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**Abstract**

The Kameda-Weiner algorithm takes a machine (nondeterministic finite automaton) as input, and provides an optimal machine (state-minimal nondeterministic finite automaton) as output. In this talk I will discuss work which provides a clear explanation of it, by translating the various syntactic constructs into more meaningful order-theoretic ones, and then composing them together to prove correctness.

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**Abstract**

A key result in computational learning theory is Dana Angluin's L* algorithm that describes how to learn a regular language, or a deterministic finite automaton (DFA), using membership and equivalence queries. In my talk I will present a generalisation of this algorithm using ideas from coalgebra and modal logic — please note, however, that prior knowledge of these topics will not be required.

In the first part of my talk I will recall how the L* algorithm works and establish a link to the notion of a filtration from modal logic. Furthermore I will provide a brief introduction to coalgebraic modal logic. In the second part of my talk I will present a generalisation of Angluin's original algorithm from DFAs to coalgebras for an arbitrary finitary set functor T in the following sense: given a (possibly infinite) pointed T-coalgebra that we assume to be regular (i.e. having an equivalent finite representation) we can learn its finite representation by (i) asking "logical queries" (corresponding to membership queries) and (ii) making conjectures to which a teacher has to reply with a counterexample (equivalence queries). This covers (known variants of) the original L* algorithm and algorithms for learning Mealy and Moore machines. Other examples are infinite streams, trees and bisimulation quotients of various types of transition systems.

Joint work with Simone Barlocco.

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**Abstract**

Quotient inductive inductive types (QIITS) are set-truncated mutually defined higher inductive types. I am going to discuss two applications of QIITs:

1. define an internal syntax of Type Theory without reference to untyped preterms;

2. define a version of the partiality monad that doesn't require countable choice.

On the one hand I think that these applications are interesting because they represent applications of HoTT which have nothing to do with homotopy theory; on the other hand they are clearly not very higher order (in the sense of truncation levels) but can be defined in the set-truncated fragment of HoTT. Hence my question: what are interesting applications of higher types which are not directly related to synthetic homotopy theory?

This talk is based on joint work with Paolo Capriotti, Nils Anders Danielsoon, Gabe Dijkstra, Ambrus Kaposi and Nicolai Kraus.

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**Abstract**

A traditional source of complaint from CS undergraduates (especially in the USA, but in other places, too) is that they are made to learn too much standard issue mathematics with little apparent relevance to computation. Differential calculus (with its usual presentational focus on physical systems) is often picked upon as the archetype. What we see in action is the fragile male ego: they are not so quick to complain about the unimportance of things they do not find difficult. All of which makes more delicious the irony that differential operators have a key role to play in understanding discrete structures, such as automata, datatypes, execution stacks, and plenty more.

The basic idea is as follows: to put your finger over any single K in the pair of words

BREKEKEKEX KOAXKOAX

you must choose either to put your finger over a single K in BREKEKEKEX and pair with KOAXKOAX intact, or to leave BREKEKEKEX intact and cover a K in KOAXKOAX. You have just followed Leibniz's rule for differentiating a product (with respect to K), and computed a one-hole context for a K in a data structure.Newton, of course, would point out that such derivatives arise as the limit of a divided difference, a concept worthy of study in more generality. I would point out that divided differences are often definable, even in situtations when neither division nor difference makes much apparent sense. Notably, Brzozowski's derivative for regular languages is a divided difference (even though it is not Leibniz's derivative).

I'll work mainly with containers (which look a lot like power series) but make sure there are plenty of concrete examples. In practice, it becomes rather useful to compute derivatives by pattern matching on types, which is especially funny as symbolic differentiation is the first example in the literature of computing anything by pattern matching at all.

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Strathclyde, room LT1415

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Strathclyde, Room MC301 (McCance Building)

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**Abstract**

Blockchains, i.e. decentralised, distributed data structures which can also carry executable code for a non-standard execution environment, introduce new models of computation. Decentralised, here, means, informally speaking, "without central control", e.g. a currency without a (central) bank, but much more. Blockchains support the recently introduced virtual currencies, a la Bitcoin, and a new class of decentralised applications, including smart contracts. In this talk we will introduce the main aspects of a blockchain, with particular reference to the Bitcoin blockchain as a paradigmatic case of such a new model of computation, and also touching on smart contracts. No previous knowledge of bitcoin/blockchain required for this introductory talk.

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**Abstract**

Statistical models in e.g. machine learning are traditionally expressed in some sort of flow charts. Writing sophisticated models succinctly is much easier in a fully fledged programming language. The programmer can then rely on generic inference algorithms instead of having to craft one for each model. Several such higher-order functional probabilistic programming languages exist, but their semantics, and hence correctness, are not clear. The problem is that the standard semantics of probability theory, given by measurable spaces, does not support function types. I will describe how to get around this.

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**Abstract**

We compare the expressive power of three programming abstractions for user-defined computational effects: Bauer and Pretnar's effect handlers, Filinski's monadic reflection, and delimited control. This comparison allows a precise discussion about the relative merits of each programming abstraction.

We present three calculi, one per abstraction, extending Levy's call-by-push-value. These comprise syntax, operational semantics, a natural type-and-effect system, and, for handlers and reflection, a set-theoretic denotational semantics. We establish their basic meta-theoretic properties: adequacy, soundness, and strong normalisation. Using Felleisen's notion of a macro translation, we show that these abstractions can macro-express each other, and show which translations preserve typeability. We use the adequate finitary set-theoretic denotational semantics for the monadic calculus to show that effect handlers cannot be macro-expressed while preserving typeability either by monadic reflection or by delimited control. We supplement our development with a mechanised Abella formalisation.

Joint work with Yannick Forster, Sam Lindley, and Matija Pretnar.

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**Abstract**

I'll report on my attempts to design a cubical type theory together with Dan Licata and Ed Morehouse during my visit to Wesleyan University, Middletown, Connecticut. We had something which seemed quite promising, but that falls apart just short of the finish line; I'll tell you about it in the hope of miraculous rescue from the audience. However, I'll start from basics so that everyone has a chance to join in in the fun. Mentions of Donald Trump will be kept to a minimum.

