Abstract
Type systems (in the form of categorial grammars) are the front runners in the quest for a formally elegant, computationally attractive and flexible theory of linguistic form and meaning. Words enact typed constants, and interact with one another via means of grammatical rules enacted by type inferences, composing larger phrases
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in the process. The end result is at the same time a parse, a proof and a program, bridging the seemingly disparate fields of linguistics, formal logics and computer science. The transition from form to meaning is traditionally handled via a series of homomorphisms that simplify nuances of the syntactic calculus to move towards a uniform semantic calculus. Alluring as this might be, it poses pragmatic considerations. For the setup to work on the semantic level, one has no choice but to start from the hardest part, namely natural language syntax. Phenomena like movement, word-order variation, discontinuities, etc. require careful treatment that needs to be both general enough to encompass the full range of grammatical utterances, yet strict enough to ward off ungrammatical derivations. This thesis takes an operational shortcut in targeting a ``deeper'' calculus of grammatical composition, engaging only minimally with surface syntax. Where previously functional functional syntactic types would be position-conscious, requiring their arguments in predetermined positions upon a binary tree, they are now agnostic to both tree structure and sequential order, alleviating the need for syntactic refinements. This simplification comes at the cost of a misalignment between provability and grammaticality: the laxer semantic calculus permits more proofs than linguistically desired. To circumvent this underspecification, the thesis takes a step away from the established norm, proposing the incorporation of unary type operators extending the function-argument axis with grammatical role labels. The new calculus produces mixed unary/n-ary trees, each unary tree denoting a dependency domain, and each n-ary tree denoting the phrases which together form that domain. Although still underspecified, these structures now subsume non-projective labeled dependency trees. More than that, they have their roots set firmly in type theory, allowing meaningful semantic interpretation. On more practical grounds and in order to investigate the formalism's adequacy, an extraction algorithm is employed to convert syntactic analyses of sentences (represented as dependency graphs) into proofs of the target logic. This gives rise to a large proofbank, a collection of sentences paired with tectogrammatic theorems and their corresponding programs, and an elaborate type lexicon, providing type assignments to one million lexical tokens within a linguistic context. The proofbank and the lexicon find use as training data in the design of a neurosymbolic proof search system, able to efficiently navigate the logic's theorem space. The system consists of three components. Component one is a supertagger responsible for assigning a type to each word — the tagger is formulated as a heterogeneous graph convolution kernel that boasts state-of-the-art accuracy. Rather than produce asignments in the form of conditional probabilities over a predefined vocabulary, it instead constructs types dynamicaly. As such, it is unconstrained by data sparsity, generalizing well to rare assignments and producing correct assignments for types never seen during training. Component two is a neural permutation module that exploits the linearity constraint of the logic in order to simplify proof search as optimal transport learning, associating resources (conditional validities) to the processes that require them (conditions). This allows for a parallel and easily optimizable implementation, unobstructed by the structure manipulation found in conventional parsers. Component three is the type system itself, responsible for navigating the produced structures and asserting their well-formedness. Results suggest performance superior to established baselines across categorial formalisms, despite the ambiguity inherent to the logic.
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