[Paper] Declarative distributed broadcast using three-valued modal logic and semitopologies

Source: arXiv - 2512.21137v1

Overview

The paper introduces a declarative way to specify distributed algorithms using a three‑valued modal logic combined with semitopologies. By expressing protocols as logical axioms rather than step‑by‑step state machines, the authors obtain specifications that are both human‑readable and amenable to formal verification. The approach is demonstrated on voting, broadcast, and agreement protocols, and it has already uncovered bugs in an industrial‑grade protocol.

Key Contributions

• A novel logical framework: Extends classic modal logic with a third truth value (unknown) and semitopological structures to model distributed system states.
• Declarative axiomatization of protocols: Shows how to capture the essence of distributed algorithms (e.g., broadcast, voting) as compact sets of logical axioms.
• Tool‑supported error detection: Applied the framework to a real‑world industrial protocol and automatically identified subtle design flaws.
• Scalable methodology: The same logical style works for simple toy protocols and for more complex, production‑level algorithms without a combinatorial explosion of cases.
• Bridge between humans and machines: Provides a specification format that is easier for engineers to read than raw code, yet precise enough for automated model‑checking or theorem‑proving tools.

Methodology

1. Three‑valued modal logic – Truth values are true, false, and unknown (⊤, ⊥, ?). The unknown value lets the logic express “I don’t know yet” which mirrors the partial knowledge typical in asynchronous networks.
2. Semitopologies – Instead of a global state space, the system is modeled as a collection of open sets representing the local views of nodes. Overlaps of opens capture communication links and shared knowledge.
3. Axiomatization – Protocol behavior is encoded as logical axioms (e.g., “if a node eventually receives a message, then it must eventually broadcast it”). No explicit transition rules are written; the axioms implicitly constrain all possible executions.
4. Verification pipeline – The axioms are fed into an off‑the‑shelf modal‑logic prover. Counter‑examples correspond to protocol violations (e.g., a node never learns the broadcast value).
5. Case studies – The authors walk through three canonical distributed problems, showing how each axiom set is derived and how the prover validates properties like termination, agreement, and resilience to message loss.

Results & Findings

• Correctness proofs for the three example protocols were obtained automatically, confirming properties such as eventual delivery (broadcast) and consensus (agreement).
• Bug discovery: When the same technique was applied to a proprietary broadcast protocol used in an industrial setting, the prover produced a counter‑example exposing a race condition that could leave some nodes permanently uninformed.
• Specification size: The declarative axioms were an order of magnitude shorter than equivalent state‑machine specifications, yet they captured all needed behaviours.
• Performance: Even with the added third truth value, the underlying provers scaled to realistic network sizes (tens of nodes) with verification times measured in seconds to minutes.

Practical Implications

• Design‑time validation: Engineers can write a concise logical spec of a new protocol and run automated checks before any code is written, catching design flaws early.
• Documentation that doubles as verification: The axioms serve as both a readable description for developers and a machine‑checkable contract for formal tools.
• Interoperability across languages: Because the spec is language‑agnostic, the same logical description can guide implementations in Go, Rust, Erlang, etc., ensuring they all adhere to the same high‑level guarantees.
• Resilience engineering: The three‑valued logic naturally models uncertain or missing information, making it easier to reason about partial failures, network partitions, and eventual consistency.
• Foundation for automated synthesis: Future tools could translate the axioms directly into skeleton code or protocol state machines, reducing boilerplate and human error.

Limitations & Future Work

• Expressiveness vs. automation trade‑off: While the three‑valued modal logic captures many distributed phenomena, certain low‑level timing constraints (e.g., real‑time deadlines) are not directly expressible.
• Scalability to very large systems: The current experiments stop at a few dozen nodes; scaling to hundreds or thousands may require abstraction techniques or compositional reasoning.
• Tool ecosystem: The approach relies on general‑purpose modal provers; tighter integration with existing distributed‑systems verification frameworks (e.g., TLA⁺, Ivy) could lower the entry barrier.
• User ergonomics: Writing axioms still demands a logical mindset; the authors suggest developing higher‑level DSLs or visual editors to make the process more developer‑friendly.

Bottom line: By treating distributed protocols as declarative logical theories, this work opens a path toward specifications that are simultaneously clear to humans and rigorous enough for machines—an attractive proposition for anyone building reliable, fault‑tolerant networked software.

Authors

• Murdoch J. Gabbay

Paper Information

• arXiv ID: 2512.21137v1
• Categories: cs.LO, cs.DC
• Published: December 24, 2025
• PDF: Download PDF