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**Abstract**

PCF is the prototypical functional programming language, with two data types (naturals and booleans), lambda-abstraction and recursion. PCF was introduced by Gordon Plotkin in his seminal "LCF Considered as a Programming Language" paper from 1977. Despite PCF's simplicity, its semantics is theoretically interesting. I will introduce PCF, its operational semantics, the "standard" domain-theoretic denotational semantics and show that the two agree on closed programs. Finally, I will discuss observational equivalence for PCF and show that the denotational semantics fails to be "fully abstract".

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**Abstract**

Hoare Logic is a logic for proving properties of programs of the form: if the initial state satisfies a precondition, then the final state satisfies a postcondition. Hoare logic proofs are structured around the structure of the program itself, making the system a compositional one for reasoning about pieces of programs. I'll introduce Hoare Logic for a little imperative language with WHILE loops. I'll then motivate Separation Logic, which enriches Hoare Logic with a Frame Rule for local reasoning.

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**Abstract**

I will try to follow on from yesterday's introduction by getting to nitty gritty of bitcoin/blockchain. I won't assume attendance of the seminar but will try not to repeat it!

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**Abstract**

In this 101 I plan to discuss omega-automata, i.e., finite automata that operate on infinite words/streams. These automata form an important tool for the specification and verification of the ongoing, possibly infinite behaviour of a system. In my talk I will provide the standard definition(s) of omega-automata and highlight what makes omega-automata difficult from a coalgebraic perspective. Finally, I am going to discuss the work by Ciancia & Venema that provides a first coalgebraic representation of a particular type of omega-automata, so-called Muller automata.

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**Abstract**

Automata learning is a well known technique to infer a finite state machine from a set of observations. One important algorithm for automata learning is the L* algorithm by Dana Angluin. In this 101, I will explain how the L* algorithm works via an example. Afterwards, I will discuss the ingredients of the algorithm both in the standard framework by Angluin and in a recently developed categorical/coalgebraic framework by Jacobs & Silva. Lastly, I plan to outline the proof of the minimality of the automaton that is built by the learning algorithm.

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**Abstract**

In this 101 I outline the syntax and semantics of classical first order predicate logic. I try to also mention some of the characteristic properties of first order logic such as compactness, the Löwenheim-Skolem theorem or locality properties in finite model theory.

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**Abstract**

Our POPL attendees will tell us about their favourite talks, the latest research gossip and show us their most scenic photos from POPL in Paris.

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**Abstract**

I explain how we can use automata and games to understand the behaviour of modal fixpoint logics.

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**Abstract**

We explore the design and implementation of Frank, a strict functional programming language with a bidirectional effect type system designed from the ground up around a novel variant of Plotkin and Pretnar's effect handler abstraction.

Effect handlers provide an abstraction for modular effectful programming: a handler acts as an interpreter for a collection of commands whose interfaces are statically tracked by the type system. However, Frank eliminates the need for an additional effect handling construct by generalising the basic mechanism of functional abstraction itself. A function is simply the special case of a Frank *operator* that interprets no commands.

Moreover, Frank's operators can be *multihandlers* which simultaneously interpret commands from several sources at once, without disturbing the direct style of functional programming with values.

Effect typing in Frank employs a novel form of effect polymorphism which avoid mentioning effect variables in source code. This is achieved by propagating an *ambient ability* inwards, rather than accumulating unions of potential effects outwards.

I'll give a tour of Frank through a selection of concrete examples.

(Joint work with Conor McBride and Craig McLaughlin)

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**Abstract**

The algebras of many-valued Lukasiewicz logics (MV algebras) as well as the theory of quantum measurement (Effect algebras) have undergone considerable development in the 1980s and 1990s; they now constitute important research fields, with connections to several contemporary areas of mathematics, logic, and theoretical computer science.

Both subjects have recently attracted considerable interest among groups of researchers in categorical logic and foundations of quantum computing. I will give a leisurely introduction to MV algebras (and their associated logics), as well as the more general world of effect algebras. If time permits, we will also illustrate some new results (with Mark Lawson, Heriot-Watt) on coordinatization of some concrete MV-algebras using inverse semigroup theory.

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Room 301, McCance building, Strathclyde

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**Abstract**

The techniques used by the generic programming community have taught us that we can greatly benefit from exposing the common internal structure of a family of objects. One can for instance derive once and for all a wealth of iterators from an abstract characterisation of recursive datatypes as fixpoints of functors.

Our previous work on type and scope preserving semantics and their properties has made us realise that numerous semantics of the lambda calculus can be presented as instances of the fundamental lemma associated to an abstract notion of 'Model'. This made it possible to avoid code duplication as well as prove these semantics' properties generically.

Putting these two ideas together, we give an abstract description of syntaxes with binding making both their recursive and scoping structure explicit. The fundamental lemma associated to these syntaxes can be instantiated to provide the user with proofs that its language is stable under renaming and substitution as well as provide a way to easily define various evaluators.

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**Abstract**

I will be discussing composition in a model of economic game theory and methods for representing the impact of choice on subsequent games. Time permitting I will also work through some games using this model.

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**Abstract**

A filter P is a consistent deductively closed set of predicates. A filter is prime when

(φ ∨ ψ) ∈ P ⇒ (φ ∈ P ∨ ψ ∈ P)

In words: if phi-or-psi is in P then phi is in P or psi is in P. Primeness gives soundness for disjunction.Using this it is not hard to construct a semantics to propositional logic in which a predicate φ "means" the set of prime filters containing it. This is a standard "trick" for building semantics and is an extremely useful proof-method.

I have developed a semantics for predicate logic and also for the lambda-calculus based on similar notions of filter, but in a

The resulting semantics is rich and interesting. In a sentence: nominal techniques help us to extend the notion of Stone representation and duality from propositional logic to full first-order logic (also with equality, if we wish, and also to other logics and calculi with variables and quantifiers).

I will give a detailed description of the filter-style conditions involved, and discuss some of what I think they tell us about predicates and quantification in logic and computation. More information can also be found in two papers here:

http://www.gabbay.org.uk/papers.html#semooc

http://www.gabbay.org.uk/papers.html#repdul

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**Abstract**

We discuss a number of semantic properties pertaining to formulas of the modal mu-calculus. For each of these properties we provide a corresponding syntactic fragment, in the sense that a mu-calculus formula \phi has the given property iff it is equivalent to a formula \phi' in the corresponding fragment. Since this formula \phi' will always be effectively obtainable from \phi, as a corollary, for each of the properties under discussion, we prove that it is decidable in elementary time whether a given mu-calculus formula has the property or not.

