This document discusses the impact of intuitionistic type theory in mathematics. It introduces identity types as a way to formalize logical equality via proofs of equality between terms. Identity types allow for a connection between term rewriting and geometric concepts like paths and homotopy. Specifically, computational paths can be used to calculate fundamental groups of topological spaces like the fundamental group of a circle. This links type theory with fields like algebraic topology and homotopy theory.
Computational Paths and the Calculation of Fundamental GroupsRuy De Queiroz
Apresentação online na Série "Lógicos em Quarentena", iniciativa conjunta da Soc. Brasileira de Lógica e do Grupo de Interesse em Lógica da Soc. Brasileira de Computação, 20/05/2020
A presentation from my preliminary defense of my Master Thesis, Sofia University “St. Kliment Ohridski”, Supervisor: Prof. Dimitar Vakarelov
A publication based on my thesis:
https://link.springer.com/chapter/10.1007/978-3-319-97879-6_11
Computational Paths and the Calculation of Fundamental GroupsRuy De Queiroz
Apresentação online na Série "Lógicos em Quarentena", iniciativa conjunta da Soc. Brasileira de Lógica e do Grupo de Interesse em Lógica da Soc. Brasileira de Computação, 20/05/2020
A presentation from my preliminary defense of my Master Thesis, Sofia University “St. Kliment Ohridski”, Supervisor: Prof. Dimitar Vakarelov
A publication based on my thesis:
https://link.springer.com/chapter/10.1007/978-3-319-97879-6_11
A presentation from my official defense of my Master Thesis, Sofia University “St. Kliment Ohridski”, Supervisor: Prof. Dimitar Vakarelov
A publication based on my thesis:
https://link.springer.com/chapter/10.1007/978-3-319-97879-6_11
Intuition – Based Teaching Mathematics for EngineersIDES Editor
It is suggested to teach Mathematics for engineers
based on development of mathematical intuition, thus, combining
conceptual and operational approaches. It is proposed to teach
main mathematical concepts based on discussion of carefully
selected case studies following solving of algorithmically generated
problems to help mastering appropriate mathematical tools.
The former component helps development of mathematical intuition;
the latter applies means of adaptive instructional technology
to improvement of operational skills. Proposed approach is applied
to teaching uniform convergence and to knowledge generation
using Computer Science object-oriented methodology.
Talk at the Melbourne Logic Seminar
on Shawn Standefer invitation
Dialectica categories for the Lambek calculus
Valeria de Paiva
The Topos Institute, Berkeley, CA
Abstract:
Dialectica categorical models of the Lambek Calculus were first presented in the Amsterdam Colloquium a long time ago. Following Lambek's lead, we approached the Lambek Calculus from the perspective of Linear Logic and adapted the Dialectica categorical models for Linear Logic to Lambek's non-commutative calculus. The old work took for granted the syntax of the Lambek calculus and only discussed the exciting possibilities of new models for the modalities that Linear Logic introduced. Many years later we find that the work on dialectica models of the Lambek calculus is still interesting and that it might inform some of the most recent work on the relationship between Categorial Grammars and notions of Distributional Semantics.
Thus we revisited the old work, making sure that the syntax details that were sketchy on the first version got completed and verified, using automated tools such as Agda and Ott. Ultimately we are interested in the applicability of the original systems to their intended uses in the construction of semantics of Natural Language. But before we can discuss it, we need to make sure that the mathematical properties that make the Lambek calculus attractive are all properly modeled and this is the main aim of this paper.
We recall the Lambek calculus with its Curry-Howard isomorphic term assignment system. We extend it with a $\kappa$ modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality $!$, as Girard did, re-introducing weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present algebraic semantics and categorical semantics, proved sound and complete for the whole system. Finally, we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.
Automated Education Propositional Logic Tool (AEPLT): Used For Computation in...CSCJournals
The Automated Education Propositional Logic Tool (AEPLT) is envisaged. The AEPLT is an automated tool that simplifies and aids in the calculation of the propositional logics of compound propositions of conjuction, disjunction, conditional, and bi-conditional. The AEPLT has an architecture where the user simply enters the propositional variables and the system maps them with the right connectives to form compound proposition or formulas that are calculated to give the desired solutions. The automation of the system gives a guarantee of coming up with correct solutions rather than the human mind going through all the possible theorems, axioms and statements, and due to fatigue one would bound to miss some steps. In addition the AEPL Tool has a user friendly interface that guides the user in executing operations of deriving solutions.
