THEIA: Learning Complete Kleene Three-Valued Logic in a Pure-Neural Modular Architecture

TL;DR

THEIA is a neural architecture that learns Kleene three-valued logic rules with high accuracy, demonstrating properties like uncertainty propagation and robustness in compositional tasks, outperforming some baseline models in speed and reliability.

Contribution

Introduces THEIA, a modular neural system that learns complete K3 logic rules without symbolic inference, showing robustness and efficiency in complex logical tasks.

Findings

THEIA achieves >99% accuracy on all 39 K3 rules.

Uncertainty-verdict asymmetric propagation is demonstrated.

The system generalizes reliably on long compositional tasks.

Abstract

We present THEIA, a 2.75M modular neural architecture that learns the complete Kleene three-valued logic (K3) truth table from task data without external symbolic inference or hand-encoded K3 gate primitives. Across 5 seeds, THEIA achieves all 39 K3 rules at >99% per-rule accuracy. K3 learnability is not the central finding: Transformer baselines also reach >99% on all 39 rules, and flat MLPs match THEIA on Phase-1 accuracy within 0.04pp. The central findings are two properties of the learned system. (1) Uncertainty-verdict asymmetric propagation. The network preserves Has-Unknown at every upstream boundary (80.0/91.1/90.8/99.7% across Arith/Order/Set/Logic vs. ~52% majority) while final-verdict decodability stays at or below a 73.4% U-vs-non-U oracle reference under linear and nonlinear MLP probes. Activation patching on non-absorbent T->U configurations flips 4,898/4,898 OR pairs (4,719/4,719 AND) across 5 seeds, ruling out residual shortcuts. (2) Reliability spectrum under discretized end-to-end training, on task structures decomposable along the engine boundaries. A mod-3 sequential composition task generalizes from 5- to 500-step eval at 99.96+-0.04% (5 seeds). Under identical Gumbel-softmax training, flat MLPs collapse to chance by 50 steps; a 2x2 ResMLP depth x expansion grid reaches >=99% on only 3/20 (config, seed) trials; a pre-LN Transformer reaches 99.24+-0.34%. The 500-step figure is dominated by straight-through discretization preventing 0.999^500 compounding; the architectural separator is sustaining Phase-1 accuracy under Phase-3 end-to-end Gumbel training, where flat MLPs fail. Auxiliary: under matched optimizer settings THEIA reaches 12/12 Kleene coverage 6.5x faster than a parameter-comparable 8L Transformer; the ratio narrows to ~3.6x under Transformer-standard tuning. We did not perform a THEIA-optimal sweep; ratios are specific-config, not asymptotic.