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Daphne Wang, Mehrnoosh Sadrzadeh, Samson Abramsky, Victor H. Cervantes

Language is contextual as meanings of words are dependent on their contexts. Contextuality is, concomitantly, a well-defined concept in quantum mechanics where it is considered a major resource for quantum computations. We investigate whether natural language exhibits any of the quantum mechanics' contextual features. We show that meaning combinations in ambiguous phrases can be modelled in the sheaf-theoretic framework for quantum contextuality, where they can become possibilistically contextual. Using the framework of Contextuality-by-Default (CbD), we explore the probabilistic variants of these and show that CbD-contextuality is also possible.

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Abhishek Dasgupta, Samson Abramsky

Anytime inference is inference performed incrementally, with the accuracy of the inference being controlled by a tunable parameter, usually time. Such anytime inference algorithms are also usually interruptible, gradually converging to the exact inference value until terminated. While anytime inference algorithms for specific domains like probability potentials exist in the literature, our objective in this article is to obtain an anytime inference algorithm which is sufficiently generic to cover a wide range of domains. For this we utilise the theory of generic inference as a basis for constructing an anytime inference algorithm, and in particular, extending work done on ordered valuation algebras. The novel contribution of this work is the construction of anytime algorithms in a generic framework, which automatically gives us instantiations in various useful domains. We also show how to apply this generic framework for anytime inference in semiring induced valuation algebras, an important subclass of valuation algebras, which includes instances like probability potentials, disjunctive normal forms and distributive lattices. Keywords: Approximation; Anytime algorithms; Resource-bounded computation; Generic inference; Valuation algebras; Local computation; Binary join trees.

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Samson Abramsky, Mehrnoosh Sadrzadeh

Language is contextual and sheaf theory provides a high level mathematical framework to model contextuality. We show how sheaf theory can model the contextual nature of natural language and how gluing can be used to provide a global semantics for a discourse by putting together the local logical semantics of each sentence within the discourse. We introduce a presheaf structure corresponding to a basic form of Discourse Representation Structures. Within this setting, we formulate a notion of semantic unification --- gluing meanings of parts of a discourse into a coherent whole --- as a form of sheaf-theoretic gluing. We illustrate this idea with a number of examples where it can used to represent resolutions of anaphoric references. We also discuss multivalued gluing, described using a distributions functor, which can be used to represent situations where multiple gluings are possible, and where we may need to rank them using quantitative measures. Dedicated to Jim Lambek on the occasion of his 90th birthday.

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