{"paper":{"title":"Model-Theoretic Characterizations of Boolean and Arithmetic Circuit Classes of Small Depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Anselm Haak, Arnaud Durand, Heribert Vollmer","submitted_at":"2017-10-05T09:23:29Z","abstract_excerpt":"In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\\textrm{NC}^1$, $\\textrm{SAC}^1$ and $\\textrm{AC}^1$ as well as their arithmetic counterparts $\\#\\textrm{NC}^1$, $\\#\\textrm{SAC}^1$ and $\\#\\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulas of first-order logic, i.e., $\\textrm{AC}^0 = \\textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering sli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.01934","kind":"arxiv","version":2},"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"}