The properties that we study have in common that they all concern the dependency of the truth of the formula at stake, on a single proposition letter p. In each case the semantic condition on \phi will be that \phi, if true at a certain state in a certain model, will remain true if we restrict the set of states where p holds, to a special subset of the state space. Important examples include the properties of complete additivity and (Scott) continuity, where the special subsets are the singletons and the finite sets, respectively.

Our proofs for these characterisation results will be automata-theoretic in nature; we will see that the effectively defined maps on formulas are in fact induced by rather simple transformations on modal automata.

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**Abstract**

Modal logic provides a simple, yet surprisingly powerful, language for specifying properties of coalgebras. In this talk I introduce the basic modal logic that is interpreted on relational structures. My aim is to provide an idea how modal logic relates to other logics, such as first-order and intuitionistic logic, and to the duality between algebraic and coalgebraic structures.

If time permits, I might also give a very informal warm-up for the modal mu-calculus which is the topic of next week's talk.

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**Abstract**

I'll give a standard overview of Simply Typed Lambda Calculus (STLC) (syntax, typing and computation rules) in a well-typed setting, and then modify it to show STLC in a bidirectional setting.

Afterwards I'll show Strong Confluence (Church-Rosser theorem) in the bidirectional setting.

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**Abstract**

I shall give a brief introduction to System F.

I will then explain how to capture our intuition about polymorphic functions behaving uniformly by relational parametricity, and talk about ongoing work to find a notion of proof-relevant parametricity.

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**Abstract**

I will give a basic introduction to data types and initial-algebra semantics. The meaning of a data type is given as the initial object in a category of types with the corresponding constructors. Initiality immediately allows the modelling of a non-dependent recursion principle. I'll show how this can be upgraded to full dependent elimination, also known as induction, by using the uniqueness of the mediating arrow; in fact, induction is equivalent to recursion plus uniqueness. All possibly unfamiliar terms in this abstract will also be explained.

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**Abstract**

The core subject of Computer Science is "generated behaviour" (quiz: who said this?). Coalgebra provides the categorical formalisation of generated behaviour. I am planning to provide a first, very basic introduction to coalgebra. This will consist of two parts: i) coinduction & corecursion as means to define & reason about the (possibly) infinite behaviour of things; ii)modal logics for coalgebras.

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**Abstract**

For a category C we consider the endomorphism category End(C) and the subcategory of automorphisms Aut(C) -> End(C). It has been observed that for C the category of finite sets, finite dimensional vector spaces, or compact metric spaces this inclusion functor admits a simultaneous left and right adjoint.

We give general criteria for the existence of such adjunctions for a broad class of categories which includes FinSet, FinVect and CompMet as special cases. This is done using the language of factorisation systems and by introducing a notion of eventual image functors which provide a general method for constructing adjunctions of this kind.

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**Abstract**

Or: how someone else found the maths to justify my dogma

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**Abstract**

Practice talk for Quantum Interactions.

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**Abstract**

Practice talk for LICS.

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi have shown that, given a suitable distribution law, a pair of Hopf algebras forms two Frobenius algebras. Coming from the perspective of quantum theory, we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise \cite{Bonchi2014a} by including non-trivial dynamics of the underlying object -- the so-called phase group -- and investigate the effects of finite dimensionality of the underlying model, and recover the system of Bonchi et al as a subtheory in the prime power dimensional case. We show that the presence of a non-trivial phase group means that the theory cannot be formalised as a distributive law.

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McCance building, Strathclyde

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**Abstract**

Practice talk for QPL.

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.

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**Abstract**

I'll show how to generalise some results from algebra (think groups, rings, R-modules etc.) to a categorical setting using factorisation systems and an appropriate notion of finiteness on the objects of a category.

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**Abstract**

Formal languages and automata are taught to every computer science student. However, the student will most likely not see the beautiful coalgebraic foundations.

In this talk, I recapitulate how infinite trees can represent formal languages (sets of strings). I explain Agda's coinduction mechanism based on copatterns and demonstrate that it allows an elegant representation of the usual language constructions like union, concatenation, and Kleene star, with the help of Brzozowski derivatives. We will also investigate how to reason about equality of languages using bisimulation and coinductive proofs.

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School of Informatics, Edinburgh

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**Abstract**

ThreadSafe is a static analysis tool for finding bugs in concurrent Java code that has been used by companies across the world to analyse and find bugs in large mission industrial applications. I will talk about how ThreadSafe works, and our experiences in applying static analysis technology to the "real world".

ThreadSafe is available from http://www.contemplateltd.com/

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**Abstract**

I will sketch an alternative approach to economic game theory based upon the computer science idea of compositionality: concretely we i) give a number of operators for building up complex and irregular games from smaller and simpler games; and ii) show how the Nash equilibrium of these complex games can be defined recursively from their simpler components. We apply compositional reasoning to sophisticated games where agents must reason about how their actions affect future games and how those future games effect the utility they receive. This forces us into a second innovation — we augment the usual lexicon of games with a dual notion to utility because, in order for games to accept utility, this utility must be generated by other games. Our third innovation is to represent our games as string diagrams so as to give a clear visual interface to manipulate them. Our fourth, and final, innovation is a categorical formalisation of these intuitive diagrams which ensures our reasoning about them is fully rigorous.

The talk will be general so as appeal to as wide an audience as possible. In particular, no knowledge of category theory will be assumed!

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**Abstract**

Motivated by the desire to understand the combinatorics of graph rewriting systems, it proved necessary to invent a formulation of graph rewriting itself that is not based on category theoretic structures as in the traditional framework, but on the concept of diagrammatic combinatorial Hopf algebras and reductions thereof. In this talk, I will present how the classical example of the Heisenberg-Weyl algebra of creation and annihilation of indistinguishable particles, which can alternatively be interpreted as the algebra of discrete graph rewriting, gave the initial clues for this novel framework. In hindsight, to pass from the special case of discrete graph rewriting to the case of general graph rewriting required every aspect of the framework of diagrammatic combinatorial Hopf algebras as a guideline for the construction, yet none of the traditional category theoretic ideas, whence one might indeed consider this reformulation as an independent formulation of graph rewriting.

