734 research outputs found

    Changing a semantics: opportunism or courage?

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    The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

    Inductive Definition and Domain Theoretic Properties of Fully Abstract

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    A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF + "parallel conditional function"), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent non-deterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given level-by-level in the finite type hierarchy. To prove full abstraction and non-dcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF^+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are non-omega-complete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omega-completeness), "naturally" continuous (with respect to existing directed "pointwise", or "natural" lubs) and also "naturally" omega-algebraic and "naturally" bounded complete -- appropriate generalisation of the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page

    Accommodating prepositional phrases in a highly modular natural language query interface to semantic web triplestores using a novel event-based denotational semantics for English and a set of functional parser combinators

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    The SemanticWeb is an emerging component of the set of technologies that will be known as Web 3.0 in the future. With the large changes it brings to how information is stored and represented to users, there is a need to re-evaluate how this information can be queried. Specifically, there is a need for Natural Language Interfaces that allow users to easily query for information on the Semantic Web. While there has been previous work in this area, existing solutions suffer from the problem that they do not support prepositional phrases in queries (e.g, “in 1958” or “with a key”). To achieve this, we improve on an existing semantics for event-based triplestores that supports prepositional phrases and demonstrate a novel method of handling the word “by”, treating it directly as a preposition in queries. We then show how this new semantics can be integrated with a parser constructed as an executable attribute grammar to create a highly modular and extensible Natural Language Interface to the Semantic Web that supports prepositional phrases in queries

    Retractions in comparing PROLOG semantics

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    We present an operational model O and a continuation based denotational model D for a uniform variant of PROLOG, including the cut operator. The two semantical definitions make use of higher order transformations Phi and Psi, respectively. We prove O and D equivalent in a novel way by comparing yet another pair of higher order transformations Phi~ and Psi~, that yield Phi and Psi, respectively, by application of a suitable abstraction operator

    Higher Order Automatic Differentiation of Higher Order Functions

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    We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Throughout, we show how the analysis extends to AD methods for computing higher order derivatives using a Taylor approximation.Comment: 34 pages, 5 figures, submitted at LMCS 2020. arXiv admin note: substantial text overlap with arXiv:2001.0220
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