Duality in Computer Science

This is a paper on Stone duality in computer science with special focus on topics with applications in formal language theory. In Section 2 we give a general overview of Stone duality in its various forms: for Boolean algebras, distributive lattices, and frames. For distributive lattices, we discuss both Stone and Priestley duality. We identify how to move between the different dualities and which dual spaces carry the Scott topology. We then focus on three themes.The first theme is additional operations on distributive lattices and Boolean algebras. Additional operations arise in denotational semantics in the form of predicate transformers. In verification they occur in the form of modal operators. They play an essential rôle in Eilenberg’s variety theorem in the form of quotient operations. Quotient operations are unary instantiations of residual operators which are dual to the operations in the profinite algebras of algebraic language theory. We discuss additional operations in Section 3.The second theme is that of hyperspaces, that is, spaces of subsets of an underlying space. Some classes of algebras may be seen as the class of algebras for a functor. In the case of predicate transformers the dual functors are hyperspace constructions such as the Plotkin, Smyth, and Hoare powerdomain constructions. The algebras-for-a-functor point of view is central to the coalgebraic study of modal logic and to the solution of domain equations. In the algebraic theory of formal languages various hyperspace-related product constructions, such as block and Schützenberger products, are used to study complexity hierarchies. We describe a construction, similar to the Schützenberger product, which is dual to adding a layer of quantification to formulas describing formal languages. We discuss hyperspaces in Section 4.The final theme is that of "equations". These are pairs of elements of dual spaces. They arise via the duality between subalgebras and quotient spaces and have provided one of the most successful tools for obtaining decidability results for classes of regular languages. The perspective provided by duality allows us to obtain a notion of equations for the study of arbitrary formal languages. Equations in language theory is the topic of Section 5.

1. Introduction

In 1936, M. H. Stone initiated duality theory in logic by presenting a dual category equivalence between the category of Boolean algebras and the category of compact Hausdorff spaces having a basis of clopen sets. Stone’s duality and its variants are central in making the link between syntactical and semantic approaches to logic. In computer science this link is central as the two sides correspond to specification languages and to spaces of computational states, respectively. The ability to translate faithfully between these two worlds has often proved itself to be a powerful theoretical tool as well as a handle for making practical problems decidable. A prime example is the seminal work (Abramsky 1991) linking program logic and domain theory via Stone duality. Other examples include the work (Plotkin 1980) and (Smyth 1983) on predicate transformers, and (Goldblatt 1989) identifying extended Stone duality as the setting for completeness issues for Kripke semantics in modal logic. Applications of Stone duality in logic and computer science generally need more than just basic Stone duality. For example, Abramsky’s work needs Stone or Priestley duality for distributive lattices and the dualisation of additional structure in the form of functors. Applications in modal logic require a duality for Boolean algebras or distributive lattices endowed with additional operations. Thus much work in duality theory has been spawned to answer questions and solve problems coming from semantics both in computer science and logic.