Logic and handling of algebraic effects

Abstract

In the thesis, we explore reasoning about and handling of algebraic effects. Those are computational effects, which admit a representation by an equational theory. Their examples include exceptions, nondeterminism, interactive input and output, state, and their combinations. In the first part of the thesis, we propose a logic for algebraic effects. We begin by introducing the a-calculus, which is a minimal equational logic with the purpose of exposing distinct features of algebraic effects. Next, we give a powerful logic, which builds on results of the a-calculus. The types and terms of the logic are the ones of Levy’s call-by-push-value framework, while the reasoning rules are the standard ones of a classical multi-sorted first-order logic with predicates, extended with predicate fixed points and two principles that describe the universality of free models of the theory representing the effects at hand. Afterwards, we show the use of the logic in reasoning about properties of effectful programs, and in the translation of Moggi’s computational ¸-calculus, Hennessy-Milner logic, and Moggi’s refinement of Pitts’s evaluation logic. In the second part of the thesis, we introduce handlers of algebraic effects. Those not only provide an algebraic treatment of exception handlers, but generalise them to arbitrary algebraic effects. Each such handler corresponds to a model of the theory representing the effects, while the handling construct is interpreted by the homomorphism induced by the universal property of the free model. We use handlers to describe many previously unrelated concepts from both theory and practice, for example CSS renaming and hiding, stream redirection, timeout, and rollback.