Neural Proof Nets

Abstract

Linear logic and the linear λ-calculus have a long standing tradition in the study of natural language form and meaning. Among the proof calculi of linear logic, proof nets are of particular interest, offering an attractive geometric representation of derivations that is unburdened by the bureaucratic complications of conventional prooftheoretic formats. Building on recent advances in set-theoretic learning, we propose a neural variant of proof nets based on Sinkhorn networks, which allows us to translate parsing as the problem of extracting syntactic primitives and permuting them into alignment. Our methodology induces a batch-efficient, end-to-end differentiable architecture that actualizes a formally grounded yet highly efficient neuro-symbolic parser. We test our approach on AEthel, a dataset of typelogical derivations for written Dutch, where it manages to correctly transcribe raw text sentences into proofs and terms of the linear λcalculus with an accuracy of as high as 70%.

Proof Net Structure for Neural Lambek Categorial Parsing

“…We see that removing all planarity information (i.e., the link filtering, the planarity-aware attention, and the planarity loss term T1 ) is disastrous; this condition has by far the largest drop in coverage. This is especially notable as LCG proof nets must be half-planar due to the non-commutativity of L*; this useful constraint is not present in type-logical grammars that do not have this property, such as that employed by Kogkalidis et al (2020).…”

“…We train our models on LCGbank, a semi-automatic conversion of CCGbank to LCG (Fowler, 2016). This conversion necessitated adjusting for instances of CCG's crossing rules that are not permitted in LCG, as well as providing fully categorial parses for the cases in CCGbank where non-categorial rules are used (e.g., unary type-changing).6 LCGbank also omits features on its categories and includes 5Although Kogkalidis et al (2020) describe their model's training as "end-to-end", their approach is perhaps better described as joint training. A truly end-to-end system would allow differentiation through the supertagger/proof frame construction, which remains a topic for further investigation.…”

“…To evaluate the effect of allowing -best linkages (Section 3.3), we evaluate all conditions with both = 1 and = 512. Note that with = 1, our baseline model is similar in design to that of Kogkalidis et al (2020), with minor differences such as model sizes and vector embedding details; this represents the closest point of comparison while controlling for our other model aspects as well as our grammar and corpus.…”

“…With this view, it becomes possible for novel categories to be produced; furthermore, the supertaggers are better able to incorporate prediction history and thereby produce grammatical outputs (Bhargava and Penn, 2020). Recently, Kogkalidis et al (2020) proposed a system for parsing a "type-logical" grammar that is essentially a modal, non-directional extension of LCG. The Dutch grammar they used is substantially different from our grammar: their connectives are both modal and non-directional; in addition, they have far more atomic categories.…”

Perspectives on Neural Proof Nets

“…In this paper I will present two different ways of combining proof net proof search with neural networks, using two different ways to split the task into two subtasks. The first approach is the 'standard' approach which has been applied to proof search in type-logical grammars in various different forms (Kogkalidis, Moortgat & Moot 2020b, De Pourtales, Rabault, Kogkalidis & Moot 2023. Since this approach has been discussed in other places, I will only present it briefly as a way to contrast it with the second, novel approach, which is the main topic of the paper.…”