Categorical models for local names

Abstract

This paper describes the construction of categorical models for the nu-calculus, a language that combines higher-order functions with dynamically created names. Names are created with local scope, they can be compared with each other and passed around through function application, but that is all.The intent behind this language is to examine one aspect of the imperative character of Standard ML: the use of local state by dynamic creation of references. The nu-calculus is equivalent to a certain fragment of ML, omitting side effects, exceptions, datatypes and recursion. Even without all these features, the interaction of name creation with higher-order functions can be complex and subtle; it is particularly difficult to characterise the observable behaviour of expressions.Categorical monads, in the style of Moggi, are used to build denotational models for the nu-calculus. An intermediate stage is the use of a computational metalanguage, which distinguishes in the type system between values and computations.The general requirements for a categorical model are presented, and two specific examples described in detail. These provide a sound denotational semantics for the nu-calculus, and can be used to reason about observable equivalence in the language. In particular a model using logical relations is fully abstract for first-order expressions.

Relating Computational Effects by ⊤ ⊤-Lifting

“…The ν-calculus structure A 2 is a semantic analogue of the dynamic name creation by a global state. We note that the interpretation A 2 − is not sound with respect to the ν-calculus axioms in [32].…”

“…On the other hand, in practical implementations of programming languages, names are represented by natural numbers, and dynamic name creation is implemented by a hidden global counter that keeps track of the next fresh name. In this section, we consider Stark's ν-calculus [32] and discuss an effect simulation problem between presheaf semantics and global counter semantics of name creation.…”

“…The ν-calculus structure A 1 extracts the ingredients that are used in the categorical semantics of the ν-calculus in [32]. The monad T 1 is Stark's dynamic name creation monad:…”

“…We show that the given relationship holds if it satisfies a set of natural conditions. We apply this result to 1) comparing two monadic semantics related by a strong monad morphism, and 2) comparing two monadic semantics of fresh name creation: Stark's new name creation monad [32], and the global counter monad. We also consider the same problem, relating semantics of computational effects, in the presence of recursive functions.…”

Correctness of Data Representations Involving Heap Data Structures

“…A number of attacks have been made on the problem: Stark's thesis [25,24], which deals with dynamic allocation but not pointers, and Ghica's and Levy's theses [4,5,7,8], which address the general semantic structure but not data representation reasoning. The recent paper of Banerjee and Naumann [2] is the first to address data representation correctness with heap variables and pointers.…”