Data types with symmetries and polynomial functors over groupoids

Polynomial functors are useful in the theory of data types, where they are often called containers. They are also useful in algebra, combinatorics, topology, and higher category theory, and in this broader perspective the polynomial aspect is often prominent and justifies the terminology. For example, Tambara's theorem states that the category of finite polynomial functors is the Lawvere theory for commutative semirings. In this talk I will explain how an upgrade of the theory from sets to groupoids is useful to deal with data types with symmetries, and provides a common generalisation of and a clean unifying framework for quotient containers (cf. Abbott et al.), species and analytic functors (Joyal 1985), as well as the stuff types of Baez-Dolan. The multi-variate setting also includes relations and spans, multispans, and stuff operators. An attractive feature of this theory is that with the correct homotopical approach - homotopy slices, homotopy pullbacks, homotopy colimits, etc. - the groupoid case looks exactly like the set case. After some standard examples, I will illustrate the notion of data-types-with-symmetries with examples from quantum field theory, where the symmetries of complicated tree structures of graphs play a crucial role, and can be handled elegantly using polynomial functors over groupoids. (These examples, although beyond species, are purely combinatorial and can be appreciated without background in quantum field theory.) Locally cartesian closed 2-categories provide semantics for 2-truncated intensional type theory. For a fullfledged type theory, locally cartesian closed \infty-categories seem to be needed. The theory of these is being developed by D.Gepner and the author as a setting for homotopical species, and several of the results exposed in this talk are just truncations of \infty-results obtained in joint work with Gepner. Details will appear elsewhere.Comment: This is the final version of my conference paper presented at the 28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); to appear in the Electronic Notes in Theoretical Computer Science. 16p

Interfaces as functors, programs as coalgebras—A final coalgebra theorem in intensional type theory

AbstractIn [P. Hancock, A. Setzer, Interactive programs in dependent type theory, in: P. Clote, H. Schwichtenberg (Eds.), Proc. 14th Annu. Conf. of EACSL, CSL’00, Fischbau, Germany, 21–26 August 2000, Vol. 1862, Springer, Berlin, 2000, pp. 317–331, URL 〈citeseer.ist.psu.edu/article/hancock00interactive.html〉; P. Hancock, A. Setzer, Interactive programs and weakly final coalgebras in dependent type theory, in: L. Crosilla, P. Schuster (Eds.), From Sets and Types to Topology and Analysis. Towards Practicable Foundations for Constructive Mathematics, Oxford Logic Guides, Clarendon Press, 2005, URL 〈www.cs.swan.ac.uk/∼csetzer/〉] Hancock and Setzer introduced rules to extend Martin-Löf's type theory in order to represent interactive programming. The rules essentially reflect the existence of weakly final coalgebras for a general form of polynomial functor. The standard rules of dependent type theory allow the definition of inductive types, which correspond to initial algebras. Coalgebraic types are not represented in a direct way. In this article we show the existence of final coalgebras in intensional type theory for these kind of functors, where we require uniqueness of identity proofs (UIP) for the set of states S and the set of commands C which determine the functor. We obtain the result by identifying programs which have essentially the same behaviour, viz. are bisimular. This proves the rules of Setzer and Hancock admissible in ordinary type theory, if we replace definitional equality by bisimulation. All proofs [M. Michelbrink, Verifications of final coalgebra theorem in: Interfaces as Functors, Programs as Coalgebras—A Final Coalgebra Theorem in Intensional Type Theory, 2005, URL 〈www.cs.swan.ac.uk/∼csmichel/〉] are verified in the theorem prover agda [C. Coquand, Agda, Internet, URL 〈www.cs.chalmers.se/∼catarina/agda/〉; K. Peterson, A programming system for type theory, Technical Report, S-412 96, Chalmers University of Technology, Göteborg, 1982], which is based on intensional Martin-Löf type theory

Laying Tiles Ornamentally: An approach to structuring container traversals

Having hardware more capable of parallel execution means that more program scheduling decisions have to be taken to utilize that hardware efficiently. To this end, compilers implement coarse-grained loop transformations in addition to traditionally used fine-grained instruction reordering. Implementors of embedded domain specific languages have to face a difficult choice: to translate operations on collections to a low-level language naively hoping that its optimizer will do the job, or to implement their own optimizer as a part of the EDSL.<br /><br />We turn ourselves to the concept of loop tiling from the imperative world and find its equivalent for recursive functions. We show the construction of a <em>tiled</em> functorial map over containers that can be naively translated to a corresponding nested loop.<br /><br />We illustrate the connection between <em>untiled</em> and tiled functorial maps by means of a type-theoretic notion of <em>algebraic ornament</em>. This approach produces an family of container traversals indexed by <em>tile sizes</em> and serves as a basis of a proof that untiled and tiled functorial maps have the same semantics.<br /><br />We evaluate our approach by designing a language of tree traversals as a DSL embedded into Haskell which compiles into C code. We use this language to implement tiled and untiled tree traversals which we benchmark under varying choices of tile sizes and shapes of input trees. For some tree shapes, we show that a tiled tree traversal can be up to 50% faster than an untiled one under a good choice of the tile size

