Date of Award

2014

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Computer and Information Science

First Advisor

Stephanie C. Weirich

Abstract

The theory of combinatorial species was developed in

the 1980s as part of the mathematical subfield of enumerative

combinatorics, unifying and putting on a firmer theoretical basis a

collection of techniques centered around generating

functions. The theory of algebraic data

types was developed, around the same time, in functional

programming languages such as Hope and Miranda, and is still used

today in languages such as Haskell, the ML family, and Scala. Despite

their disparate origins, the two theories have striking

similarities. In particular, both constitute algebraic frameworks in

which to construct structures of interest. Though the similarity has

not gone unnoticed, a link between combinatorial species and algebraic

data types has never been systematically explored. This dissertation

lays the theoretical groundwork for a precise—and, hopefully,

useful—bridge bewteen the two theories. One of the key

contributions is to port the theory of species from a classical,

untyped set theory to a constructive type theory. This porting process

is nontrivial, and involves fundamental issues related to equality and

finiteness; the recently developed homotopy type

theory is put to good use formalizing these issues in a

satisfactory way. In conjunction with this port, species as general

functor categories are considered, systematically analyzing the

categorical properties necessary to define each standard species

operation. Another key contribution is to clarify the role of species

as labelled shapes, not containing any data, and to

use the theory of analytic functors to model labelled

data structures, which have both labelled shapes and data associated

to the labels. Finally, some novel species variants are considered,

which may prove to be of use in explicitly modelling the memory layout

used to store labelled data structures.

Share

COinS