Intensional and Extensional Semantics of Bounded and Unbounded Nondeterminism

We give extensional and intensional characterizations of nondeterministic functional programs: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which compute them. A fundamental result establishes that the extensional and intensional representations of non-deterministic programs are equivalent, by showing how to construct a 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 and must-testing denotational semantics for a sequential functional language 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 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

English article usage as a window on the meanings of same, identical and similar

We propose an explanation for a traditional puzzle in English linguistics involving the use of articles with the nominal modifiers same, identical and similar. Same can only take the definite article the, whereas identical and similar take either the or a. We argue that there is a fundamental difference in the manner in which a comparison is made with these modifiers. Identical and similar involve direct comparisons between at least two entities and an assertion of either full property matching (identical), or partial property matching (similar). The comparison with same proceeds differently: what is compared is not linguistic entities directly, but definite descriptions of these entities that can be derived through logical entailments. John and Mary live in the same house entails the house that John lives in is the (same) house that Mary lives in. There must be a pragmatic equivalence between these entailed definite descriptions, ranging from full referential equivalence to a possibly quite minimal overlap in semantic and real-world properties shared by distinct referents. These differences in meaning and article cooccurrence reveal the sensitivity of syntax to semantic and pragmatic properties, without which all and only the grammatical sentences of a language cannot be predicted

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Just below the surface: developing knowledge management systems using the paradigm of the noetic prism

In this paper we examine how the principles embodied in the paradigm of the noetic prism can illuminate the construction of knowledge management systems. We draw on the formalism of the prism to examine three successful tools: frames, spreadsheets and databases, and show how their power and also their shortcomings arise from their domain representation, and how any organisational system based on integration of these tools and conversion between them is inevitably lossy. We suggest how a late-binding, hybrid knowledge based management system (KBMS) could be designed that draws on the lessons learnt from these tools, by maintaining noetica at an atomic level and storing the combinatory processes necessary to create higher level structure as the need arises. We outline the “just-below-the-surface” systems design, and describe its implementation in an enterprise-wide knowledge-based system that has all of the conventional office automation features

Perspectival Indexicality in Fiction

In everyday language use, the content of an indexical sentence is determined by the parameters of the context in which it occurs. In fictional discourse, however, indexical sentences seem to behave in a nonstandard way. This paper attempts to show that the difference can be best explained by using the concept of fictional perspective

New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized

The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation fails to identify two `obviously' equal terms. If only the machine could decide a richer theory! We propose a way to decide theories which supplement evaluation with `$\nu$-rules', rearranging the neutral parts of normal forms, and report a successful initial experiment. We study a simple -calculus with primitive fold, map and append operations on lists and develop in Agda a sound and complete decision procedure for an equational theory enriched with monoid, functor and fusion laws