# 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.