A presentation from my official defense of my Master Thesis, Sofia University “St. Kliment Ohridski”, Supervisor: Prof. Dimitar Vakarelov
A publication based on my thesis:
https://link.springer.com/chapter/10.1007/978-3-319-97879-6_11
Intuition – Based Teaching Mathematics for EngineersIDES Editor
It is suggested to teach Mathematics for engineers
based on development of mathematical intuition, thus, combining
conceptual and operational approaches. It is proposed to teach
main mathematical concepts based on discussion of carefully
selected case studies following solving of algorithmically generated
problems to help mastering appropriate mathematical tools.
The former component helps development of mathematical intuition;
the latter applies means of adaptive instructional technology
to improvement of operational skills. Proposed approach is applied
to teaching uniform convergence and to knowledge generation
using Computer Science object-oriented methodology.
Talk at the Melbourne Logic Seminar
on Shawn Standefer invitation
Dialectica categories for the Lambek calculus
Valeria de Paiva
The Topos Institute, Berkeley, CA
Abstract:
Dialectica categorical models of the Lambek Calculus were first presented in the Amsterdam Colloquium a long time ago. Following Lambek's lead, we approached the Lambek Calculus from the perspective of Linear Logic and adapted the Dialectica categorical models for Linear Logic to Lambek's non-commutative calculus. The old work took for granted the syntax of the Lambek calculus and only discussed the exciting possibilities of new models for the modalities that Linear Logic introduced. Many years later we find that the work on dialectica models of the Lambek calculus is still interesting and that it might inform some of the most recent work on the relationship between Categorial Grammars and notions of Distributional Semantics.
Thus we revisited the old work, making sure that the syntax details that were sketchy on the first version got completed and verified, using automated tools such as Agda and Ott. Ultimately we are interested in the applicability of the original systems to their intended uses in the construction of semantics of Natural Language. But before we can discuss it, we need to make sure that the mathematical properties that make the Lambek calculus attractive are all properly modeled and this is the main aim of this paper.
We recall the Lambek calculus with its Curry-Howard isomorphic term assignment system. We extend it with a $\kappa$ modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality $!$, as Girard did, re-introducing weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present algebraic semantics and categorical semantics, proved sound and complete for the whole system. Finally, we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.
Automated Education Propositional Logic Tool (AEPLT): Used For Computation in...CSCJournals
The Automated Education Propositional Logic Tool (AEPLT) is envisaged. The AEPLT is an automated tool that simplifies and aids in the calculation of the propositional logics of compound propositions of conjuction, disjunction, conditional, and bi-conditional. The AEPLT has an architecture where the user simply enters the propositional variables and the system maps them with the right connectives to form compound proposition or formulas that are calculated to give the desired solutions. The automation of the system gives a guarantee of coming up with correct solutions rather than the human mind going through all the possible theorems, axioms and statements, and due to fatigue one would bound to miss some steps. In addition the AEPL Tool has a user friendly interface that guides the user in executing operations of deriving solutions.
Propositional Equality, Identity Types and HomotopiesRuy De Queiroz
Palestra, em 05/02/2015, no Workshop de Matemática Aplicada, parte do VII Workshop de Verão em Matemática da Universidade de Brasília (UnB) realizado entre os dias 2 e 13 de fevereiro de 2015.
ASSESSING SIMILARITY BETWEEN ONTOLOGIES: THE CASE OF THE CONCEPTUAL SIMILARITYIJwest
In ontology engineering, there are many cases where assessing similarity between ontologies is required, this is the case of the alignment activities, ontology evolutions, ontology similarities, etc. This paper presents a new method for assessing similarity between concepts of ontologies. The method is based on the
set theory, edges and feature similarity. We first determine the set of concepts that is shared by two ontologies and the sets of concepts that are different from them. Then, we evaluate the average value of similarity for each set by using edges-based semantic similarity. Finally, we compute similarity between
ontologies by using average values of each set and by using feature-based similarity measure too.
