# A foundation for computation

## Description

Domains were discovered in the late 1960's by Dana Scott, where they were first studied in the more restrictive form of a continuous lattice. Scott's motivation was a desire to provide a formal semantics for the lambda calculus. Since then, domains have been used to provide mathematical models for many different types of computational phenomena including, of course, programming languages The motivations of the present work differ significantly from those encountered in traditional applications of domain theory. We seek to develop a mathematical setting in which the question 'Does this program work, and if so, how does it compare to others which solve the same problem?' may be sensibly formalized and answered. Above all else, we seek a simple theory which is easy to apply To this end, the mathematical setting chosen is that of a domain with a measurement. Roughly speaking, a domain is a collection of informative objects, a measurement expresses the amount of information each object in the domain possesses, and an algorithm is a function between domains with measurements This thesis is the study and application of domains with measurements. While many of the results and ideas introduced were arrived at in an attempt to provide a framework for computation, it has become increasingly clear that they also have value in the more general arena of mathematics and applied science. The extent to which this is the case can then be regarded a secondary goal of this work