CMU-CS-11-105Computer Science Department School of Computer Science, Carnegie Mellon University
CMU-CS-11-105
Daniel R. Licata February 2011 Ph.D. Thesis
Keywords: Type theory, category theory, functional programming,
dependent types, higher-dimensional type theory, abstract syntax,
binding and scope, derivability, admissibilityThis dissertation describes progress on programming with domain-specific specification logics in dependently typed programming languages. Domain-specific logics are a promising way to verify software, using a logic tailored to a style of programming or an application domain. Examples of domain-specific logics include separation logic, which has been used to verify imperative programs, and authorization logics, which have been used to verify security properties in security-typed languages. The first goal of the research described here is to show that it is possible to define, study, automate, and use domain-specific logics within a dependently typed programming language. We demonstrate this fact with a significant new example, showing how to embed a security-typed language using dependent types. This example suggests that better support for programming with logics in type theory will facilitate this style of program verification. The central notion in logic is consequenceâentailment from premises to conclusionsâand two notions of consequence are necessary for programming with logics: derivability, which captures uniform reasoning, and admissibility, which captures inductive proofs and functional programs. Presently, derivability is better supported in LF-based proof assistants, such as Twelf, Delphin, and Beluga, whereas admissibility is better supported in proof assistants based on Martin-Löf of type theory, such as Coq, Agda, and Epigram. Our second contribution is to show that it is possible to implement, within a dependently typed programming language, a logical framework that allows derivability and admissibility to be mixed in novel and interesting ways. The above framework is simply-typed, which makes it suitable for programming with abstract syntax but not logical derivations. Our third contribution is to generalize this framework to dependent types, which we accomplish as an instance of a more general problem: We describe Directed Type Theory (DTT), a new notion of dependent type theory, inspired by higher-dimensional category theory, which equips each type with a notion of transformation on its elements. The structural properties of a logic arise as a special case, by considering a type of contexts equipped with an appropriate notion of transformation. DTT is an exciting development independently of our application, as it generalizes recent connections between type theory, homotopy theory, and category theory to the asymmetric case. 218 pages
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