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.
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
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.
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
Analogy is one of the most studied representatives of a family of non-classical forms of reasoning working across different domains, usually taken to play a crucial role in creative thought and problem-solving. In the first part of the talk, I will shortly introduce general principles of computational analogy models (relying on a generalization-based approach to analogy-making). We will then have a closer look at Heuristic-Driven Theory Projection (HDTP) as an example for a theoretical framework and implemented system: HDTP computes analogical relations and inferences for domains which are represented using many-sorted first-order logic languages, applying a restricted form of higher-order anti-unification for finding shared structural elements common to both domains. The presentation of the framework will be followed by a few reflections on the "cognitive plausibility" of the approach motivated by theoretical complexity and tractability considerations.
In the second part of the talk I will discuss an application of HDTP to modeling essential parts of concept blending processes as current "hot topic" in Cognitive Science. Here, I will sketch an analogy-inspired formal account of concept blending —developed in the European FP7-funded Concept Invention Theory (COINVENT) project— combining HDTP with mechanisms from Case-Based Reasoning.
Extending the knowledge level of cognitive architectures with Conceptual Spac...Antonio Lieto
Extending the knowledge level of cognitive architectures with Conceptual Spaces (+ a case study with Dual-PECCS: a hybrid knowledge representation system for common sense reasoning). Talk given at Stockholm, September 2016.
"Objective fiction: the semantic construction of web reality" talks about current challenges for semantic technologies, and the Semantic Web in particular, focusing on cognitive and social dimensions of human semantics.
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...Antonio Lieto
We claim that Conceptual Spaces offer a lingua franca that allows to unify and generalize many aspects of the symbolic, sub-symbolic and diagrammatic approaches (by overcoming some of their typical problems) and to integrate them on a common ground. In doing so we extend and detail some of the arguments explored by Gardenfors [23] for defending the need of a conceptual, intermediate, representation level between
the symbolic and the sub-symbolic one. Additionally, we argue that Conceptual Spaces could offer a unifying framework for interpreting many kinds of diagrammatic and analogical representations. As a consequence, their adoption could also favor the integration of diagrammatical representation and
reasoning in Cognitive Architectures
Lecture 2: From Semantics To Semantic-Oriented ApplicationsMarina Santini
From the "Natural Language Processing" LinkedIn group:
John Kontos, Professor of Artificial Intelligence
I wonder whether translating into formal logic is nothing more than transliteration which simply isolates the part of the text that can be reasoned upon using the simple inference mechanism of formal logic. The real problem I think lies with the part of text that CANNOT be translated one the one hand and the one that changes its meaning due to civilization advances. My own proposal is to leave NL text alone and try building inference mechanisms for the UNTRANSLATED text depending on the task requirements.
All the best
John"
Analogy is one of the most studied representatives of a family of non-classical forms of reasoning working across different domains, usually taken to play a crucial role in creative thought and problem-solving. In the first part of the talk, I will shortly introduce general principles of computational analogy models (relying on a generalization-based approach to analogy-making). We will then have a closer look at Heuristic-Driven Theory Projection (HDTP) as an example for a theoretical framework and implemented system: HDTP computes analogical relations and inferences for domains which are represented using many-sorted first-order logic languages, applying a restricted form of higher-order anti-unification for finding shared structural elements common to both domains. The presentation of the framework will be followed by a few reflections on the "cognitive plausibility" of the approach motivated by theoretical complexity and tractability considerations.
In the second part of the talk I will discuss an application of HDTP to modeling essential parts of concept blending processes as current "hot topic" in Cognitive Science. Here, I will sketch an analogy-inspired formal account of concept blending —developed in the European FP7-funded Concept Invention Theory (COINVENT) project— combining HDTP with mechanisms from Case-Based Reasoning.
Extending the knowledge level of cognitive architectures with Conceptual Spac...Antonio Lieto
Extending the knowledge level of cognitive architectures with Conceptual Spaces (+ a case study with Dual-PECCS: a hybrid knowledge representation system for common sense reasoning). Talk given at Stockholm, September 2016.
"Objective fiction: the semantic construction of web reality" talks about current challenges for semantic technologies, and the Semantic Web in particular, focusing on cognitive and social dimensions of human semantics.
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...Antonio Lieto
We claim that Conceptual Spaces offer a lingua franca that allows to unify and generalize many aspects of the symbolic, sub-symbolic and diagrammatic approaches (by overcoming some of their typical problems) and to integrate them on a common ground. In doing so we extend and detail some of the arguments explored by Gardenfors [23] for defending the need of a conceptual, intermediate, representation level between
the symbolic and the sub-symbolic one. Additionally, we argue that Conceptual Spaces could offer a unifying framework for interpreting many kinds of diagrammatic and analogical representations. As a consequence, their adoption could also favor the integration of diagrammatical representation and
reasoning in Cognitive Architectures
Lecture 2: From Semantics To Semantic-Oriented ApplicationsMarina Santini
From the "Natural Language Processing" LinkedIn group:
John Kontos, Professor of Artificial Intelligence
I wonder whether translating into formal logic is nothing more than transliteration which simply isolates the part of the text that can be reasoned upon using the simple inference mechanism of formal logic. The real problem I think lies with the part of text that CANNOT be translated one the one hand and the one that changes its meaning due to civilization advances. My own proposal is to leave NL text alone and try building inference mechanisms for the UNTRANSLATED text depending on the task requirements.