The new framework results in a number of surprising results even directly from the formulation itself: besides the two main variants of graph rewriting known in the literature (DPO and SPO rewriting), there exist two more natural variants in the new framework. For all four variants, graph rewriting rules are encoded in so-called rule diagrams, with their composition captured in the form of diagrammatic compositions followed by one of four variants of reduction operations. Besides the general structure theory of the resulting algebras aka the rule algebras, one of the most important results to date of this framework in view of applications is the possibility to formulate stochastic graph rewriting systems in terms of the canonical representations of the rule algebras. Notably, this is closely analogous to the formulation of chemical reaction systems in terms of the canonical representation of the Heisenberg-Weyl algebra aka the bosonic Fock space. The presentation will not assume any prior knowledge of the audience on the particular mathematics required for this construction, and will be given on the whiteboard. The work presented is the result of a collaboration with Vincent Danos and Ilias Garnier (ENS Paris/LFCS University of Edinburgh), and (in an earlier phase) with Tobias Heindel (University of Copenhagen).

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**Abstract**

In this talk, we explore the fundamental category-theoretic structure needed to model relational parametricity (i.e., the fact that polymorphic programs preserve all relations) for the polymorphic lambda calculus (a.k.a. System F). Taken separately, the notions of categorical model of impredicative polymorphism and relational parametricity are well-known (lambda2-fibrations and reflexive graph categories, respectively). Perhaps surprisingly, simply combining these two structures results in a notion that only enjoys the expected properties in case the underlying category is well-pointed. This rules out many categories of interest (e.g. functor categories) in the semantics of programming languages.

To circumvent this restriction, we modify the definition of fibrational model of impredicative polymorphism by adding one further ingredient to the structure: comprehension in the sense of Lawvere. Our main result is that such comprehensive models, once further endowed with reflexive-graph-category structure, enjoy the expected consequences of parametricity. This is proved using a type-theoretic presentation of the category-theoretic structure, within which the desired consequences of parametricity are derived. Working in this type theory requires new techniques, since equality relations are not available, so that standard arguments that exploit equality need to be reworked.

This is joint work with Neil Ghani and Alex Simpson, and a dry run for a talk in Cambridge the week after.

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**Abstract**

String diagrams give a powerful graphical syntax for morphisms in symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory.

An important role in many such approaches is played by equational theories of diagrams, which can be oriented and used as rewrite systems. In this talk, I'll lay the foundations for this form of rewriting by interpreting diagrams combinatorially as typed hypergraphs and establish the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hypergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure.

If there's time, I'll also discuss some of the results we have in developing the rewrite theory of hypergraphs for SMCs, namely termination proofs via graph metrics and strongly convex critical pair analysis.

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**Abstract**

In this talk, we consider extending Lawvere theories to allow enrichment in a base category such as CMonoid, Poset or Cat. In doing so, we see that we need to alter the formulation in a fundamental way, using the notion of cotensor, a kind of limit that is hidden in the usual accounts of ordinary category theory but is fundamental to enriched category theory. If time permits, we will briefly consider the specific issues that arise when one has two-dimensional structure in the enriching category, as exemplified by Poset and Cat.

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**Abstract**

In 1963, Bill Lawvere characterised universal algebra in category theoretic terms. His formulation being category theoretic was not its central contribution: more fundamental was its presentation independence. Three years later, monads were proposed as another category theoretic formulation of universal algebra. Overall, the former are technically better but the relationship is particularly fruitful and the latter are more prominent, cf Betamax vs VHS. So we study Lawvere theories carefully in the setting of universal algebra and in relation to monads.

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**Abstract**

I have recently begun to learn about the Cubical Type Theory of Coquand et al., as an effective computational basis for Voevodsky's Univalent Foundations, inspired by a model of type theory in cubical sets. It is in some ways compelling in its simplicity, but in other ways intimidating in its complexity. In order to get to grips with it, I have begun to develop my own much less subtle variation on the theme. If I am lucky, I shall get away with it. If I am unlucky, I shall have learned more about why Cubical Type Theory has to be as subtle as it is.

My design separates Coquand's all-powerful "compose" operator into smaller pieces, dedicated to more specific tasks, such as transitivity of paths. Each type path Q : S = T, induces a notion of value path s {Q} t, where either s : S, or s is •, "blob", and similarly, t : T or t = •. A "blob" at one end indicates that the value at that end of the path is not mandated by the type. This liberalisation in the formation of "equality" types allows us to specify the key computational use of paths between types, extrusion:

if Q : S = T and s : S, then s ⌢• Q : s {Q} •

That is, whenever we have a value s at one end of a type path Q : S = T, we can extrude that value across the type path, getting a value path which is s at the S end, but whose value at the T end is not specified in advance of explaining how to compute it. Extrusion gives us a notion of coercion-by-equality which is coherent by construction. It is defined by recursion on the structure of type paths. Univalence can be added to the system by allowing the formation of types interpolating two equivalent types, with extrusion demanding the existence of the corresponding interpolant values, computed on demand by means of the equivalence.

So far, there are disconcerting grounds for optimism, but the whole of the picture has not yet emerged: I may just have pushed the essential complexity into one corner, or the whole thing may be holed below the waterline. But if it does turn out to be nonsense, it will be nonsense for an interesting reason.

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**Abstract**

I will present work in progress on a (co)algebraic framework that allows to uniformly study dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL). Underlying our framework is the basic observation that the program/game constructs of PDL/GL arise from monad structure, and that the axioms of these logics correspond to compatibility requirements between the modalities and this monad structure. So far we have a general soundness and completeness result for PDL-like logics wrt T-coalgebras for a monad T. I will discuss our completeness theorem, its limitations and plans for extending our results. [For the latter we might require the help of koalas, wallabies and wombats.]

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**Abstract**

Infinity-categories simultaneously generalise topological spaces and categories. Intuitively, a (weak) infinity-category should have objects, morphisms, 2-morphisms, 3-morphisms, ... and identity morphisms and composition which is suitably unital and associative up to a higher (invertible) morphism (the number 1 in (infinity, 1)-category means that k-morphisms for k > 1 are invertible) . The trouble begins when one naively tries to make these coherence conditions precise; already 4-categories famously requires 51 pages to define explicitly. Instead, one typically turns to certain "models" of infinity-categories that encode all this data implicitly, usually as some kind of simplicial object with additional properties. I will introduce two such models: quasicategories and complete Segal spaces. If time allows, I will also discuss hopes and dreams about internalising these notions in Type Theory, which should give a satisfactory treatment of category theory in Type Theory without assuming Uniqueness of Identity Proofs.

