Interactive Programs in Dependent Type Theory

We propose a representation of interactive systems in dependent type theory. This is meant as a basis for an execution environment for dependently typed programs, and for reasoning about their construction; there may be wider applications. The inspiration is the ‘I/O-monad’ of Haskell. The fundamental notion is an I/O-tree; its definition is parameterised over a general notion of dependently typed, command-response interactions called a world. I/O-trees represent strategies for one of the parties in a command/response interaction – the notion is not confined to functional programming. The definitions make essential use of the expressive strength of dependent typing. We present I/O-trees in two forms that we call ‘non-normalising’ and ‘normalising’. The first form, which is simpler, is suitable for Turing-complete functional programming languages with general recursion. The second is definable within (ordinary) normalising type theory and we identify programs written in it as ‘normalising I/O-programs’. We define new looping constructs (while and repeat), and a new refinement construct (redirect), which permits the implementation of libraries. We introduce a bisimulation relation between interactive programs, with respect to which we prove the monad laws and defining equations of while.