Extensional and Intensional Semantics of Bounded and Unbounded Nondeterminism

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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 {\lambda}-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.
Original languageEnglish
Pages (from-to)11:1-11:37
JournalLogical Methods in Computer Science
Issue number4
Publication statusPublished - 24 Nov 2021


  • Biorders
  • Denotational Semantics
  • Nondeterminism
  • Sequential algorithms

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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