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**Abstract**

I've been working with Jules Hedges on a compositional model of game theory. After briefly reminding you of the model, I'll discuss where we are at – namely the definition of morphisms between games, and the treatment of choice and iteration of games. I'm hoping you will be able to shed some light on this murky area. There is a draft paper if anyone is interested.

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**Abstract**

The probabilistic calculus introduced in the eponymous paper by Robin Adams and Bart Jacobs is inspired by quantum theory by considering that conditional probabilities can be seen as side-effect-free measurements. A type-theoretic treatment of this semantic observation leads, once equipped with suitable computation rules, to the ability to do exact conditional inference.

I will present the type theory and the accompanying computation rules proposed in the paper and discuss some of the interesting open questions I will be working on in the near future.

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**Abstract**

Coherence spaces are a simplification of Scott domains, introduced by Girard to give a semantics to the polymorphic lambda-calculus. While investigating the structure of coherence spaces, Girard noticed that the denotation of the function type in coherence spaces can be decomposed into two independent constructions: a linear ("use-once") function space, and a many-uses-into-one-use modality. And so Linear Logic was discovered.

Coherence spaces are interesting because they model computation at a low level in terms of interchange of atomic 'tokens' of information. This makes them a useful tool for understanding several different computational phenomena.

In this talk, I'll show how coherence spaces naturally model session types, via Wadler's interpretation of Classical Linear Logic as a session-typed pi-calculus, and how that interpretation extends to an interpretation of a dependently typed version of session types.

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**Abstract**

I would like to speak about the categorical structure of the category of von Neumann algebras, with as morphisms normal, completely positive, unital linear maps. For some years my colleagues and I have worked on identifying basic structures in this category, and while surprisingly many things do not exist or do not work in this category (it's not a topos or even an extensive category, there's no epi-mono factorisation system, there is no dagger, colimits — if they exist at all — are horrendous...), we did find some structure (the products behave reasonable in some sense, there is a 'quotient', and 'comprehension', and we have a universal property for the minimal Stinespring dilation, and a universal property for M_2—the qubit). There is no deep category theory involved by any standards, and I promise I will spare you the functional analysis, so it should be a light talk.

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**Abstract**

Due to popular demand I am going to give a brief introduction to Marc Pauly's Coalition Logic, a propositional modal logic that allows to reason about the power of coalitions in strategic games. I will provide motivation and basic definitions. Furthermore I am planning to discuss how the logic can be naturally viewed as a coalgebraic logic and what we gain from the coalgebraic perspective. Finally — if (preparation) time permits — I am going to say how the logic can be applied to the area of mechanism design.

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**Abstract**

In societies the notion of a law is not given by nature. Instead social dynamics are driven by the theories the citizens have about the dynamics of the social system. Obviously self-referential mathematical structures, developed in computer science, are candidates to be applied in social sciences for this foundational issue. We will see a prototypical game theoretical problem where several computer scientific tools can help to discuss these structures. It is a long standing problem in economics and of human kind and their scarce recourses: "should we go to a bar that is always so overcrowded"?

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**Abstract**

Research in the proof theory of dynamic logics has recently gained momentum. However, features which are essential to these logics prevent standard proof-theoretic methodologies to apply straightforwardly. In this talk, I will discuss the main properties proof systems should enjoy in order to serve as suitable environments for an inferential theory of meaning (proof-theoretic semantics). Then, I'll identify the main challenges to the inferential semantics research agenda posed by the very features of dynamic logics which make them so appealing and useful to applications. Finally, I'll illustrate a methodology generating multi-type display calculi, which has been successful on interesting case studies (dynamic epistemic logic, propositional dynamic logic, monotone modal logic).

References

1. S. Frittella, G. Greco, A. Kurz, A. Palmigiano, V. Sikimić, A Proof-Theoretic Semantic Analysis of Dynamic Epistemic Logic, Journal of Logic and Computation, Special issue on Substructural logic and information dynamics (2014), DOI:10.1093/logcom/exu063.

2. S. Frittella, G. Greco, A. Kurz, A. Palmigiano, V. Sikimić, Multi-type Display Calculus for Dynamic Epistemic Logic, Journal of Logic and Computation, Special issue on Substructural logic and information dynamics (2014), DOI:10.1093/logcom/exu068.

3. S. Frittella, G. Greco, A. Kurz, A. Palmigiano, Multi-type Display Calculus for Propositional Dynamic Logic, Special issue on Substructural logic and information dynamics (2014), DOI:10.1093/logcom/exu064.

4. S. Frittella, G. Greco, A. Kurz, A. Palmigiano, V. Sikimić, Multi-type Sequent Calculi, Proc. Trends in Logic XIII, A. Indrzejczak, J. Kaczmarek, M. Zawidski eds, p 81-93, 2014.

5. G. Greco, A. Kurz, A. Palmigiano, Dynamic Epistemic Logic Displayed, Proc. Fourth International Workshop on Logic, Rationality and Interaction (LORI 2013) Eds: Huaxin Huang, Davide Grossi, Olivier Roy eds, 2013.

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**Abstract**

Escalation is the behavior of players who play forever in the same game. Such a situation is a typical field for application of coinduction which is the tool conceived for reasoning in infinite mathematical structures. In particular, we use coinduction to study formally the game called --dollar auction--, which is considered as the paradigm of escalation. Unlike what is admitted since 1971, we show that, provided one assumes that the other agent will always stop, bidding is rational, because it results in a subgame perfect equilibrium. We show that this is not the only rational strategy profile (the only subgame perfect equilibrium). Indeed if an agent stops and will stop at every step, whereas the other agent keeps bidding, we claim that he is rational as well because this corresponds to another subgame perfect equilibrium. In the infinite dollar auction game the behavior in which both agents stop at each step is not a Nash equilibrium, hence is not a subgame perfect equilibrium, hence is not rational. Fortunately, the notion of rationality based on coinduction fits with common sense and experience. Finally the possibility of a rational escalation in an arbitrary game can be expressed as a predicate on games and the rationality of escalation in the dollar auction game can be proved as a theorem which we have verified in the proof assistant COQ. In this talk we will recall the principles of infinite extensive games and use them to introduce coinduction and equilibria (Nash equilibrium, subgame perfect equilibrium). We will show how one can prove that the two strategy profiles presented above are equilibria and how this leads to a "rational" escalation in the dollar auction. We will show that escalation may even happen in much simpler game named 0,1-game.

