arXiv subjects

Christian Williams

Publications and source records attributed to Christian Williams.

3 recordsLinked to original sources

arXiv2021

Native Type Theory

Native type systems are those in which type constructors are derived from term constructors, as well as the constructors of predicate logic and intuitionistic type theory. We present a method to construct native type systems for a broad class of languages, lambda-theories with equality, by embedding such a theory into the internal language of its topos of presheaves. Native types provide total specification of the structure of terms; and by internalizing transition systems, native type systems serve to reason about structure and behavior simultaneously. The construction is functorial, thereby providing a shared framework of higher-order reasoning for many languages, including programming languages.

cs.LO↗

arXiv2020

Abstract Congruence Criteria for Weak Bisimilarity

We introduce three general compositionality criteria over operational semantics and prove that, when all three are satisfied together, they guarantee weak bisimulation being a congruence. Our work is founded upon Turi and Plotkin's mathematical operational semantics and the coalgebraic approach to weak bisimulation by Brengos. We demonstrate each criterion with various examples of success and failure and establish a formal connection with the simply WB cool rule format of Bloom and van Glabbeek. In addition, we show that the three criteria induce lax models in the sense of Bonchi et al.

cs.LO↗

arXiv2019

Enriched Lawvere Theories for Operational Semantics

Enriched Lawvere theories are a generalization of Lawvere theories that allow us to describe the operational semantics of formal systems. For example, a graph enriched Lawvere theory describes structures that have a graph of operations of each arity, where the vertices are operations and the edges are rewrites between operations. Enriched theories can be used to equip systems with operational semantics, and maps between enriching categories can serve to translate between different forms of operational and denotational semantics. The Grothendieck construction lets us study all models of all enriched theories in all contexts in a single category. We illustrate these ideas with the SKI-combinator calculus, a variable-free version of the lambda calculus.

math.CT↗