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For the case that $k=1$ (i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson ({\\em Computational Complexity'02}) demonstrate that $\\SZK$ exactly characterizes languages having 1-bit proof systems with\"non-trivial\" soundness (i.e., $1/2 < s \\leq 1-2\\epsilon$). We demonstrate that for the case that $k\\geq 2$, 1-bit $k$-prover games exhibit a significantly richer structure:"},"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":"1301.2729","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-01-12T23:02:05Z","cross_cats_sorted":[],"title_canon_sha256":"fed9d259192d07f3d88974a0f870913647e24153d5a9b4b6ed10f2d4b7922b5d","abstract_canon_sha256":"3c1e0fb04dffb5fab75fa4fa346336ad93a8e11b02685029c57bee02334b2442"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:35.423372Z","signature_b64":"mduIJNhnitk/v0FaWy+WeqL9hkCitLbk4MND+BiJsgdcalTaIfEymyMDxNMS50N45deqtQT544x2hZNbj9EDBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06542fb36f8eef1b24334c79ad05a38a008173a774dc7f628b2475d1c88e0216","last_reissued_at":"2026-05-18T03:36:35.422643Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:35.422643Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Power of Many One-Bit Provers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Johan H{\\aa}stad, Per Austrin, Rafael Pass","submitted_at":"2013-01-12T23:02:05Z","abstract_excerpt":"We study the class of languages, denoted by $\\MIP[k, 1-\\epsilon, s]$, which have $k$-prover games where each prover just sends a \\emph{single} bit, with completeness $1-\\epsilon$ and soundness error $s$. For the case that $k=1$ (i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson ({\\em Computational Complexity'02}) demonstrate that $\\SZK$ exactly characterizes languages having 1-bit proof systems with\"non-trivial\" soundness (i.e., $1/2 < s \\leq 1-2\\epsilon$). 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