Tools for Reasoning about Effectful Declarative Programs

Abstract

In the pure functional language Haskell, nearly all side-effects that a function can produce have to be noted in its type. This includes input/output, propagation of a state, and nondeterminism. If no side-effects are noted, such a function acts like a mathematical function, i.e., mapping arguments to unique results. In that case, expressions in a program can be reasoned about like mathematical expressions. In addition to this socalled equational reasoning, the type system also enables type based reasoning. One example are free theorems - equations between expressions that are true only due to the types of the expressions involved. Some such statements serve as formal justification for optimization strategies in compilers.<br /> The thesis at hand investigates two generalizations of such methods for programs not free of side-effects, i.e., effectful programs. First, effectful traversals of data structures are being studied. The most important contribution in this part is that a data structure can be lawfully traversed if, and only if, it is isomorphic to a polynomial functor. This result links the widespread interface of traversing to a clear intuition regarding the structure and behavior of the data type. Furthermore, tools are presented facilitating convenient proofs about effectful traversals.<br /> Second, free theorems for the functional-logic language Curry are derived. Due to the close relationship between both languages, Curry can be understood as Haskell with built-in nondeterminism, i.e., a built-in side-effect. Equational and type based reasoning can both be adapted to Curry to a certain degree. In particular, short cut fusion - a very fertile runtime optimization - is enabled for Curry.