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**Abstract**

Commutative Frobenius algebras play an important role in both Topological Quantum Field Theory and Categorical Quantum Mechanics; in the first case they correspond to 2D TQFTs, while in the second they are non-degenerate observables. I will consider the case of "special" Frobenius algebras, and their associated group of phases. This gives rise to a free construction from the category of abelian groups to the PROP generated by this Frobenius algebra. Of course a theory with only one observable is not very interesting. I will consider how two such PROPs should be combined, and show that if the two algebras (i) jointly form a bialgebra and (ii) their units are "mutually real"; then they jointly form a Hopf algebra. This gives a "free" model of a pair of strongly complementary observables. I will also consider which unitary maps must exist in such models.

Slides are here if you want a preview.

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Edinburgh

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**Abstract**

We introduce a notion of type and scope preserving semantics generalising Goguen and McKinna's "Candidates for Substitution" approach to defining one traversal generic enough to be instantiated to renaming first and then substitution. Its careful distinction of environment and model values as well as its variation on a structure typical of a Kripke semantics make it capable of expressing renaming and substitution but also various forms of Normalisation by Evaluation as well as, perhaps more surprisingly, monadic computations such as a printing function.

We then demonstrate that expressing these algorithms in a common framework yields immediate benefits: we can deploy some logical relations generically over these instances and obtain for instance the fusion lemmas for renaming, substitution and normalisation by evaluation as simple corollaries of the appropriate fundamental lemma. All of this work has been formalised in Agda.

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Dundee

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**Abstract**

Social Choice functions are procedures used to aggregate the preferences of individuals into a collective decision. We outline two recent abstract approaches to SCFs: a recent sheaf treatment of Arrow's Theorem by Abramsky and a modal logic studied by Ulle Endriss and myself. We show how to relate the categorical modelling of Social Choice problems to said work in Modal Logic. This insight prompts a number of research questions, from the relevance of sheaf-like condition to the modelling of properties of SCFs on varying electorates.

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**Abstract**

The Polymorphic Blame Calculus (PBC) of Ahmed et al. (2011) combines polymorphism, as in System F, with type dynamic and runtime casts, as in the Blame Calculus. The PBC is carefully designed to ensure relational parametricity, that is, to ensure that type abstractions do not reveal their abstracted types. The operational semantics of PBC uses runtime sealing and syntactic barriers to enforce parametricity. However, it is an open question as to whether these mechanisms actually guarantee parametricity for the PBC. Furthermore, there is some question regarding what parametricity means in the context of the PBC, as we have examples that are morally parametric but not technically so. This talk will review the design of the PBC with respect to ensuring parametricity and hopefully start a discussion regarding what parametricity should mean for the PBC.

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**Abstract**

This is a sequel to my last 101 where I spoke about describing cellular automata as algebras of a comonad on Set. I'll describe how we can make sense of "generalised cellular automata" (probabilistic/non-deterministic/quantum, for example) as comonads on other categories via distributive laws of monads and comonads.

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**Abstract**

We give a technique to construct a final coalgebra out of modal logic. An element of the final coalgebra is a set of modal formulas. The technique works for both the finite and the countable powerset functors. Starting with a corecursive algebra, we coinductively obtain a suitable subalgebra. We see - first with an example, and then in the general setting of modal logic on a dual adjunction - that modal theories form a corecursive algebra, so that this construction may be applied.

We generalize the framework to categories other than Set, and look at examples in Poset and in the opposite category of Set.

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**Abstract**

Relational parametricity is a fundamental concept within theoretical computer science and the foundations of programming languages, introduced by John Reynolds. His fundamental insight was that types can be interpreted not just as functors on the category of sets, but also as equality preserving functors on the category of relations. This gives rise to a model where polymorphic functions are uniform in a suitable sense; this can be used to establish e.g. representation independence, equivalences between programs, or deriving useful theorems about programs from their type alone.

The relations Reynolds considered were proof-irrelevant, which from a type theoretic perspective is a little limited. As a result, one might like to extend his work to deal with proof-relevant, i.e. set-valued relations. However naive attempts to do this fail: the fundamental property of equality preservation cannot be established. Our insight is that just as one uses parametricity to restrict definable elements of a type, one can use parametricity of proofs to ensure equality preservation. The idea of parametricity for proofs can be formalised using the idea of 2-dimensional logical relations. Interestingly, these 2-dimensional relations have clear higher dimensional analogues where (n+1)-relations are fibered over a n-cube of n-relations. Thus the story of proof relevant logical relations quickly expands into one of higher dimensional structures similar to the cubical sets which arises in Homotopy Type Theory. Of course, there are also connections to Bernardy and Moulin's work on internal parametricity.

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**Abstract**

There exists various possible methods to distribute seats proportionally between states (or parties) in a parliament. Hamilton's method (also known as the method of largest reminder) was abandoned in the USA because of some drawbacks, in particular the possibility of the Alabama paradox, but it is still in use in many other countries.

In recent work (joint with Svante Janson) we give, under certain assumptions, a closed formula for the probability that the Alabama paradox occurs given the vector p_1,...,p_m of relative sizes of the states.

From the theorem we deduce a number of consequences. For example it is shown that the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1/e. For random (uniformly distributed) relative sizes p_1,...,p_m the expected number of states to suffer from the Alabama paradox converges to slightly more than a third of this, or approximately 0.335/e=0.123, as m -> infinity.

I will assume no prior knowledge of electoral mathematics, but begin by giving a brief background to various methods suggested and used for the distribution of seats proportionally in a parliament (it's all in the rounding).

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**Abstract**

We define a category whose morphisms are 'games relative to a continuation', designed to allow games to be built recursively from simple components. The resulting category has interesting structure similar to (but weaker than) compact closed, and comes with an associated string diagram language.

