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14:00 | A homotopy type theory for directed homotopy theory |

16:00 | ABSTRACT. In his 1977 Turing Award address, John Backus introduced the model of functional programming called "FP". FP is a descendant of the Herbrand-Godel notion of recursive definablity and the ancestor of the programming language Haskell. One reason that FP is attractive is that it provides "an algebra of functional programs" However, Backus did not believe that basic FP was powerful enough; "FP systems have a number of limitations..... If the set of primitive functions and functional forms is weak, it may not be able to express every computable function." and he moved on to stronger systems. It turns out that, in this respect, Backus was mistaken. Here we shall show that FP can compute every partial recursive function. Indeed we shall show that FP can simulate the behavior of an arbitrary Turing machine. Our method for doing this is the following. We first observe that the equational theory of Cartesian monoids is a fragment of FP. Cartesian monoids are rather simple algebraic structures of which you know many examples. They also travel under many assumed names such as Cantor algebras, Jonsson-Tarski algebras, and Freyd-Heller monoids. This theory, together with fixed points for all Cartesian monoid polynomials, is contained in FP. Now Cartesian monoids with fixed points can be studied using rewrite techniques, learned from lambda calculus, including confluence and standarization. Turing machines can then be simulated; transitions corresponding to fixed points and computations corresponding to standard reductions to normal form. |

16:30 | String data structures for Chinese monoids ABSTRACT. The structure of Chinese monoid appeared in the classification of monoids with the growth function coinciding with that of the plactic monoid. In this work, we deal with the presentations of the Chinese monoid from the rewriting theory perspective using the notion of string data structures. We define a string data structure associated to the Chinese monoid using the insertion algorithm on Chinese staircases. As a consequence, we construct a finite semi-quadratic convergent presentation of the Chinese monoid and we extend it into a finite coherent presentation of this monoid. |

17:00 | SPEAKER: Christophe Cordero ABSTRACT. The associative operad is the quotient of the magmatic operad by the operad congruence identifying the two binary trees of degree $2$. We introduce here a generalization of the associative operad depending on a nonnegative integer $d$, called $d$-comb associative operad, as the quotient of the magmatic operad by the operad congruence identifying the left and the right comb binary trees of degree $d$. We study the case $d = 3$ and provide an orientation of its space of relations by using rewrite systems on trees and the Buchberger algorithm for operads to obtain a convergent rewrite system. |

17:30 | Normal forms for planar connected string diagrams SPEAKER: Antonin Delpeuch ABSTRACT. In the graphical calculus of planar string diagrams, equality is generated by the left and right exchange moves, which swaps the heights of adjacent vertices. We show that for connected diagrams the left- and right-handed exchanges each give strongly normalizing rewrite strategies. We show that these strategies terminate in O(n^3) steps where n is the number of vertices. We also give an algorithm to directly construct the normal form, and hence determine isotopy, in O(n∙m) time, where m is the number of edges. |

Workshops dinner at Balliol College. Drinks reception from 7.45pm, to be seated by 8:15 (pre-booking via FLoC registration system required; guests welcome).