Practical Livelock Analysis in Parameterized Unidirectional Rings
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913079022845952 |
|---|---|
| author | Farahat, Aly |
| author_facet | Farahat, Aly |
| contents | We develop a practical framework for livelock analysis in self-disabling unidirectional ring protocols. Klinkhamer and Ebnenasir established that livelock detection for parameterized rings is $Σ^0_1$-complete and livelock-freedom verification is $Π^0_1$-complete, via reduction from the periodic domino problem. We observe that lifting the analysis from the transition space to an \emph{equivariant product space} -- the space of transition-witness pairs -- reveals structure that supports effective verification. We construct a \emph{product transition graph} (at most $|T|^2$ nodes) that captures all livelocks: every livelock maps into this graph as a witness-closed subgraph. The maximal such subgraph $G^*(T)$ is computable in polynomial time ($O(|T|^8)$ worst case) via monotone fixed-point iteration. When $G^*(T) = \emptyset$, the protocol is \emph{provably livelock-free} for all ring sizes -- a sound and complete livelock-freedom verifier. When $G^*(T) \neq \emptyset$, we apply a backtracking search that backward-propagates each simple cycle through $G^*$ until the chain either closes into a torus (confirming a livelock) or dies (no livelock from that cycle). This two-phase algorithm -- polynomial-time pruning followed by finite combinatorial verification -- produces three outcomes: Free, Livelock, or Inconclusive. Across 4{,}349 protocols tested (including an adversarial protocol derived from Klinkhamer and Ebnenasir's tiling construction and Kari's 14-tile aperiodic set converted via their SE gadget), the algorithm is conclusive in every case with zero errors.
We further demonstrate that the algorithm extends to non-self-disabling protocols via a protocol transformation. This extends the algorithm's applicability to all parameterized unidirectional ring protocols.
Python implementation and usage instructions are at URL: https://github.com/cosmoparadox/mathematical-tools. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21443 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Practical Livelock Analysis in Parameterized Unidirectional Rings Farahat, Aly Distributed, Parallel, and Cluster Computing Discrete Mathematics Formal Languages and Automata Theory Logic in Computer Science 68Q85 (Primary), 68W15 (Secondary) We develop a practical framework for livelock analysis in self-disabling unidirectional ring protocols. Klinkhamer and Ebnenasir established that livelock detection for parameterized rings is $Σ^0_1$-complete and livelock-freedom verification is $Π^0_1$-complete, via reduction from the periodic domino problem. We observe that lifting the analysis from the transition space to an \emph{equivariant product space} -- the space of transition-witness pairs -- reveals structure that supports effective verification. We construct a \emph{product transition graph} (at most $|T|^2$ nodes) that captures all livelocks: every livelock maps into this graph as a witness-closed subgraph. The maximal such subgraph $G^*(T)$ is computable in polynomial time ($O(|T|^8)$ worst case) via monotone fixed-point iteration. When $G^*(T) = \emptyset$, the protocol is \emph{provably livelock-free} for all ring sizes -- a sound and complete livelock-freedom verifier. When $G^*(T) \neq \emptyset$, we apply a backtracking search that backward-propagates each simple cycle through $G^*$ until the chain either closes into a torus (confirming a livelock) or dies (no livelock from that cycle). This two-phase algorithm -- polynomial-time pruning followed by finite combinatorial verification -- produces three outcomes: Free, Livelock, or Inconclusive. Across 4{,}349 protocols tested (including an adversarial protocol derived from Klinkhamer and Ebnenasir's tiling construction and Kari's 14-tile aperiodic set converted via their SE gadget), the algorithm is conclusive in every case with zero errors. We further demonstrate that the algorithm extends to non-self-disabling protocols via a protocol transformation. This extends the algorithm's applicability to all parameterized unidirectional ring protocols. Python implementation and usage instructions are at URL: https://github.com/cosmoparadox/mathematical-tools. |
| title | Practical Livelock Analysis in Parameterized Unidirectional Rings |
| topic | Distributed, Parallel, and Cluster Computing Discrete Mathematics Formal Languages and Automata Theory Logic in Computer Science 68Q85 (Primary), 68W15 (Secondary) |
| url | https://arxiv.org/abs/2603.21443 |