Species and functors and types, oh my!

Yorgey, Brent A.

doi:10.1145/1863523.1863542

Cited by 7 publications

(5 citation statements)

References 18 publications

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“…One suggestion is to bring the method of characteristics as used, for example in [22], into common parlance for sequence work. Another is to find a smooth passage from the level of the present paper to results obtained using the language of Species, for example those in [4,70] (an introduction to Species for Haskell programmers is [89]).…”

Section: Discussionmentioning

confidence: 99%

In Praise of Sequence (Co-)Algebra and its implementation in Haskell

Clenaghan

2018

Preprint

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What is Sequence Algebra? This is a question that any teacher or student of mathematics or computer science can engage with. Sequences are in Calculus, Combinatorics, Statistics and Computation. They are foundational, a step up from number arithmetic. Sequence operations are easy to implement from scratch (in Haskell) and afford a wide variety of testing and experimentation. When bits and pieces of sequence algebra are pulled together from the literature, there emerges a claim for status as a substantial pre-analysis topic. Here we set the stage by bringing together a variety of sequence algebra concepts for the first time in one paper. This provides a novel economical overview, intended to invite a broad mathematical audience to cast an eye over the subject. A complete, yet succinct, basic implementation of sequence operations is presented, ready to play with. The implementation also serves as a benchmark for introducing Haskell by mathematical example.

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Section: Discussionmentioning

confidence: 99%

In Praise of Sequence (Co-)Algebra and its implementation in Haskell

Clenaghan

2018

Preprint

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“…Boltzmann samplers can be used to generate objects from a wide range of combinatorial structures, with uniform distribution over values of an approximate or exact size (Flajolet et al 1994(Flajolet et al , 2007. Yorgey has explored the relation between a class of combinatorial objects called species and ADTs (Yorgey 2010(Yorgey , 2014. This work can potentially be used for uniform random generation of algebraic types as well as some more complex structures involving symmetries.…”

Section: Related Workmentioning

confidence: 99%

Generating constrained random data with uniform distribution

Claessen¹,

Duregård²,

Pałka³

2015

J. Funct. Prog.

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We present a technique for automatically deriving test data generators from a given executable predicate representing the set of values we are interested in generating. The distribution of these generators is uniform over values of a given size. To make the generation efficient, we rely on laziness of the predicate, allowing us to prune the space of values quickly. In contrast, implementing test data generators by hand is labour intensive and error prone. Moreover, handwritten generators often have an unpredictable distribution of values, risking that some values are arbitrarily underrepresented. We also present a variation of the technique that has better performance, but where the distribution is skewed in a limited, albeit predictable way. Experimental evaluation of the techniques shows that the automatically derived generators are much easier to define than handwritten ones, and their performance, while lower, is adequate for some realistic applications.

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“…In fact it has been in the air for some time (see for example [14], and more recently [12], [47]) that species should be a good framework for data type theory. It is the contention of the present contribution that polynomial functors over groupoids provide a clean unifying framework: in the setting of groupoids, the essential distinction between 'analytic' and 'polynomial' evaporates (3.7), and the functors can be represented by diagrams with combinatorial content (3) just as polynomials over sets, as we proceed to explain.…”

Section: Incorporating Symmetriesmentioning

confidence: 99%

Data Types with Symmetries and Polynomial Functors over Groupoids

Kock

2012

Electronic Notes in Theoretical Computer Science

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Polynomial functors (over Set or other locally cartesian closed categories) 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 [45], [18]. In this talk I will explain how an upgrade of the theory from sets to groupoids (or other locally cartesian closed 2-categories) is useful to deal with data types with symmetries, and provides a common generalisation of and a clean unifying framework for quotient containers (in the sense of Abbott et al.), species and analytic functors (Joyal 1985), as well as the stuff types of Baez and 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 perturbative 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 a 2-truncated version of Martin-Löf intensional type theory. For a fullfledged type theory, locally cartesian closed ∞-categories seem to be needed. The theory of these is being developed by David Gepner and the author as a setting for homotopical species, and several of the results exposed in this talk are just truncations of ∞-results obtained in joint work with Gepner. Details will appear elsewhere.

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Species and functors and types, oh my!

Cited by 7 publications

References 18 publications

In Praise of Sequence (Co-)Algebra and its implementation in Haskell

In Praise of Sequence (Co-)Algebra and its implementation in Haskell

Generating constrained random data with uniform distribution

Data Types with Symmetries and Polynomial Functors over Groupoids

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