computational paths, identity type and groupoid model
TRANSCRIPT
Computational Paths, identitytype, and the groupoid model
LSFA 2015 Arthur Freitas Ramos
Ruy de Queiroz
Anjolina de Oliveira
Motivation
• Mathematics becoming increasingly abstract and complex
• Automatic Proof Checkers
• Type Theory: Identity type as a bridge between computer science andmath
• Objective: To propose a more intuitive approach to the identity type
Intensional Identity Type
• Witness p as proof of propositional equality between two objects of thesame type
• Connection between extensionality and intensionality
• Homotopy Type Theory: Semantic connection with homotopy (paths)
• HOFMANN - STREICHER 1994: Groupoid structure refutes the principle ofthe unicity of proofs of equality
• LUMSDAINE 2009: weak structure
Identity Type: Formal Construction
Identity Type: New Approach
• We propose a new approach to the identity type
• Objective: to be more intuitive than the classic approach
• Based on Computational Paths, entity originally proposed by Gabbay and Ruy de Queiroz in 1994
• Paths as part of the syntax of type theory: algebra of paths (or calculus of paths)
• Main objectives of this work: Detailed explanation of our new approach andproof of essential properties, such as the groupoid structure
Beta Equality
• Given terms P e Q if we say that they are if:
• reduction:
• reduction together with : theory of
Theory of Lambda-Beta-Eta Equality
• Axiom:
• Inference rules:
Equality Theory for the Product Type
Computational Paths
• Composition of axioms and inference rules s that establishes thepropositional equality between two terms a : A and b : A
• Notation a =s b : A
• Composition done by applications of the transitivity
Computational Paths: Example
• Path between and :
From we obtain
From we obtain
To obtain the final path between and , we just need toconcatenate both paths using the transitivity, obtaining:
Type Identity as the Type of ComputationalPaths: Formalization
Type Identity as the Type of ComputationalPaths: Formalization
Usage example: Symmetry
• Construction of :
Usage example: Transitivity
• Construction of :
Term Rewriting System – LNDEQ-TRS
• Reduction between different paths
• Simple examples: and ; and
• Anjolina (1994) and Ruy & Anjolina (2011): Term rewriting system –LNDEQ-TRS
• Total of 39 reduction rules – 7 essential to the current work
Reductions Involving Symmetry andReflexivity
• Obtained rules:
Reductions Involving Transitivity
• Obtained rules:
Reductions Involving Transitivity
• Obtained rule:
rw-equality
• Each LNDEQ-TRS is known as rw – rule
• From s to t in 1 rule:
• From s to t in multiple rules:
• Rw-equality s =rw t: sequence R0, ....., Rn , with such that:
• Rw-equality is an equivalence class (since it has been defined as a transitive, symmetric and reflexive closure)
LNDRW-TRS2 – Redundancies between Paths ofPaths• Redundancies caused by rw-equality
• There is a version for each redunction previously showed
• Exemple:
rw2-equality
• Rw2-equality: similar to rw-equality
• Rw2 is an equivalence class (analogous to rw)
• Special rule cd2:
Category Arw Induced by Computational Paths
• Objects: terms a: A
• Morphisms: Paths s between objects a,b: A. iff a =s b
• Composition:
• Identity:
• Weak category: Equality holds only up to rw-equality
• Associativity:
• Identity Laws:
ARW is a Weak Groupoid
• Every arrow is an isomorphism
• We need to show that every morphism s has an inverse morphism t
• Set :
Higher Strucure: 2 - Arw
• Category A2rw(a,b) for each pair of objects Arw
• Objects of A2rw(a,b) are paths s: a =s b and morphisms between paths s,r are sets of rw-equalities s =rw r
• Associativity and transitivity hold weakly up to rw2-equality(analogous to Arw)
• Considering equivalence classes of rw2, equalities hold “on the nose” on the second level. Structure [2 – Arw]
• Is [2 – Arw] a bicategory? Is it a weak 2-grupoid?
Bicategory
• Horizontal Composition
• Associativity and identity of the horizontal composition
• Interchange law
• Coherence law: Mac Lane’s pentagon and triangle
[2 – Arw] is a Bicategory
• Horizontal composition :
Given:
We define as:
[2 – Arw] is a Bicategory
• Associativity assoc of : Natural isomorphism between
Given by the isomorphism of each component:
• Identity :
We only need to check each component:
Analogous to : Use trr
[2 – Arw] is a Bicategory
• Interchange law:
[2 – Arw] is a Bicategory
• Coherence laws:
[2 – Arw]: Results
• We have showed that [2 – Arw] is a bicategory
• From the fact that [2 – Arw] is a bicategory, that A2rw is a weakgroupoid and Arw a groupoid, we conclude that [2 – Arw] is a weak 2-groupoid
• It is possible to think of weak groupoids with higher number of levels. Eventually, we can think of a weak groupoid with an infinite numberof levels, the weak
Conclusion
• We have proposed a new approach based on computer
• Computer paths are present in the syntax of type theory, opposite tobeing only a semantical interpretation
• Using computational paths, it is possible to induce higher groupoids.
• We have obtained results compatible with the ones obtained byHofmann-Streicher for the traditional identity type
Future Work
• Mapping of all possible rw2-rules
• The study of induced weak groupoids with order higher than 2
• Possibility of obtaining, to our approach based on computationalpaths, results similar to the ones obtained by Lumsdaine. In otherwords, to prove that it is possible to induce a