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https://github.com/andreasabel/wksh-tt-lambda-2018

Workshop on Type Theory and Lambda Calculus
https://github.com/andreasabel/wksh-tt-lambda-2018

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Workshop on Type Theory and Lambda Calculus

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# Workshop on Type Theory and Lambda Calculus



Thursday, 6th September 2018, 10.00-16.00



Room ED, EDIT Building, Chalmers Johanneberg campus, Gothenburg, Sweden



## Schedule (preliminary)



| Time | Talk |

| --- | --- |

| 09.45 | Coffee in the lunch room |

| 10.15 | Rasmus Møgelberg, Bisimulation as path type for guarded recursive types |

| 11.15 | Andy Pitts, Crisp Type Theory in Agda-flat |

| --- | --- |

| 12.00 | Lunch break |

| --- | --- |

| 13.15 | Hugo Herbelin, Proving with side effects |

| 14.00 | Bob Atkey, Syntax and Semantics of Quantitative Type Theory  |

| 14.45 | Coffee break in the lunch room |

| 15.15 | Ken-Etsu Fujita, A constructive proof of the Church—Rosser theorem |

| --- | --- |

| 16.00 | End |



## Talks and abstracts



### Rasmus Møgelberg (ITU Copenhagen), Bisimulation as path type for guarded recursive types



In type theory, coinductive types are used to represent processes, 

and are thus crucial for the formal verification of non-terminating 

reactive programs in proof assistants based on type theory, such 

as Coq and Agda. Currently, programming and reasoning about 

coinductive types is difficult for two reasons: The

need for recursive definitions to be productive, and the lack of 

coincidence of the built-in identity types and the important notion 

of bisimilarity. 



Guarded recursion in the sense of Nakano has recently been 

suggested as a possible approach for dealing with the problem of 

productivity, allowing this to be encoded in types. Indeed, 

coinductive types can be encoded using a combination of guarded

recursion and universal quantification over clocks. 



This talk studies the notion of bisimilarity for guarded recursive types 

in Ticked Cubical Type Theory,

an extension of Cubical Type Theory with guarded recursion. 

As a worked example we study a guarded

notion of labelled transition systems, and show that path equality coincides 

with an adaptation of the usual notion 

of bisimulation for processes. In particular, this implies that guarded recursion

can be used to give simple equational reasoning proofs of 

bisimilarity.



If I have time I will also describe a general result stating that, 

for any functor, an abstract, category theoretic 

notion of bisimilarity for the final guarded coalgebra 

is equivalent (in the sense of homotopy type theory)

to path equality (the primitive notion of equality in cubical type 

theory). This work should be seen as a step towards obtaining 

bisimilarity as path equality for coinductive types using the 

encodings mentioned above.



### Andy Pitts (Cambridge), Crisp Type Theory in Agda-flat



I will describe how recent work by Licata, Orton, Spitters

and myself axiomatizing cubical sets models of univalent foundations

forced us to use a modal type theory, implemented by Andrea Vezzosi as

an exension of Agda called agda-flat.



### Hugo Herbelin (INRIA Paris), Proving with side effects



We shall explore how to use memory assignment to express forcing in

direct style, the same way as classical logic can be seen as direct

style for the double-negation translation. In particular, our

attention shall be retained by a new and short proof with memory

assignment of Gödel's completeness theorem.



### Bob Atkey (Strathclyde University), Syntax and Semantics of Quantitative Type Theory 



I'll present Quantitative Type Theory, a Type Theory that records usage information for each variable in a judgement, based on a previous system by McBride. The usage information is used to give a realizability semantics using a variant of Linear Combinatory Algebras, refining the usual realizability semantics of Type Theory by accurately tracking resource behaviour. We define the semantics in terms of Quantitative Categories with Families, a novel extension of Categories with Families for modelling resource sensitive type theories. 



### Ken-etsu Fujita (Gunma University), A constructive proof of the Church—Rosser theorem



My motivation behind this talk comes from a quantitative analysis of

reduction systems

based on the two viewpoints, computational cost and computational orbit.



In the first part, we show that an upper bound function for the

Church—Rosser theorem

of type-free lambda-calculus with beta-reduction must be in the fourth

level of the

Grzegorczyk hierarchy. That is, the number of reduction steps to arrive

at a common

reduct is bounded by a function in the smallest Grzegorczyk class

properly extending

that of elementary functions. At this level we also find common reducts

for the confluence

property. The proof method developed here can be applied not only to

type-free lambda-calculus

with beta-eta-reduction but also to typed lambda-calculi such as Pure

Types Systems.



In the second part, we propose a formal system of reduction paths for

parallel reduction,

wherein reduction paths are generated from a quiver by means of three

path-operators,

concatenation, monotonicity, and cofinality. Next, we introduce an

equational theory

and reduction rules for the reduction paths, and show that the rules on

paths are

terminating and confluent so that normal paths are obtained. Following

the notion of

normal paths, a graphical representation of reduction paths is provided,

based on which

unique path and universal common-reduct properties are established.

Finally, transformation

rules from a conversion sequence to a reduction path leading to the

universal common-reduct

are given, and path matrices are also defined as block matrices of

adjacency matrices in

order to count reduction orbits.