This work investigates three notions of program equivalence for a higher-order functional language with recursion and general algebraic effects, in which programs are written in continuation-passing style. Our main contribution is the following: we define a logic whose formulas express program properties and show that, under certain conditions which we identify, the induced program equivalence coincides with a contextual equivalence. Moreover, we show that this logical equivalence also coincides with an applicative bisimilarity. We exemplify our general results with the nondeterminism, probabilistic choice, global store and I/O effects.
This dissertation is concerned with the study of program equivalence and algebraic effects as they arise in the theory of programming languages. Algebraic effects represent impure behaviour in a functional programming language, such as input and output, exceptions, nondeterminism etc. all treated in a generic way. Program equivalence aims to identify which programs can be considered equal in some sense. This question has been studied for a long time but has only recently been extended to languages with algebraic effects, which are a newer development. Much work remains to be done in order to understand program equivalence in the presence of algebraic effects. In particular, there is no characterisation of contextual equivalence using a logic.We define a logic whose formulas express properties of higher-order programs with algebraic effects. We then investigate three notions of program equivalence for algebraic effects: logical equivalence induced by the aforementioned logic, applicative bisimilarity and contextual equivalence. For the programming language used in this dissertation, we prove that they all coincide.Therefore, the main novel contribution of the dissertation is defining the first logic for algebraic effects whose induced program equivalence coincides with contextual equivalence.
The propositions-as-types correspondence is ordinarily presented as linking the metatheory of typed λ-calculi and the proof theory of intuitionistic logic. Griffin observed that this correspondence could be extended to classical logic through the use of control operators. This observation set off a flurry of further research, leading to the development of Parigot's λµ-calculus. In this work, we use the λµ-calculus as the foundation for a system of proof terms for classical first-order logic. In particular, we define an extended call-by-value λµ-calculus with a type system in correspondence with full classical logic. We extend the language with polymorphic types, add a host of data types in 'direct style', and prove several metatheoretical properties. All of our proofs and definitions are mechanised in Isabelle/HOL, and we automatically obtain an interpreter for a system of proof terms cum programming language-called µML-using Isabelle's code generation mechanism. Atop our proof terms, we build a prototype LCF-style interactive theorem prover-called µTPfor classical first-order logic, capable of synthesising µML programs from completed tactic-driven proofs. We present example closed µML programs with classical tautologies for types, including some inexpressible as closed programs in the original λµ-calculus, and some example tactic-driven µTP proofs of classical tautologies.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.