{"paper":{"title":"Short Presburger arithmetic is hard","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.LO","math.LO"],"primary_cat":"math.CO","authors_text":"Danny Nguyen, Igor Pak","submitted_at":"2017-08-28T03:54:10Z","abstract_excerpt":"We study the computational complexity of short sentences in Presburger arithmetic (Short-PA). Here by \"short\" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of Short-PA sentences with $m+2$ alternating quantifiers is $\\Sigma_{P}^m$-complete or $\\Pi_{P}^m$-complete, when the first quantifier is $\\exists$ or $\\forall$, respectively. Counting versions and restricted systems are also analyzed. Further application are given to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.08179","kind":"arxiv","version":3},"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"}