Differentiable Modal Logic for Multi-Agent Diagnosis, Orchestration and Communication

TL;DR

This tutorial demonstrates differentiable modal logic (DML), implemented via Modal Logical Neural Networks (MLNNs), enabling systems to learn trust networks, causal chains, and regulatory boundaries from behavioral data alone.

Abstract

As multi-agent AI systems evolve from simple chatbots to autonomous swarms, debugging semantic failures requires reasoning about knowledge, belief, causality, and obligation, precisely what modal logic was designed to formalize. However, traditional modal logic requires manual specification of relationship structures that are unknown or dynamic in real systems. This tutorial demonstrates differentiable modal logic (DML), implemented via Modal Logical Neural Networks (MLNNs), enabling systems to learn trust networks, causal chains, and regulatory boundaries from behavioral data alone. We present a unified neurosymbolic debugging framework through four modalities: epistemic (who to trust), temporal (when events cause failures), deontic (what actions are permitted), and doxastic (how to interpret agent confidence). Each modality is demonstrated on concrete multi-agent scenarios, from discovering deceptive alliances in diplomacy games to detecting LLM hallucinations, with complete implementations showing how logical contradictions become learnable optimization objectives. Key contributions for the neurosymbolic community: (1) interpretable learned structures where trust and causality are explicit parameters, not opaque embeddings; (2) knowledge injection via differentiable axioms that guide learning with sparse data (3) compositional multi-modal reasoning that combines epistemic, temporal, and deontic constraints; and (4) practical deployment patterns for monitoring, active control and communication of multi-agent systems. All code provided as executable Jupyter notebooks.

Figures (10)

• Figure 1: Temporal Trust Dynamics in Alliance Discovery. Evolution of trust from all agents toward Turkey across 35 interaction steps. Each subplot shows one agent's trust trajectory. Sharp drops (marked with "Lie Detected") indicate moments when Turkey's actions violated communicated intent, triggering immediate trust collapse. France, Germany, and England independently learn to distrust Turkey through repeated contradiction detection, while Turkey's self-trust (dashed crimson line) and Italy remains stable. The horizontal dotted line at 0.5 indicates neutral trust threshold. Note the event-driven nature of updates: trust remains stable during honest interactions and collapses only upon logical contradiction.
• Figure 2: Training Dynamics. Contradiction loss during training (log scale).
• Figure 3: Causality Scores. Learned causal attribution across the pre-crash timeline.
• Figure 4: Learned Deontic Boundary. Learned decision boundary in the duration--size feature space.
• Figure 5: Reliability Diagrams: Confidence vs. Empirical Accuracy. Each subplot shows one agent's calibration curve, with bubble size proportional to sample count in each confidence bin. The black dashed diagonal represents perfect calibration. Top row: Well-calibrated agents (0, 1) show strong diagonal alignment. Agent 1 lies above the diagonal, indicating under-confidence (accuracy exceeds reported confidence). Middle row: Agent 2 (moderate hallucinator) exhibits systematic overconfidence in the 0.6--0.8 range, where reported confidence exceeds empirical accuracy by $\sim$20 points. Bottom row: Agent 4 (severe hallucinator) shows catastrophic miscalibration, reporting 0.8+ confidence despite $<$50% accuracy in those bins. The bottom-right panel aggregates all agents, demonstrating that population-level calibration masks individual heterogeneity, highlighting the need for agent-specific modeling.