ASSESSING SIMILARITY BETWEEN ONTOLOGIES: THE CASE OF THE CONCEPTUAL SIMILARITYdannyijwest
In ontology engineering, there are many cases where assessing similarity between ontologies is required, this is the case of the alignment activities, ontology evolutions, ontology similarities, etc. This paper presents a new method for assessing similarity between concepts of ontologies. The method is based on the set theory, edges and feature similarity. We first determine the set of concepts that is shared by two ontologies and the sets of concepts that are different from them. Then, we evaluate the average value of similarity for each set by using edges-based semantic similarity. Finally, we compute similarity between ontologies by using average values of each set and by using feature-based similarity measure too.
Application of Boolean pre-algebras to the foundations of Computer ScienceMarcelo Novaes
Senior thesis
Field: Mathematical Logic
Supervisor: Steffen Lewitzka
University: Universidade Federal da Bahia (UFBA)
Abstract:
"Increasing the expressiveness of a logical system is a goal of many fields in Computer Science such as Formal Systems, Knowledge construction, Linguistics, Universal Logic and Model Theory. The increasing of this expressiveness can be reached by the use of non-Fregean Logic, a non-classical logic. In non-Fregean Logic, formulas with the same truth value can have different denotations or meanings (also called situations). This concept breaks the Frege Axiom, the reason for the name non-Fregean Logic. Recently, it was shown that there is an equivalence between Boolean pre-algebras and non-Fregean logic models. This fact linked fields which were already using Boolean pre-algebras to represent their semantic models. In this thesis, an investigation on this equivalence is done and applications are exposed in the fields of Modal Logic, Truth Theory, Logic with Quantifiers and Epistemic Logic."
The full thesis can be found at http://repositorio.ufba.br/ri/handle/ri/1938
Chapter 1 Logic and ProofPropositional Logic SemanticsPropo.docxcravennichole326
Chapter 1: Logic and Proof
Propositional Logic Semantics
Propositional variables: p, q, r, s, ... (stand for simple sentences)
T: any proposition that is always true
F: any proposition that is always false
Compound propositions: formed from propositional variables and logical operators (all binary except negation):
Negation ¬
Conjunction ∧
Disjunction ∨
Implication →
Biconditional ↔
Exclusive Or ⊕
Truth Tables: assign all possible T, F to all possible variables, and determines all possible T, F of compound propositions; with n variables there are 2n rows in the table
Negation changes T to F and vice versa
Conjunction is only T if both conjuncts are T
Disjunction is only F is both disjuncts are F
Implication is only F is the antecedent is T and the consequent is F
Biconditional is only true if they have the same tvalue
Exclusive Or is only T if they differ in tvalue
Two (compound) propositions are equivalent (≡) iff they always have the same tvalue (see also below)
English translations:
Conjunction: and, but, although, yet, still, ...
Disjunctions: or, unless
Implication: if, if ... then, only if, when, implies, entails, follows from, is sufficient, is necessary, when, whenever
Biconditional: if and only if, just in case, is necessary and sufficient
A set of propositions is consistent iff there is some assignment of tvalue that makes all T
A set of propositions is inconsistent iff there is no assignment of tvalue that makes all T
A tautology is a compound propositions that is always T
A contradiction is a compound propositions that is always F
A contingency is a compound propositions that is sometimes T, sometimes F
A compound proposition is satisfiable iff some assignment of tvalues make it T
A compound proposition is unsatisfiable iff no assignment of tvalues make it T
Two compound propositions p and q are logically equivalent iff p ↔ q is a tautology
Common equivalences:
DeMorgan’s Laws (Dem)
¬(p ∨ q)≡¬p ∧ ¬q
¬(p ∧ q)≡¬p ∨ ¬q
Identity Laws (Id)
p ∧ T ≡p
p ∨ F ≡p
Domination Laws (Dom)
p ∨ T ≡T
p ∧ F ≡F
Idempotent Laws (Idem)
p ∨ T ≡T
p ∧ p ≡p
Double Negation Law (DN)
¬(¬p) ≡ p
Negation Laws (Neg)
p ∨ ¬p ≡T
p ∧ ¬p ≡F
Commutative Laws (Comm)
p ∨ q ≡q ∨ p
p ∧ q ≡q ∧ p
Associative Laws (Assoc)
(p ∨ q) ∨ r ≡p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡p ∧ (q ∧ r)
Distributive Laws (Dist)
p ∨ (q ∧ r) ≡
(p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡
(p ∧ q) ∨ (p ∧ r)
Absorption Laws (Abs)
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Conditional Laws (Cond)
p →q≡ ¬p ∨ q
¬(p →q)≡ p ∧ ¬q
Biconditional Law (Bicond)
p ↔ q ≡ (p →q) ∧ (q →p)
Quantifier Negation (QNeg)
¬ ∀x P ( x ) ≡ ∃x ¬ P ( x )
¬ ∃x P ( x ) ≡ ∀x ¬ P ( x )
Predicate and Relational Logic (Quantificational Logic, First Order Logic): Semantics
Variables: x, y, z, ...