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**Abstract**

Let's say you have a database of people's private information. For SCIENCE, or some other reason, you want to let third parties query your data to learn aggregate information about the people described in the database. However, you have a duty to the people whose information your database contains not to reveal any of their individual personal information. How do you determine which queries you will let third parties execute, and those you will not?

"Differential Privacy" defines a semantic condition on probabilistic queries that identifies queries that are safe to execute, up to some "privacy budget".

I'll present the definition of differential privacy, talk a bit about why it is better than some 'naive' alternatives (e.g., anonymisation), and also describe how the definition can be seen as an instance of relational parametricity.

A good place to read about the definition of differential privacy is the book "The Algorithmic Foundations of Differential Privacy" by Cynthia Dwork and Aaron Roth.

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**Abstract**

Classical computation, invertible computation, probabilistic computation, and quantum computation, form increasingly more sophisticated labelled transition systems. How can we approximate a transition system by less sophisticated ones? Considering all ways to get probabilistic information out of a quantum system leads to domain-theoretic ideas, that also apply in the accompanying Boolean logic. I will survey to what extent these domains characterise the system, leading with examples from quantum theory, in a way that is accessible to a broad audience of computer scientists, mathematicians, and logicians.

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**Abstract**

In pure mathematics, cyclic homology is an invariant of associative algebras that is motivated by algebra, topology and even mathematicial physics. However, when studied from an abstract point of view it turns out to be an invariant of a pair of a monad and a comonad that are related by a mixed distributive law, and I speculate that this could lead to some potential applications in computer science.

(based on joint work with Niels Kowalzig and Paul Slevin)

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**Abstract**

Type : Set

Type = String -> Bool

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**Abstract**

The topic comes from theory of infinitary proofs, and cut-elimination. In essence it is about nicely-behaved maps from higher "infinities" to lower ones, as the infinitary proofs are er, infinite, and can be thought of as glorified transfinite iterators. What might nice behaviour mean?

You can think of it as how to fit an uncountable amount of beer into a bladder whose capacity is merely countable. (Or maybe even finite.)

The most ubiquitous form of infinity is the regular cardinal, iepassing from a container F to F + (mu F -> _), where mu is the W-type operation. I'll "explain" regular collapsing as being all about diagonalisation.

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Royal College Building, room RC512

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**Abstract**

I gave an SPLS talk, which was mostly propaganda, about why people should stop claiming that totality loses Turing completeness. There was some technical stuff, too, about representing a recursive definition as a construction in the free monad whose effect is calling out to an oracle for recursive calls: that tells you what it is to be recursive without prejudicing how to run it. I'm trying to write this up double-quick as a paper for the miraculously rubbery MPC deadline, with more explicit attention to the monad morphisms involved. So I'd be grateful if you would slap down the shtick and make me more morphic. The punchline is that the Bove-Capretta domain predicate construction is a (relative) monad morphism from the free monad with a recursion oracle to the (relative) monad of Dybjer-Setzer Induction-Recursion codes. But it's worth looking at other monad morphisms (especially to the Delay monad) along the way.

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**Abstract**

What structure is required of a set so that computations in a given notion of computation can be run statefully this with set as the state space? Some answers: To be able to serve stateful computations, a set must carry the structure of a lens; for running interactive I/O computations statefully, a "responder-listener" structure is necessary etc. I will observe that, in general, to be a runner of computations for an algebraic theory (defined as a set equipped with a monad morphism between the corresponding monad and the state monad for this set) is the same as to be a comodel of this theory, ie a coalgebra of the corresponding comonad. I will work out a number of instances. I will also compare runners to handlers.

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**Abstract**

It would be a great shame if dependently-typed programming (DTP) restricted us to only writing very clever programs that were a priori structurally recursive and hence obviously terminating. Put another way, it is a lot to ask of the programmer to provide the program and its termination proof in one go, programmers should also be supported in working step-by-step. I will show a technique that lowers the barrier of entry, from showing termination to only showing productivity up front, and then later providing the opportunity to show termination (convergence). I will show an example of a normaliser for STLC represented in Agda as a potentially non-terminating but nonetheless productive corecursive function targeting the coinductive delay monad.

(Joint work with Andreas Abel)

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**Abstract**

I've managed to prove a theorem that I've been chasing for a while. The trouble, of course, was *stating* it. I'll revisit the motivation for extending type systems with an analysis of not just *what* things are but *where*, *when*, *whose*, etc. The idea is that typed constructions occur in one of a preordered set of worlds, with scoping restricted so that information flows only "upwards" from one world to another. Worlds might correspond to "at run time" and "during typechecking", or to computation in distinct cores, or in different stages, etc. What does the dependent function space mean in this setting? For a long time, I thought that each world had its own universal quantifier, for abstracting over stuff from that world. Failure to question this presumption is what led to failure to state a theorem I could prove. By separating quantifiers from worlds, I have fixed the problem. I'll show how to prove the key fact: if I can build something in one world and then move it to another, then it will also be a valid construction once it has arrived at its destination.

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We will read the paper A Model of Type Theory in Cubical Sets by Marc Bezem, Thierry Coquand and Simon Huber. Thierry's Variation on cubical sets might also be useful. *Administrative details*: meet for lunch at 12am for those who want to, reading group starts at 2pm.

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**Abstract**

Neil and Clemens will report back from the Lorentz Center Workshop on Logics for Social Behaviour.

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**Abstract**

Monoidal categories are essentially 2-dimensional things, so why on earth would we represent them using a linear string of symbols? I'll talk about how to use string diagrams for monoidal categories, graph rewriting for reasoning within them, and how the syntax can be extended to handle certain kinds of infinitary expressions with the infamous !-box. If there's time I'll finish with some half-baked (eh... basically still looking for the on switch of the oven...) ideas of how to generalise them.

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**Abstract**

Kevin will be giving an informal talk about some of the stuff he has been learning about. He'll give the definition of a cellular automaton and then talk about how this definition can be phrased in terms of a comonad.

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**Abstract**

Logical relations are widely used to study various properties of typed lambda calculi. By extending them to the lambda calculus with monadic types, we can gain understanding of the properties on functional programming languages with computational effects. Among various constructions of logical relations for monads, I will talk about a categorical TT-lifting, which is a semantic analogue of Lindley and Stark's leapfrog method.