All the best
John"
EDLD813 Paul Gruhn - My Research AutobiographyPaul Gruhn
In my my EDLD813 Learning Theory course, we were required to present an autobiography. Who you are, what you believe shapes how you do research. Which theoretical framework you choose to build your research upon, comes out of knowing who you are.
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.
ANYTIME YOU COMPLETE A PAPERESSAY in this course you must follow .docxfestockton
ANYTIME YOU COMPLETE A PAPER/ESSAY in this course you must follow the APA rules. Use these requirements to attain full credit regardless if they are all listed out in the directions within the course.
· Set paper with 1-inch margins all around. Spacing ‘before’ and ‘after’ set at 0. Entire document including reference list and running is double spaced, Times New Roman, font 12 with all paragraphs indented on the first line by 1/2 inch.
· One title page with APA running heads and page numbers and the title, your name, school, professor’s name and credentials, date. Video for running head directions https://www.youtube.com/watch?v=8u47x2dvQHs
· There is NO ABSTRACT in papers for this course. At the top of page 2 you will repeat the title of your assignment (not in bold, but centered) and then write a brief introduction paragraph of the ENTIRE paper (main sections should be mentioned; THIS INCLUDES ANY TOPICS FOR CASE STUDY SECTIONS).
· Intro is followed by a Level 1 subheading (bold and centered) for the first half of the assignment. This week it’s Nursing Past Related to Current Profession. Any question/point you are addressing under this heading should be marked clearly with Level 2 subheading which are bolded and flush left.
· Immediately after the first section above without any spaces, you will also use another Level 1 subheading (bold and centered) prior to the second half of the assignment which is the case study. This week it’s Professional Nursing Organizations. Again, differentiate which question/point you are answering by using a Level 2 subheading (bold and at the left margin).
· After both sections are discussed at length – there will be ONE Conclusion - needed for all papers as the last Level 1 subheading bold and centered that summarizes the entire paper/knowledge gained
· There will be ONE alphabetized reference page for all sources set “hanging” with references in APA format. All citations need a reference!
· All references listed are cited correctly in APA format in the text! Points are docked for incorrect citations and not meeting the source requirement!
· Should use 3rd person the majority of the time but it is OK to use 1st person when describing a personal experience related to a specific question.
Journal of Theoretical and Philosophical Criminology, Vol 1 (1) 2009
Quantitative versus Qualitative Methods: Understanding Why Quantitative Methods are
Predominant in Criminology and Criminal Justice
George E. Higgins
University of Louisville
Abstract
The development of knowledge is important for criminology and criminal justice. Two
predominant types of methods are available for criminologists’ to use--quantitative and
qualitative methods. A debate is presently taking place in the literature as to which of these
methods is the proper method to provide knowledge in criminology and criminal justice. The
present study outlines the key issues for both methods and suggests that a criminologist’ resea ...
Social constructivism as a philosophy of mathematicsPaul Ernest
This talk presents the outlines of Social constructivism as a philosophy of mathematics by the originator of this theory. It is relevant to philosophy of mathematics, social constructivism, mathematics education
Similar to Propositional equality, identity types, and computational paths (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
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
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.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
Toxic effects of heavy metals : Lead and Arsenicsanjana502982
Heavy metals are naturally occuring metallic chemical elements that have relatively high density, and are toxic at even low concentrations. All toxic metals are termed as heavy metals irrespective of their atomic mass and density, eg. arsenic, lead, mercury, cadmium, thallium, chromium, etc.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
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
Propositional equality, identity types, and computational paths
1. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Propositional Equality, Identity Types
and Computational Paths
Ruy de Queiroz
(joint work with Anjolina de Oliveira)
Centro de Inform´atica
Universidade Federal de Pernambuco (UFPE)
Recife, Brazil
Univ. Lisboa
25 Jul 2017
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
2. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Homotopy Type Theory
Univalent Foundations of Mathematics
Institute for Advanced Study, Princeton
approx. 600p.
Open-source book: The Univalent Foundations Program
27 main participants. 58 contributors
Available on GitHub. Latest version October 3, 2016
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
3. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Homotopy Type Theory
Univalent Foundations of Mathematics
“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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
4. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Homotopy Type Theory
Origins of Research Programme
Vladimir Voevodsky, (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)
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
5. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Homotopy Type Theory
Henk Barendregt on the Evolution of Type Theory
“Type theory as coming originally from Whitehead-Russell
and simplified and essentially extended by Ramsey
(simplifying), Church (adding lambda terms), de Bruijn
(adding dependent types), Scott (adding inductive types
with recursion), Girard (adding higher order types),
Martin-L¨of (showing the natural position and power of
intuitionism) all lead to proof-checking based on type theory
with successes like the full formalization of the 4CT and the
Feit-Thompson theorem by Gonthier and collaborators and
the forthcoming one of the Kepler conjecture by Hales and
collaborators.
