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Let the even (online Kolmogorov) complexity of an n-bitstring $x_1x_2... x_n$ be the length of a shortest program that computes $x_2$ on input $x_1$, computes $x_4$ on input $x_1x_2x_3$, etc; and similar for odd complexity. We show that for all n there exist an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence $x_2x_"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1307.4007","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2013-07-15T16:22:29Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"64c1e92d143390309933f5a2235a0a4c368bc3efb145d8f2d37988bbdc751fa4","abstract_canon_sha256":"c9a9ed5f4fe958c48817cdceb5807529ddc20fbc544f3b596886337014158bec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:59.330510Z","signature_b64":"k6fr13rQBP3+dcDSbvHExboDCqBQO8x9XKREvnJnOMR9kmH59A2AEVhYfUy1mcuJbvl4JIb0cpp+I6ow+04HCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81cffedcc136af1ed37dfb39636cb9c22661487b8b2caf9fd99c04abd79d3f77","last_reissued_at":"2026-05-18T03:02:59.329724Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:59.329724Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymmetry of the Kolmogorov complexity of online predicting odd and even bits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Bruno Bauwens","submitted_at":"2013-07-15T16:22:29Z","abstract_excerpt":"Symmetry of information states that $C(x) + C(y|x) = C(x,y) + O(\\log C(x))$. We show that a similar relation for online Kolmogorov complexity does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring $x_1x_2... x_n$ be the length of a shortest program that computes $x_2$ on input $x_1$, computes $x_4$ on input $x_1x_2x_3$, etc; and similar for odd complexity. We show that for all n there exist an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. 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