After reviewing some fundamental properties of the categorical TT-lifting, we apply it to the problem of relating two monadic semantics of a call-by-value functional programming language with computational effects. This kind of problem has been considered in various forms: for example, the relationship between monadic style and continuation passing style representations of call-by-value programs was studied around '90s. We give a set of sufficient conditions to solve the problem of relating two monadic semantics affirmatively. These conditions are applicable to a wide range of such problems.

Boardroom (LT1101d)

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**Abstract**

All kinds of semantics are syntax directed: the semantics follows from the syntax. Some varieties of semantics are syntax and type directed. In this talk, I'll discuss syntax, type, *and* analysis directed semantics (analysis-directed semantics for short!), for analyses other than types. An analysis-directed semantics maps from terms coupled with derivations of a static program analysis into some semantic domain. For example, the simply-typed lambda calculus with an effect system maps to the category generated by a strong parametric effect monad (due to Katsumata) and a bounded-linear logic-like analysis (described as a coeffect systems) maps to a category generated by various structures related to monoidal comonads. I'll describe a general technique for building analysis-directed semantics where semantic objects and analysis objects have the same structure and are coupled by lax homomorphisms between them. This aids proving semantic properties: the proof tree of an equality for two program analyses implies the rules needed to prove equality of the programs' denotations.

Boardroom (LT1101d)

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**Abstract**

The topic is the unholy trinity of eta, zeta, and xi. I'll indicate how Curry managed to give a finite combinatorial axiomatisation of this nastiness, by anticipating (almost-but-not-*quite*) McBride et al's applicative functors.

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Dundee

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**Abstract**

Neil will talk about partial progress made during the summer on higher dimensional parametricity and the cubical structures that seem to arise.

Details will be kept to a minimum and, of course, concepts stressed.

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Heriot Watt

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**Abstract**

Conor has offered to talk to us about what he has been thinking about recently. He says this includes models, views, and dependent types.

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**Abstract**

Or: My summer with Steve

Or: How Christine and Frank were right, after all

Or: Inductive types for the price of function extensionality and impredicative Set

Christine Paulin-Mohring and Frank Pfenning suggested to use impredicative encodings of inductive types in the Calculus of Constructions, but this was later abandoned, since it is "well-known" that induction principles, i.e. dependent elimination, can not be derived for this encoding. It now seems like it is possible to give a variation of this encoding for which the induction principle is derivable after all. The trick is to use identity types to cut down the transformations of type (Pi X : Set) . (F(X) -> X) -> X to the ones that are internally strongly dinatural, making use of a formula for a "generalised Yoneda Lemma" by Uustalu and Vene.

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**Abstract**

Ohad gave an informal overview of his current draft, with the following abstract:

Haskell incorporates computational effects modularly using sequences of monad transformers, termed monad stacks. The current practice is to find the appropriate stack for a given task using intractable brute force and heuristics. By restricting attention to algebraic stack combinations, we provide a linear-time algorithm for generating all the appropriate monad stacks, or decide no such stacks exist. Our approach is based on Hyland, Plotkin, and Power's algebraic analysis of monad transformers, who propose a graph-theoretical solution to this problem. We extend their analysis with a straightforward connection to the modular decomposition of a graph and to cographs, a.k.a. series-parallel graphs.

We present an accessible and self-contained account of this monad-stack generation problem, and, more generally, of the decomposition of a combined algebraic theory into sums and tensors, and its algorithmic solution. We provide a web-tool implementing this algorithm intended for semantic investigations of effect combinations and for monad stack generation.

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**Abstract**

This 101 is intended to be a brainstorming session on possible links between the theory of coalgebras and the theory of databases. I will outline some ideas in this direction and I am looking forward to your feedback.

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**Abstract**

In Intuitionistic Multiplicative Linear Logic, the right introduction rule for tensors implies picking a 2-partition of the set of assumptions and use each component to inhabit the corresponding tensor's subformulas. This makes a naive proof search algorithm intractable. Building a notion of resource availability in the context and massaging the calculus into a more general one handling both resource consumption and a notion of "leftovers" of a subproof allows for a well-structured well-typed by construction proof search mechanism.

Here is an Agda file implementing the proof search algorithm.

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**Abstract**

Guillaume presented parts of Hedges' paper Monad transformers for backtracking search (accepted to MSFP 2014). The paper extends Escardo and Oliva's work on the selection and continuation monads to the corresponding monad transformers, with applications to backtracking search and game theory.

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**Abstract**

Stuart spoke about Lagrange inversion, a species-theoretic attempt to discuss the existence of solutions to equations defining species.

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**Abstract**

Neil spoke about how adding structured quotients to containers gives rise to a larger class of data types.

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**Abstract**

Tim gave a brief introduction to Synthetic Differential Geometry. This is an attempt to treat smooth spaces categorically so we can extend the categorical methods used in the discrete world of computer science to the continuous work of physics.

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**Abstract**

Conor talked about worlds (aka phases, aka times, ...): why one might bother, and how we might go about equipping type theory with a generic notion of permitted information flow.

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**Abstract**

Miles has kindly agreed to come through and tell us about Operads, thus revisiting the topic of his PhD and the city in which he did it.

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**Abstract**

In 1987, Felleisen showed how to add control operators (for things like exceptions and unconditional jumps) to the untyped lambda-calculus. In 1990, Griffin idly wondered what would happen if one did the same in a typed lambda calculus. The answer came out: the inhabited types become the theorems of classical logic.

I will present the lambda mu-calculus, one of the cleanest attempts to add control operators to a type theory. We'll cover the good news: the inhabited types are the tautologies of minimal classical logic, and Godel's Double Negation translation from classical to intuitionistic logic turns into the CPS translation.

And the bad news: control operators don't play well with other types. Add natural numbers (or some other inductive type), and you get inconsistency. Add Sigma-types, and you get degeneracy (any two objects of the same type are definitionally equal). It gets worse: add plus-types, and you break Subject Reduction.

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**Abstract**

I chose to go through (parts of) Hatcliff and Danvy's paper "A Generic Account of Continuation-Passing Styles" (POPL 94) which gives a nice factorization of various CPS transforms in terms of:

- embeddings from STLC to Moggi's computational meta-language (either call-by-value, call-by-name, or whatever you can come up with)
- followed by a generic CPS transform transporting terms from ML back to STLC

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