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We introduce an expressive class of syntactic reductions, which we call\n                    <jats:italic>natural reductions<\/jats:italic>\n                    . Natural reductions are specified by introducing atomic blocks and global rendezvous points in the parameterized program\u2019s thread template. We study the problem of deciding whether a given natural reduction is sound wrt. a given (semi-)commutativity relation. In the case that there is no synchronization between threads, we present a sound and complete polynomial-time algorithm. In the case where synchronization is considered, we provide a general lower bound for the problem (parametric in the synchronization mechanism), and show that the problem is\n                    <jats:sc>coNP<\/jats:sc>\n                    -hard already for a simple mechanism like locking.\n                  <\/jats:p>","DOI":"10.1007\/978-3-032-32519-8_2","type":"book-chapter","created":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T07:18:05Z","timestamp":1784791085000},"page":"19-41","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the\u00a0Complexity of\u00a0Checking Soundness of\u00a0Natural Reductions"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2727-8865","authenticated-orcid":false,"given":"Constantin","family":"Enea","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9005-2653","authenticated-orcid":false,"given":"Azadeh","family":"Farzan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4885-0728","authenticated-orcid":false,"given":"Dominik","family":"Klumpp","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,24]]},"reference":[{"key":"2_CR1","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"290","DOI":"10.1007\/978-3-642-37036-6_17","volume-title":"Programming Languages and Systems","author":"A Bouajjani","year":"2013","unstructured":"Bouajjani, A., Emmi, M., Enea, C., Hamza, J.: Verifying Concurrent Programs against Sequential Specifications. 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