{"paper":{"title":"Parameterized circuit complexity of model checking first-order logic on sparse structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DM","authors_text":"Micha{\\l} Pilipczuk, Sebastian Siebertz, Szymon Toru\\'nczyk","submitted_at":"2018-05-09T12:49:23Z","abstract_excerpt":"We prove that for every class $C$ of graphs with effectively bounded expansion, given a first-order sentence $\\varphi$ and an $n$-element structure $\\mathbb{A}$ whose Gaifman graph belongs to $C$, the question whether $\\varphi$ holds in $\\mathbb{A}$ can be decided by a family of AC-circuits of size $f(\\varphi)\\cdot n^c$ and depth $f(\\varphi)+c\\log n$, where $f$ is a computable function and $c$ is a universal constant. This places the model-checking problem for classes of bounded expansion in the parameterized circuit complexity class $para-AC^1$. On the route to our result we prove that the ba"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03488","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}