =s b : A
where one is to read: a is equal to b because of ‘s’ (‘s’ being the
rewrite reason); ‘s’ is a term denoting a sequence of equality
identifiers (β, η, ξ, etc.), i.e. a composition of rewrites. In other
words, ‘s’ is the (explicit) computational path from a to b.
(This formal entity is missing in both of Martin-Löf’s
formulations of Identity Types.)

HoTT Book
Path terms are not in the syntax, thus: Encode-Decode Method
“To characterize a path space, the first step is to define
a comparison fibration “code” that provides a more ex-
plicit description of the paths.”
(...)
“There are several different methods for proving that
such a comparison fibration is equivalent to the paths
(we show a few different proofs of the same result
in §8.1). The one we have used here is called
the encode-decode method: the key idea is to de-
fine decode generally for all instances of the fibra-
tion (i.e. as a function Π(x:A+B)code(x) → (inl(a0) =
x)), so that path induction can be used to analyze
decode(x, encode(x, p)).” (section 2.12, p. 95)

Propositional Equality
Id-introduction
a =s b : A
s(a, b) : IdA(a, b)
Id-elimination
m : IdA(a, b)
[a =g b : A]
h(g) : C
J(m, λg.h(g)) : C
Id-reduction
a =s b : A
s(a, b) : IdA(a, b)
Id-intr
[a =g b : A]
h(g) : C
J(s(a, b), λg.h(g)) : C
Id-elim
▷β
[a =s b : A]
h(s/g) : C

Propositional Equality: A Simple Example of a Proof
By way of example, let us prove
ΠxA
ΠyA
(IdA(x, y) → IdA(y, x))
[p : IdA(x, y)]
[x =t y : A]
y =σ(t) x : A
(σ(t))(y, x) : IdA(y, x)
J(p, λt(σ(t))(y, x)) : IdA(y, x)
λp.J(p, λt(σ(t))(y, x)) : IdA(x, y) → IdA(y, x)
λy.λp.J(p, λt(σ(t))(y, x)) : ΠyA(IdA(x, y) → IdA(y, x))
λx.λy.λp.J(p, λt(σ(t))(y, x)) : ΠxAΠyA(IdA(x, y) → IdA(y, x))

Normal form for the rewrite reasons
Strategy:
Analyse possibilities of redundancy
Construct a rewriting system
Prove termination and confluence

Definition (equation)
An equation in our LNDEQ is of the form:
s =r t : A
where s and t are terms, r is the identifier for the rewrite reason, and
A is the type (formula).
Definition (system of equations)
A system of equations S is a set of equations:
{s1 =r1
t1 : A1, . . . , sn =rn
tn : An}
where ri is the rewrite reason identifier for the ith equation in S.

Definition (rewrite reason)
Given a system of equations S and an equation s =r t : A, if
S ⊢ s =r t : A, i.e. there is a deduction/computation of the
equation starting from the equations in S, then the rewrite
reason r is built up from:
(i) the constants for rewrite reasons: { ρ, σ, τ, β, η, ν, ξ, µ };
(ii) the ri’s;
using the substitution operations:
(iii) subL;
(iv) subR;
and the operations for building new rewrite reasons:
(v) σ, τ, ξ, µ.

Definition (general rules of equality)
The general rules for equality (reflexivity, symmetry and
transitivity) are defined as follows:
x : A
x =ρ x : A
(reflexivity)
x =t y : A
y =σ(t) x : A
(symmetry)
x =t y : A y =u z : A
x =τ(t,u) z : A
(transitivity)

Definition (subterm substitution)
The rule of “subterm substitution” is split into two rules:
x =r C[y] : A y =s u : A′
x =subL(r,s) C[u] : A
x =r w : A′ C[w] =s u : A
C[x] =subR(r,s) u : A
where C[x] is the context in which the subterm x appears

Reductions
Definition (reductions involving ρ and σ)
x =ρ x : A
x =σ(ρ) x : A
▷sr x =ρ x : A
x =r y : A
y =σ(r) x : A
x =σ(σ(r)) y : A
▷ss x =r y : A
Associated rewritings:
σ(ρ) ▷sr ρ
σ(σ(r)) ▷ss r

Reductions
Definition (reductions involving τ)
x=r y:D y=σ(r)x:D
x=τ(r,σ(r))x:D ▷tr x =ρ x : D
y=σ(r)x:D x=r y:D
y=τ(σ(r),r)y:D ▷tsr y =ρ y : D
u=r v:D v=ρv:D
u=τ(r,ρ)v:D ▷rrr u =r v : D
u=ρu:D u=r v:D
u=τ(ρ,r)v:D ▷lrr u =r v : D
Associated rewrites: τ(r, σ(r)) ▷tr ρ, τ(σ(r), r) ▷tsr ρ, τ(r, ρ) ▷rrr r,
τ(ρ, r) ▷lrr r.

Reductions
Definition
βrewr -×-reduction
x =r y : A z : B
⟨x, z⟩ =ξ1(r) ⟨y, z⟩ : A × B
× -intr
FST(⟨x, z⟩) =µ1(ξ1(r)) FST(⟨y, z⟩) : A
× -elim
▷mx2l1 x =r y : A
Associated rewriting:
µ1(ξ1(r)) ▷mx2l1 r

Reductions
Definition
βrewr -×-reduction
x =r x′ : A y =s z : B
⟨x, y⟩ =ξ∧(r,s) ⟨x′, z⟩ : A × B
× -intr
FST(⟨x, y⟩) =µ1(ξ∧(r,s)) FST(⟨x′, z⟩) : A
× -elim
▷mx2l2 x =r x′ : A
Associated rewriting:
µ1(ξ∧(r, s)) ▷mx2l2 r

Categorical Interpretation of Computational Paths
Computational Paths form a Weak Category
Theorem
For each type A, computational paths induce a weak
categorical structure Arw where:
objects: terms a of the type A, i.e., a : A
morphisms: a morphism (arrow) between terms a : A and
b : A are arrows s : a → b such that s is a computational
path between the terms, i.e., a =s b : A.
Corollary
Arw has a weak groupoidal structure.

Publications
Recent publications:
1 R. J. G. B. de Queiroz, A. G. de Oliveira and A. F. Ramos.
Propositional equality, identity types, and direct computational
types. Special issue of South American Journal of Formal Logic
(ISSN: 2446-6719) entitled “Logic and Applications: in honor to
Francisco Miraglia by the occasion of his 70th birthday”, M.
Coniglio & H. L. Mariano (eds.), 2(2):245–296, December 2016.
2 T. L. M. de Veras, A. F. Ramos, R. J. G. B. de Queiroz, A. G. de
Oliveira. An alternative approach to the calculation of
fundamental groups based on labeled natural deduction.
arXiv:1906.09107
Oliveira. A Topological Application of Labelled Natural
Deduction. arXiv:1906.09105

Publications
Recent publications (cont’d):
Oliveira. On the Calculation of Fundamental Groups in
Homotopy Type Theory by Means of Computational Paths.
arXiv:1804.01413
2 A. F. Ramos, R. J. G. B. de Queiroz, A. G. de Oliveira. On The
Identity Type as The Type of Computational Paths. EBL’14
special issue of Logic Journal of the IGPL, Oxford Univ Press,
Published online 26 June 2017.
3 A. F. Ramos, R. J. G. B. de Queiroz, A. G. de Oliveira. On the
Groupoid Model of Computational Paths. arXiv:1506.02721

Homotopic Foundations of the Theory of Computation

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