Twisted Cubes and their Applications in Type Theory

This thesis captures the ongoing development of twisted cubes, which is a modification of cubes (in a topological sense) where its homotopy type theory does not require paths or higher paths to be invertible. My original motivation to develop the twisted cubes was to resolve the incompatibility between cubical type theory and directed type theory. The development of twisted cubes is still in the early stages and the intermediate goal, for now, is to define a twisted cube category and its twisted cubical sets that can be used to construct a potential definition of (infinity, n)-categories. The intermediate goal above leads me to discover a novel framework that uses graph theory to transform convex polytopes, such as simplices and (standard) cubes, into base categories. Intuitively, an n-dimensional polytope is transformed into a directed graph consists 0-faces (extreme points) of the polytope as its nodes and 1-faces of the polytope as its edges. Then, we define the base category as the full subcategory of the graph category induced by the family of these graphs from all n-dimensional cases. With this framework, the modification from cubes to twisted cubes can formally be done by reversing some edges of cube graphs. Equivalently, the twisted n-cube graph is the result of a certain endofunctor being applied n times to the singleton graph; this endofunctor (called twisted prism functor) duplicates the input, reverses all edges in the first copy, and then pairwisely links nodes from the first copy to the second copy. The core feature of a twisted graph is its unique Hamiltonian path, which is useful to prove many properties of twisted cubes. In particular, the reflexive transitive closure of a twisted graph is isomorphic to the simplex graph counterpart, which remarkably suggests that twisted cubes not only relate to (standard) cubes but also simplices.Comment: PhD thesis (accepted at the University of Nottingham), 162 page

Twisted Cubes and their Applications in Type Theory

This thesis captures the ongoing development of twisted cubes, which is a modification of cubes (in a topological sense) where its homotopy type theory does not require paths or higher paths to be invertible. My original motivation to develop the twisted cubes was to resolve the incompatibility between cubical type theory and directed type theory. The development of twisted cubes is still in the early stages and the intermediate goal, for now, is to define a twisted cube category and its twisted cubical sets that can be used to construct a potential definition of (infinity, n)-categories. The intermediate goal above leads me to discover a novel framework that uses graph theory to transform convex polytopes, such as simplices and (standard) cubes, into base categories. Intuitively, an n-dimensional polytope is transformed into a directed graph consists 0-faces (extreme points) of the polytope as its nodes and 1-faces of the polytope as its edges. Then, we define the base category as the full subcategory of the graph category induced by the family of these graphs from all n-dimensional cases. With this framework, the modification from cubes to twisted cubes can formally be done by reversing some edges of cube graphs. Equivalently, the twisted n-cube graph is the result of a certain endofunctor being applied n times to the singleton graph; this endofunctor (called twisted prism functor) duplicates the input, reverses all edges in the first copy, and then pairwisely links nodes from the first copy to the second copy. The core feature of a twisted graph is its unique Hamiltonian path, which is useful to prove many properties of twisted cubes. In particular, the reflexive transitive closure of a twisted graph is isomorphic to the simplex graph counterpart, which remarkably suggests that twisted cubes not only relate to (standard) cubes but also simplices

Twisted Cubes and their Applications in Type Theory

This thesis captures the ongoing development of twisted cubes, which is a modification of cubes (in a topological sense) where its homotopy type theory does not require paths or higher paths to be invertible. My original motivation to develop the twisted cubes was to resolve the incompatibility between cubical type theory and directed type theory. The development of twisted cubes is still in the early stages and the intermediate goal, for now, is to define a twisted cube category and its twisted cubical sets that can be used to construct a potential definition of (infinity, n)-categories. The intermediate goal above leads me to discover a novel framework that uses graph theory to transform convex polytopes, such as simplices and (standard) cubes, into base categories. Intuitively, an n-dimensional polytope is transformed into a directed graph consists 0-faces (extreme points) of the polytope as its nodes and 1-faces of the polytope as its edges. Then, we define the base category as the full subcategory of the graph category induced by the family of these graphs from all n-dimensional cases. With this framework, the modification from cubes to twisted cubes can formally be done by reversing some edges of cube graphs. Equivalently, the twisted n-cube graph is the result of a certain endofunctor being applied n times to the singleton graph; this endofunctor (called twisted prism functor) duplicates the input, reverses all edges in the first copy, and then pairwisely links nodes from the first copy to the second copy. The core feature of a twisted graph is its unique Hamiltonian path, which is useful to prove many properties of twisted cubes. In particular, the reflexive transitive closure of a twisted graph is isomorphic to the simplex graph counterpart, which remarkably suggests that twisted cubes not only relate to (standard) cubes but also simplices