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arXiv · 1710.10203

Extensional and Intensional Semantics of Bounded and Unbounded Nondeterminism

James Laird
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Abstract

We give extensional and intensional characterizations of functional programs with nondeterminism: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which compute them. A fundamental result establishes that these extensional and intensional representations are equivalent, by showing how to construct the unique sequential algorithm which computes a given monotone and stable function, and describing the conditions on sequential algorithms which correspond to continuity with respect to each order. We illustrate by defining may-testing and must-testing denotational semantics for sequential functional languages with bounded and unbounded choice operators. We prove that these are computationally adequate, despite the non-continuity of the must-testing semantics of unbounded nondeterminism. In the bounded case, we prove that our continuous models are fully abstract with respect to may-testing and must-testing by identifying a simple universal type, which may also form the basis for models of the untyped λ-calculus. In the unbounded case we observe that our model contains computable functions which are not denoted by terms, by identifying a further "weak continuity" property of the definable elements, and use this to establish that it is not fully abstract.

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James Laird. 2021-11-23. Extensional and Intensional Semantics of Bounded and Unbounded Nondeterminism. https://doi.org/10.46298/lmcs-17(4%3A11)2021

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