Predicates/Relations, Propositional Functions: P(x), M(x), Q(x,y), S(x,y,z), ...
Constants: a, b, c, 0, -1, 4, Socrates, ...
Domain (U): set of things the variables range over
Propositional functions are neither T nor F; however, if all the variables are re ...
Similar to Computational Paths and the Calculation of Fundamental Groups (20)
Connections between Logic and Geometry via Term RewritingRuy De Queiroz
Invited talk at the 11th International Conference on Logic and Applications - LAP 2022, held as a hybrid meeting hosted by the Inter University Center Dubrovnik, Croatia, September 26 - 29, 2022.
Contribuição ao painel sobre os aspectos jurídicos, tecnológicos e compliance, "A Nova Lei Geral de Proteção de Dados", com Amália Câmara, Marcílio Braz Jr, e Carlos Sampaio, Auditório da FCAP-UPE, 15/03/2019
Apresentação realizada no Sexto Prospecta | Recife:
a pesquisa científica com foco na responsabilidade social, evento organizado pelo Instituto Futuro / UFPE, em 26/10/2017
Linguagem, lógica e a natureza da matemáticaRuy De Queiroz
"Linguagem, Lógica e a Natureza da Matemática", palestra ministrada no IV Congresso da Associação de Linguagem & Direito - Linguagem & Direito: rios sem discurso?", de 28 a 30/09/2016, na UNICAP, em Recife.
Apresentado no VIII Congresso de Direito da Informática e Telecomunicações (TELECON), realizado no Auditório do Fórum Des. Rodolfo Aureliano, Recife, 22/09/2016
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
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genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
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11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
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Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
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Computational Paths and the Calculation of Fundamental Groups
1. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Computational Paths and
the Calculation of Fundamental Groups
Ruy de Queiroz
(joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica
Universidade Federal de Pernambuco (UFPE)
Recife, Brazil
Col´oquio da Graduac¸ ˜ao - DMat-UFPE
09 Outubro 2020
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
2. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory
2020 Rolf Schock prize in logic and philosophy
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
3. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory
2020 Rolf Schock prize in logic and philosophy
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
4. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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-L¨of’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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
5. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory in Mathematics
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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
6. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory in Mathematics
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
framework 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 λ-terms are the same if they are convertible to one another
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
7. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory in Mathematics
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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
8. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory in Mathematics
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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
9. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory in Mathematics
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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
10. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory in Mathematics
Calculation of fundamental groups of surfaces
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
11. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory in Mathematics
Fundamental Group of Circle S1
We show a surprising connection between the labelled
natural deduction theory developed by now, which has
been developed by the use of computational paths as
labels and the use of LNDEQ-TRS as the logical rules
between these computational paths. We now show a
surprising connection: it is possible to use this system
to obtain one of the main results of algebraic topology:
the calculation of the fundamental group of the circle.
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
12. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Impact of Intuitionistic Type Theory in Mathematics
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
13. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
14. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
15. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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).
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
16. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
17. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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-L¨of came up with Type Theory extending
Howard’s Formulae-as-Types with Natural Numbers and
Universes.
(1973) P. Martin-L¨of came up with Intuitionistic Type Theory
extending Type Theory with Identity Types.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
18. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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-L¨of 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-L¨of’s
type theory was the identity types “x = y”.”
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
19. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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-L¨of 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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
20. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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. ”
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
21. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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.”
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
22. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
23. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Homotopy Type Theory
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.”
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
24. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Homotopy Type Theory
Origins of Research Programme
Vladimir Voevodsky (IAS, Princeton) (b. 4/06/1966 – d. 30/09/2017)
(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.”
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
25. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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.”
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
26. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Algebraic Topology and Fundamental Groups
Calculation of Fundamental Groups
“One of the main ideas of algebraic topology is to consider two
spaces to be equivalent if they have ‘the same shape’ in a sense that
is much broader than homeomorphism.” (...)
