Slightly Non-Linear Higher-Order Tree Transducers

2024-02-08

AI Summary

This paper investigates the expressive power of tree-to-tree transducers based on affine λ-terms (affine λ-transducers). The central questions are: (i) whether affine λ-transducers are expressively equivalent to reversible tree-walking transducers under the affine restriction, and (ii) whether their finite nonlinearity extension captures exactly the class of invisible pebble tree transducers. To address these, we introduce a novel methodology: specializing the Interaction Abstract Machine (IAM) to affine λ-terms of low-exponential depth, integrated with geometric interaction semantics, MSO relabeling, and higher-order tree automata techniques. Our main contributions are threefold: (1) affine λ-transducers precisely characterize reversible tree-walking transducers; (2) their finite nonlinearity extension is expressively equivalent to invisible pebble tree transducers; and (3) we fully resolve the long-standing Nguyen–Pradic conjecture on implicit automata inexpressibility, revealing a fundamental correspondence among affinity, reversibility, and pebble capabilities.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Transparent, Interpretable, Explainable ML
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We investigate the tree-to-tree functions computed by"affine $lambda$-transducers": tree automata whose memory consists of an affine $lambda$-term instead of a finite state. They can be seen as variations on Gallot, Lemay and Salvati's Linear High-Order Deterministic Tree Transducers. When the memory is almost purely affine ($ extit{`a la}$ Kanazawa), we show that these machines can be translated to tree-walking transducers (and with a purely affine memory, we get a reversible tree-walking transducer). This leads to a proof of an inexpressivity conjecture of Nguy^en and Pradic on"implicit automata"in an affine $lambda$-calculus. We also prove that a more powerful variant, extended with preprocessing by an MSO relabeling and allowing a limited amount of non-linearity, is equivalent in expressive power to Engelfriet, Hoogeboom and Samwel's invisible pebble tree transducers. The key technical tool in our proofs is the Interaction Abstract Machine (IAM), an operational avatar of Girard's geometry of interaction, a semantics of linear logic. We work with ad-hoc specializations to $lambda$-terms of low exponential depth of a tree-generating version of the IAM.

Problem

• Study tree-to-tree functions via affine λ-transducers.
• Translate affine λ-transducers to tree-walking transducers.
• Prove equivalence to invisible pebble tree transducers.

Innovation

• Affine λ-transducers computation
• Reversible tree-walking transducers
• Interaction Abstract Machine usage