[Paper Review] Domain Theory and the Logic of Observable Properties

Samson Abramsky, Goré, Rajeev|arXiv (Cornell University)|Dec 1, 2011

Logic, programming, and type systems81 references72 citations

TL;DR

This paper establishes a foundational duality between denotational semantics and logic in theoretical computer science by using Stone duality to unify domain theory, process semantics, and program logics. It introduces a metalanguage where types represent computational universes and terms represent programs, showing that the denotational and logical interpretations are Stone duals—ensuring harmony between semantics and logic and enabling systematic derivation of logics from domain equations.

ABSTRACT

The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: - Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. - The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. - Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable properties---i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme: 1. A metalanguage is introduced, comprising - types = universes of discourse for various computational situations. - terms = programs = syntactic intensions for models or points. 2. A standard denotational interpretation of the metalanguage is given, assigning domains to types and domain elements to terms. 3. The metalanguage is also given a {\em logical} interpretation, in which types are interpreted as propositional theories and terms are interpreted via a program logic, which axiomatizes the properties they satisfy. 4. The two interpretations are related by showing that they are Stone duals of each other. Hence, semantics and logic are guaranteed to be in harmony with each other, and in fact each determines the other up to isomorphism. This opens the way to a whole range of applications. Given a denotational description of a computational situation in our meta-language, we can turn the handle to obtain a logic for that situation.

Motivation & Objective

• To unify disparate developments in theoretical computer science—domain theory, process semantics, and program logics—under a common mathematical foundation.
• To resolve the long-standing gap between denotational semantics and logical reasoning about programs by establishing a duality between them.
• To provide a systematic method for deriving logical systems from domain-theoretic descriptions of computational systems.
• To demonstrate that observable properties (decided in finite time) are naturally captured by geometric logic, aligning with computational intuition.
• To lay the groundwork for a general, compositional proof theory for program logics by exploiting duality in domain equations.

Proposed method

• Formalize a metalanguage with types (universes of discourse) and terms (programs) for modeling computational systems.
• Provide a standard denotational interpretation assigning domains to types and domain elements to terms.
• Introduce a logical interpretation where types are propositional theories and terms are interpreted via an axiomatized program logic.
• Establish that the denotational and logical interpretations are Stone duals—each determines the other up to isomorphism.
• Use domain equations (e.g., for bisimulation and lazy λ-calculus) to generate both operational models and their corresponding logics.
• Apply the duality to derive full abstraction results and compositional proof systems for logics like Hennessy-Milner logic.

Experimental results

Research questions

• RQ1 How can denotational semantics and program logic be formally unified in a way that ensures harmony and mutual determination?
• RQ2 Can observable properties of computational processes be systematically captured by a geometric logic grounded in domain theory?
• RQ3 To what extent can a logic be automatically derived from a domain equation describing a computational system?
• RQ4 How does Stone duality provide a bridge between topological spaces (semantics) and lattices of properties (logic) in computation?
• RQ5 Can this duality framework support full abstraction and compositional proof systems for process and functional languages?

Key findings

• The denotational and logical interpretations of the metalanguage are shown to be Stone duals, guaranteeing that semantics and logic are in perfect harmony and mutually reconstructible.
• The logic of observable properties is geometric, meaning it captures only properties decidable in finite time, aligning with computational realizability.
• A domain equation for bisimulation yields a fully abstract denotational semantics for SCCS, and the logic derived from it faithfully interprets Hennessy-Milner logic.
• For the lazy λ-calculus, the framework yields a domain equation and a corresponding logic that supports compositional reasoning via applicative transition systems.
• The approach enables the automatic generation of a logic from a denotational model, demonstrating that the logic is not an ad hoc addition but a dual counterpart to the semantics.
• The framework suggests a path toward a unified logical foundation for both functional and concurrent computation, with potential for extending to topos theory and linear logic.