Now, there are some difficulties with types (...). For this
reason there is work in progress by Voevodsky and
collaborators to modify this theory.”
(Barendregt 2014, FOM list)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
6. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
HoTT Approach to Spaces: New Proofs
“Homotopy type theory is a synthetic theory of ∞-groupoids” (Michael Shulman)
“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)
• π4(S3
) = Z2 (Brunerie – almost)
• 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)
Some of these are new proofs.”
(Homotopy type theory: towards Grothendieck’s dream, Michael
Shulman, 2013 International Category Theory conference, Sydney.)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
7. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
8. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 1983)
homotopy types ←→ ∞-groupoids
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
9. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Geometry and Logic
Spaces
Approaching the notion of space in mathematics:
1 topological spaces
2 metric spaces
3 manifolds
4 schemes
5 stacks
6 homotopy spaces (starts from points and paths, thus, less
committed to set-theoretic foundations)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
10. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Geometry and Logic
Vladimir Voevodsky
“From an observation by Grothendieck:
formalism of higher equivalences (theory of grupoids)
=
homotopy theory (theory of shapes up to a
deformation)
combined with some other ideas leads to an encoding of
mathematics in terms of the homotopy theory. Unlike the usual
encodings in terms of set theory this one respects
equivalences.”
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
11. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
12. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Algebraic Structure: Groupoids
Steve Awodey
“A groupoid is like a group, but with a partially-defined
composition operation. Precisely, a groupoid can be defined as
a category in which every arrow has an inverse. A group is thus
a groupoid with only one object. Groupoids arise in topology as
generalized fundamental groups, not tied to a choice of
basepoint.”
(Type Theory and Homotopy, 2010.)
“A groupoid is a generalized group, with the multiplication being
only a partial operation – or equivalently, a category in which
every arrow has an inverse.”
(Univalence as a Principle of Logic, 2016.)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
13. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Equality in λ-Calculus: definitional vs. propositional
Proofs of equality as paths
Church’s (1936) original paper:
NB: equality as the reflexive, symmetric and transitive closure of
1-step contraction: rewriting paths. An algebra of paths (with α, β, η,
µ, ν, ξ, ρ, σ, τ)? E.g. σ(σ(r)) = r, τ(τ(t, r), s) = τ(t, τ(r, s)).
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
14. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Equality in λ-Calculus: definitional vs. propositional
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
15. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
16. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
17. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
18. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
19. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Type Theory and its Derivations-as-Terms
Interpretation
Howard on Curry-Howard
“ [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)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
20. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
21. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
22. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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”. (LICS ’94, pp. 208–212.)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
23. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Identity Types
Identity Types as Topological Spaces
According to B. van den Berg and R. Garner (“Topological and
simplicial models of identity types”, ACM Transactions on
Computational Logic, Jan 2012),
“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.”.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
24. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
25. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
26. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
27. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Identity Types: Iteration
The homotopy interpretation (Awodey (2016))
point, path, homotopy, ...
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
28. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
29. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
The Homotopy Interpretation
Steve Awodey (2016)
“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 (...). The
interpretation was shown to be complete in the logical sense by
Gambino and Garner. 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. The next thing one might ask is, how much general homo-
topy theory can be expressed in this way? ”
(A proposition is the (homotopy) type of its proofs, Jan 2016.)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
30. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
31. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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
∀xD
.P(x) a function that turns an element a
into a proof of P(a)
∃xD
.P(x) an element a (witness)
and a proof of P(a)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
32. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
Brouwer–Heyting–Kolmogorov Interpretation: Formally
Canonical proofs rather than truth-values
a proof of the proposition: has the canonical form of:
A ∧ B p, q where p is a proof of A and
q is a proof of B
A ∨ B inl(p) where p is a proof of A or
inr(q) where q is a proof of B
(‘inl’ and ‘inr’ abbreviate
‘into the left/right disjunct’)
A → B λx.b(x) where b(p) is a proof of B
provided p is a proof of A
∀xD
.P(x) Λx.f(x) where f(a) is a proof of P(a)
provided a is an arbitrary individual chosen
from the domain D
∃xD
.P(x) f(a), a where a is a witness
from the domain D, f(a) is a proof of P(a)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
33. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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
what happens with equality statements?
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
34. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
35. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
36. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
37. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
38. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
39. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
40. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
41. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
42. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
43. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
44. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
45. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
46. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
HoTT Book
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.”
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
47. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
48. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
49. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
50. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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.” (Hofmann & Streicher 1996)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
51. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
52. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
53. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
54. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
55. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
56. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
57. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 equations: 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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
58. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
59. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
60. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths
61. What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality The Functional Interpretation of Propositional Equality N
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 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 Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Computational Paths