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A formal approach to Menger's theorem

Bonacina Roberta, Misselbeck-Wessel Daniel

Reports on Mathematical Logic

2022

Vol. 57

45--51

Menger’s graph theorem equates the minimum size of a separating set for non-adjacent vertices a and b with the maximum number of disjoint paths between a and b. By capturing separating sets as models of an entailment relation, we take a formal approach to Menger’s result. Upon showing that inconsistency is characterised by the existence of sufficiently many disjoint paths, we recover Menger’s theorem by way of completeness.

Graded hyponymy for compositional distributional semantics

Bankova D., Coecke B., Lewis M., Marsden D.

Journal of Language Modelling

2018

Vol. 6, No. 2

225--260

The categorical compositional distributional model of natural language provides a conceptually motivated procedure to compute the meaning of a sentence, given its grammatical structure and the meanings of its words. This approach has outperformed other models in mainstream empirical language processing tasks, but lacks an effective model of lexical entailment. We address this shortcoming by exploiting the freedom in our abstract categorical framework to change our choice of semantic model. This allows us to describe hyponymy as a graded order on meanings, using models of partial information used in quantum computation. Quantum logic embeds in this graded order.