We introduce a new formal model -- based on the mathematical construct of sheaves -- for representing contradictory information in textual sources. This model has the advantage of letting us (a) identify the causes of the inconsistency; (b) measure how strong it is; (c) and do something about it, e.g. suggest ways to reconcile inconsistent advice. This model naturally represents the distinction between contradictions and disagreements. It is based on the idea of representing natural language sentences as formulas with parameters sitting on lattices, creating partial orders based on predicates shared by theories, and building sheaves on these partial orders with products of lattices as stalks. Degrees of disagreement are measured by the existence of global and local sections. Limitations of the sheaf approach and connections to recent work in natural language processing, as well as the topics of contextuality in physics, data fusion, topological data analysis and epistemology are also discussed.
The work on contextuality in computer science and physics exemplified by @cite_31 and previously cited works. It shows the application of sheaves and cohomology to modeling of incompatible measurements and assignments of values to variables. Cohomology is also discussed as a tool for sensor data fusion in the cited works of Robinson. These approaches are likely to also be applicable to language understanding, and, again, we are planning to look into cohomological models in a near future. @cite_17 gives a detailed exposition of using cohomology to find obstacles to global sections.
@cite_31 @cite_17
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Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We generalise obstructions to higher cohomology groups and show that they give rise to a refinement of the notion of cohomological contextuality: different "levels" of contextuality are organised in a hierarchy of logical implications. Finally, we present an alternative description of the first cohomology group in terms of torsors, resulting in a new interpretation of the cohomological obstructions. Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions. Contextuality in quantum physics provides a key resource for quantum information and computation. The topological approach in [Abramsky and Brandenburger, New J. Phys., 2011, , CSL 2015, 2015] characterizes contextuality as "global inconsistency" coupled with "local consistency", revealing it to be a phenomenon also found in many other fields. This has yielded a logical method of detecting and proving the "global inconsistency" part of contextuality. Our goal is to capture the other, "local consistency" part, which requires a novel approach to logic that is sensitive to the topology of contexts. To achieve this, we formulate a logic of local inference by using context-sensitive theories and models in regular categories. This provides a uniform framework for local consistency, and lays a foundation for high-level methods of detecting, proving, and moreover using contextuality as computational resource. Centering theory is the best-known framework for theorizing about local coherence and salience; however, its claims are articulated in terms of notions which are only partially specified, such as "utterance," "realization," or "ranking." A great deal of research has attempted to arrive at more detailed specifications of these parameters of the theory; as a result, the claims of centering can be instantiated in many different ways. We investigated in a systematic fashion the effect on the theory's claims of these different ways of setting the parameters. Doing this required, first of all, clarifying what the theory's claims are (one of our conclusions being that what has become known as "Constraint 1" is actually a central claim of the theory). Secondly, we had to clearly identify these parametric aspects: For example, we argue that the notion of "pronoun" used in Rule 1 should be considered a parameter. Thirdly, we had to find appropriate methods for evaluating these claims. We found that while the theory's main claim about salience and pronominalization, Rule 1—a preference for pronominalizing the backward-looking center (CB)—is verified with most instantiations, Constraint 1–a claim about (entity) coherence and CB uniqueness—is much more instantiation-dependent: It is not verified if the parameters are instantiated according to very mainstream views ("vanilla instantiation"), it holds only if indirect realization is allowed, and is violated by between 20 and 25 of utterances in our corpus even with the most favorable instantiations. We also found a trade-off between Rule 1, on the one hand, and Constraint 1 and Rule 2, on the other: Setting the parameters to minimize the violations of local coherence leads to increased violations of salience, and vice versa. Our results suggest that "entity" coherence—continuous reference to the same entities—must be supplemented at least by an account of relational coherence.