“The fundamental group of a space X will be defined so that its
elements are loops in X starting and ending at a fixed basepoint
x0 ∈ X, but two such loops are regarded as determining the same
element of the fundamental group if one loop can be continuously
deformed to the other within the space X”. (...)
“One can often show that two spaces are not homeomorphic by
showing that their fundamental groups are not isomorphic, since it will
be an easy consequence of the definition of the fundamental group
that homeomorphic spaces have isomorphic fundamental groups.”
Algebraic Topology, Allan Hatcher, Cornell Univ, 2001
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
27. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Computational Paths: New Proofs
Calculation of Fundamental Groups
Calculation of Fundamental Groups of Surfaces:
circle
cylinder
M¨obius 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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
28. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Geometry and Logic
Alexander Grothendieck
Alexander Grothendieck
b. 28 March 1928, Berlin, Prussia, Germany
d. 13 November 2014 (aged 86), Saint-Girons, Ari`ege, France
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
29. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
30. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
∞-Groupoids for the foundations of mathematics
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 generalization 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 & M M Kapranov, Comm. of the Moscow Math. Soc., 1990)
“Let me try to explain what homotopy category has to do with the
foundations of mathematics. First of all I want to suggest a
modification of the usual thesis stating that categories are higher level
analogs of sets. We will take a slightly different position. We will
consider groupoids to be the next level analogs of sets.” (“Notes on
homotopy λ-calculus”, Started Jan. 18, Feb. 11, 2006.)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
31. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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.)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
32. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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?
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
33. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
34. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Brouwer–Heyting–Kolmogorov Interpretation
What is a proof of an equality statement?
a proof of the proposition: is given by:
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)?
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
35. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
36. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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 Escard`o: 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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
37. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
What is a proof of an equality statement?
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-L¨of’s Intuitionistic Type Theory (Log Coll ’73, published 1975)
with the so-called Identity Type
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
38. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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-L¨of 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
syntax”.
(M Hofmann, T Streicher, “The groupoid model refutes uniqueness of
identity proofs”. In Logic in Computer Science, 1994 (LICS’94), pp.
208–212.)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
39. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Identity Types
Identity Types as Topological Spaces
“All of this work can be seen as an elaboration of the
following basic idea: that in Martin-L¨of 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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
40. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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
somewhat 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.”
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
41. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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, modulo
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...”
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
42. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Identity Types: Iteration
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
43. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Identity Types: Iteration
The homotopy interpretation (Awodey (2016))
point, path, homotopy, ...
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
44. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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
foundational 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
directly express statements about such objects as, for
example, the ones that my work on 2-theories was
about.”
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
45. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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
combinatorial 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.)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
46. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
47. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
48. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Brouwer–Heyting–Kolmogorov Interpretation
What is a proof of an equality statement?
a proof of the proposition: is given by:
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?
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
49. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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.)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
50. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Propositional Equality: The Uniqueness of Identity Proofs
“We will call UIP (Uniqueness of Identity Proofs) the
following property. If a1, a2 are objects of type A then for
any proofs p and q of the proposition “a1 equals a2” there is
another 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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
51. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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. (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?
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
52. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Equality
Propositional equality
Definition (Hindley & Seldin 2008)
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
53. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
54. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of 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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
55. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computation
Equality in Type Theory
Martin-L¨of’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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
56. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computation
Definitional Equality
Definition (Hindley & Seldin 2008)
(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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
57. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computation
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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
58. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
59. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Equality in Martin-L¨of’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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
60. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Equality in Martin-L¨of’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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
61. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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-L¨of’s
formulations of Identity Types.)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
62. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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 explicit 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 define
decode generally for all instances of the fibration (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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
63. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
64. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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))
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
65. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
Strategy:
Analyse possibilities of redundancy
Construct a rewriting system
Prove termination and confluence
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
66. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
67. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
68. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
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)
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
69. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
70. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
71. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
72. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
73. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
74. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
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.
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
75. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
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
3 T. L. M. de Veras, A. F. Ramos, R. J. G. B. de Queiroz, A. G. de
Oliveira. A Topological Application of Labelled Natural
Deduction. arXiv:1906.09105
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups
76. What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
The Functional Interpretation of Direct Computations
Publications
Recent publications (cont’d):
1 T. L. M. 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
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
Ruy de Queiroz (joint work with A. de Oliveira, A. Ramos, T. Veras)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Computational Paths and the Calculation of Fundamental Groups