Non-commutative Logic for Compositional Distributional Semantics

Abstract

Distributional models of natural language use vectors to provide a contextual foundation for meaning representation. These models rely on large quantities of real data, such as corpora of documents, and have found applications in natural language tasks, such as word similarity, disambiguation, indexing, and search. Compositional distributional models extend the distributional ones from words to phrases and sentences. Logical operators are usually treated as noise by these models and no systematic treatment is provided so far. In this paper, we show how skew lattices and their encoding in upper triangular matrices provide a logical foundation for compositional distributional models. In this setting, one can model commutative as well as non-commutative logical operations of conjunction and disjunction. We provide theoretical foundations, a case study, and experimental results for an entailment task on real data.

What is a noncommutative topos?

“…In this section we will recall the main structural results on noncommutative (complete) Heyting algebras obtained in [1].…”

“…In [1] noncommutative Heyting algebras were introduced and studied. A skew lattice is an algebra (L, ∧, ∨) where ∧ and ∨ are idempotent and associative binary operations satisfying the identities…”

“…The main structural result on noncommutative Heyting algebras, [1,Thm. 3.5] asserts that if (H, ∧, ∨, →, 0, t) is a noncommutative Heyting algebra, then (1) (t ↓ , ∧, ∨, →, 0, t) is a Heyting algebra with a unique top element t, isomorphic to H/D.…”

“…First, we will construct complete noncommutative Heyting algebras. By a result of [1] complete noncommutative Heyting algebras are exactly noncommutative frames (together with a distinguished element in the top D-class), where a noncommutative frame is a strongly distributive, join complete skew lattice that satisfies the infinite distributive laws.…”

“…Let then the equivalence classes H/D with the induced structures are isomorphic to Ω as Heyting algebras. H is an example of a noncommutative complete Heyting algebra as introduced and studied in [1]. We can view H as the set of opens of a noncommutative topological space with commutative shadow X.…”