{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/strong-eth-breaks-with-merlin-and-arthur","title":"Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation","arxiv_id":"1601.04743","date":"2016-01-18","proceeding":null,"authors":["Ryan Williams"],"abstract":"We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\\mathbb F}$, and any inputs $a_1,\\ldots,a_K \\in {\\mathbb F}^n$, $\\bullet$ the Prover sends the Verifier the values $C(a_1), \\ldots, C(a_K) \\in {\\mathbb F}$ and a proof of $\\tilde{O}(K \\cdot d)$ length, and $\\bullet$ the Verifier tosses $\\textrm{poly}(\\log(dK|{\\mathbb F}|/\\varepsilon))$ coins and can check the proof in about $\\tilde{O}(K \\cdot(n + d) + s)$ time, with probability of error less than $\\varepsilon$. For small degree $d$, this \"Merlin-Arthur\" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in $c^{n}$ time (for various $c < 2$) for the Permanent, $\\#$Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of $0$-$1$ linear programs. In general, the value of any polynomial in Valiant's class ${\\sf VP}$ can be certified faster than \"exhaustive summation\" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. We also give a three-round (AMA) proof system for quantified Boolean formulas running in $2^{2n/3+o(n)}$ time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in $n^{k/2+O(1)}$-time for counting $k$-cliques in graphs. We point to some potential future directions for refuting the Nondeterministic Strong ETH.","url_abs":"http://arxiv.org/abs/1601.04743v1","url_pdf":"http://arxiv.org/pdf/1601.04743v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"strong-eth-breaks-with-merlin-and-arthur","repo_url":"https://github.com/scipr-lab/dizk","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}