Right Divisibility in Erasing Semi-Thue Systems: A Minimal View of Intruder Deduction

2026-08-04

AI Summary

This study addresses the intruder deduction problem in symbolic security protocols—specifically, whether an adversary can derive a target message from observed communications. By reducing terms involving unary function symbols to words, the paper formulates this problem for the first time as the right-divisibility problem in semi-Thue systems, leveraging techniques from term rewriting and formal language theory. The main contributions establish that right-divisibility is decidable for convergent rewrite systems equipped with prefix or suffix erasing rules, while demonstrating its undecidability in variable-promoting systems. These results delineate both the potential and the limitations of extending the approach to richer equational theories.

Abstract

The intruder deduction problem is central to symbolic security-protocol analysis: it asks whether an attacker can derive a target message from observed messages using (cryptographic) operators available to the attacker. Although convergent rewrite systems provide canonical normal forms, deduction modulo convergent theories remains undecidable in general, and existing decidable fragments are often shaped by practical cryptographic examples. In this paper, we study deduction from a minimal structural perspective. When all function symbols are unary, terms collapse to words and deduction becomes a right-divisibility problem for semi-Thue systems: given words $u$ and $v$ decide whether there exists $w$ such that $wu \equiv_S v$. We investigate this problem for several classes of semi-Thue systems and prove, to the best of our knowledge, new decidability results for convergent prefix-erasing and convergent suffix-erasing systems. We then extend this perspective to term rewriting systems whose rules erase contexts while lifting selected subterms or variables. Although these classes suggest possible decidable generalisations beyond the unary setting, we show that deduction is already undecidable for a convergent simultaneous variable-lifting system. This exposes both the potential and the limits of extending the right-divisibility results to richer equational theories.

Problem

• intruder deduction
• semi-Thue systems
• right divisibility
• convergent rewrite systems
• decidability

Innovation

• right-divisibility
• semi-Thue systems
• intruder deduction
• convergent rewriting
• decidability