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Currently, programming and reasoning about \ncoinductive types is difficult for two reasons: The\nneed for recursive definitions to be productive, and the lack of \ncoincidence of the built-in identity types and the important notion \nof bisimilarity. \n\nGuarded recursion in the sense of Nakano has recently been \nsuggested as a possible approach for dealing with the problem of \nproductivity, allowing this to be encoded in types. Indeed, \ncoinductive types can be encoded using a combination of guarded\nrecursion and universal quantification over clocks. \n\nThis talk studies the notion of bisimilarity for guarded recursive types \nin Ticked Cubical Type Theory,\nan extension of Cubical Type Theory with guarded recursion. \nAs a worked example we study a guarded\nnotion of labelled transition systems, and show that path equality coincides \nwith an adaptation of the usual notion \nof bisimulation for processes. In particular, this implies that guarded recursion\ncan be used to give simple equational reasoning proofs of \nbisimilarity.\n\nIf I have time I will also describe a general result stating that, \nfor any functor, an abstract, category theoretic \nnotion of bisimilarity for the final guarded coalgebra \nis equivalent (in the sense of homotopy type theory)\nto path equality (the primitive notion of equality in cubical type \ntheory). This work should be seen as a step towards obtaining \nbisimilarity as path equality for coinductive types using the \nencodings mentioned above.\n\n### Andy Pitts (Cambridge), Crisp Type Theory in Agda-flat\n\nI will describe how recent work by Licata, Orton, Spitters\nand myself axiomatizing cubical sets models of univalent foundations\nforced us to use a modal type theory, implemented by Andrea Vezzosi as\nan exension of Agda called agda-flat.\n\n### Hugo Herbelin (INRIA Paris), Proving with side effects\n\nWe shall explore how to use memory assignment to express forcing in\ndirect style, the same way as classical logic can be seen as direct\nstyle for the double-negation translation. In particular, our\nattention shall be retained by a new and short proof with memory\nassignment of Gödel's completeness theorem.\n\n### Bob Atkey (Strathclyde University), Syntax and Semantics of Quantitative Type Theory \n\nI'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. \n\n### Ken-etsu Fujita (Gunma University), A constructive proof of the Church—Rosser theorem\n\nMy motivation behind this talk comes from a quantitative analysis of\nreduction systems\nbased on the two viewpoints, computational cost and computational orbit.\n\nIn the first part, we show that an upper bound function for the\nChurch—Rosser theorem\nof type-free lambda-calculus with beta-reduction must be in the fourth\nlevel of the\nGrzegorczyk hierarchy. That is, the number of reduction steps to arrive\nat a common\nreduct is bounded by a function in the smallest Grzegorczyk class\nproperly extending\nthat of elementary functions. At this level we also find common reducts\nfor the confluence\nproperty. The proof method developed here can be applied not only to\ntype-free lambda-calculus\nwith beta-eta-reduction but also to typed lambda-calculi such as Pure\nTypes Systems.\n\nIn the second part, we propose a formal system of reduction paths for\nparallel reduction,\nwherein reduction paths are generated from a quiver by means of three\npath-operators,\nconcatenation, monotonicity, and cofinality. Next, we introduce an\nequational theory\nand reduction rules for the reduction paths, and show that the rules on\npaths are\nterminating and confluent so that normal paths are obtained. Following\nthe notion of\nnormal paths, a graphical representation of reduction paths is provided,\nbased on which\nunique path and universal common-reduct properties are established.\nFinally, transformation\nrules from a conversion sequence to a reduction path leading to the\nuniversal common-reduct\nare given, and path matrices are also defined as block matrices of\nadjacency matrices in\norder to count reduction orbits.\n\n\n\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fandreasabel%2Fwksh-tt-lambda-2018","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fandreasabel%2Fwksh-tt-lambda-2018","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fandreasabel%2Fwksh-tt-lambda-2018/lists"}