[ { "@id": "https://w3id.org/np/RAohAXrzytP8w7CINkKZLoog3BxIIUpggxow_mnZxeidQ/Head", "@graph": [ { "@id": "https://w3id.org/np/RAohAXrzytP8w7CINkKZLoog3BxIIUpggxow_mnZxeidQ", "http://www.nanopub.org/nschema#hasAssertion": [ { "@id": "https://w3id.org/np/RAohAXrzytP8w7CINkKZLoog3BxIIUpggxow_mnZxeidQ/assertion" } ], "http://www.nanopub.org/nschema#hasProvenance": [ { "@id": "https://w3id.org/np/RAohAXrzytP8w7CINkKZLoog3BxIIUpggxow_mnZxeidQ/provenance" } ], "http://www.nanopub.org/nschema#hasPublicationInfo": [ { "@id": "https://w3id.org/np/RAohAXrzytP8w7CINkKZLoog3BxIIUpggxow_mnZxeidQ/pubinfo" } ], "@type": [ "http://www.nanopub.org/nschema#Nanopublication" ] } ] }, { "@id": "https://w3id.org/np/RAohAXrzytP8w7CINkKZLoog3BxIIUpggxow_mnZxeidQ/assertion", "@graph": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch05-15--claim-test", "http://purl.org/dc/terms/description": [ { "@value": "analysis.py encodes each constraint system as a bitmask of allowed length-w words. Satisfiability is decided by iterating a reachability vector over the k^(w-1) de Bruijn vertices for 4V + 1 edge steps (existence of a valid word of length 4V + w). Independently, every period word u with |u| <= V is enumerated (sum of k^p over p = 1..V candidates, each stored as the bitmask of its cyclic length-w windows) and u^infinity is accepted iff its window mask is contained in the allowed set. The two indicators are compared for all 138,856 systems (exhaustive for (k,w) = (2,2), (2,3), (2,4), (3,2), (4,2); seeded samples of 2000 and 5000 for (2,5) and (3,3)); minimal periods are tabulated. Run time about 3 s." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 1 fraction of satisfiable constraint systems that have a periodic solution with period <= k^(w-1) (n = 129939, exact for the exhaustive families; the two sampled families are 2000 and 5000 seeded draws). Interpretation: Wolfram's 'local set of constraints' on a one-dimensional array of coloured cells (p. 210-211: each cell's neighbourhood must be one of a given set of allowed configurations) is a one-dimensional subshift of finite type: a set of allowed words of a fixed window length w over k colours. 'Can be satisfied at all' is read as the existence of a bi-infinite arrangement all of whose windows are allowed; by compactness this is equivalent to the existence of arbitrarily long valid finite words, and operationally it is tested by the existence of a valid word of length 4V + w (any valid word longer than V must revisit a de Bruijn vertex, so this length is far beyond the threshold, and the check is done by a reachability iteration, not a cycle search). 'Simple and purely repetitive arrangement' is read as a periodic configuration, found by brute-force enumeration of all period words u with |u| <= V = k^(w-1) and checking all cyclic windows. All constraint systems are enumerated exhaustively for (k, w) in {(2,2), (2,3), (2,4), (3,2), (4,2)} and sampled with a fixed seed for (2,5) and (3,3)." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch05-15: supports, confidence 0.97" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch05-15/result.json" } ], "http://www.w3.org/ns/prov#wasAttributedTo": [ { "@id": "https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1" } ], "https://w3id.org/delphi/ontology/admissibleDatasets": [ { "@value": "Planned datasets: sft-k2-w2-all, sft-k2-w3-all, sft-k2-w4-all, sft-k3-w2-all, sft-k4-w2-all, sft-k2-w5-sample, sft-k3-w3-sample" } ], "https://w3id.org/delphi/ontology/assessmentType": [ { "@value": "claim-test/0.1.0" } ], "https://w3id.org/delphi/ontology/dataKind": [ { "@value": "computational" } ], "https://w3id.org/delphi/ontology/disconfirmingObservation": [ { "@value": "Pre-registered: at least one constraint system admits a valid word of length 4V + w but no periodic arrangement with period <= V (this would refute the claim as stated, since the shortest possible periodic solution is bounded by V); or the two satisfiability indicators disagree in the other direction, which would indicate a broken test." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "Satisfiability was operationalised by the existence of a valid word of length 4V + w; by compactness this is equivalent to the existence of a bi-infinite arrangement, but the equivalence itself relies on finiteness of the vertex set, so the two indicators are not logically independent even though they are computed by different algorithms." }, { "@value": "The claim is a theorem (a non-empty one-dimensional shift of finite type contains a periodic point, from a cycle in its de Bruijn graph); the computation confirms it exhaustively for all two-colour constraints with windows up to 4 and all four-colour constraints with window 2, and by sampling for windows 5 (two colours) and 3 (three colours). The maximal minimal period found (8 for k = 2, w = 4; exactly the two de Bruijn-cycle constraint sets) matches the bound V = k^(w-1)." }, { "@value": "Wolfram's 'simple' is read as 'periodic'; the observed minimal periods are small (at most V), but 'simple' is not otherwise quantified." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered: across all enumerated or sampled constraint systems, every system that admits a valid word of length 4V + w (the operational test of satisfiability, independent of any cycle argument) also admits a periodic arrangement u^infinity with |u| <= V whose every cyclic window of length w is allowed; and every system with no such long valid word has no periodic arrangement either. Zero violations in both directions." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered: at least one constraint system admits a valid word of length 4V + w but no periodic arrangement with period <= V (this would refute the claim as stated, since the shortest possible periodic solution is bounded by V); or the two satisfiability indicators disagree in the other direction, which would indicate a broken test." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.97", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-1" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-2" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-3" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-4" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-5" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-6" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-7" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "For every one-dimensional local constraint system (a set of allowed length-w colour words over k colours) that admits any bi-infinite satisfying arrangement, there is a purely periodic satisfying arrangement, and its period can be taken at most V = k^(w-1)." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py encodes each constraint system as a bitmask of allowed length-w words. Satisfiability is decided by iterating a reachability vector over the k^(w-1) de Bruijn vertices for 4V + 1 edge steps (existence of a valid word of length 4V + w). Independently, every period word u with |u| <= V is enumerated (sum of k^p over p = 1..V candidates, each stored as the bitmask of its cyclic length-w windows) and u^infinity is accepted iff its window mask is contained in the allowed set. The two indicators are compared for all 138,856 systems (exhaustive for (k,w) = (2,2), (2,3), (2,4), (3,2), (4,2); seeded samples of 2000 and 5000 for (2,5) and (3,3)); minimal periods are tabulated. Run time about 3 s." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered: across all enumerated or sampled constraint systems, every system that admits a valid word of length 4V + w (the operational test of satisfiability, independent of any cycle argument) also admits a periodic arrangement u^infinity with |u| <= V whose every cyclic window of length w is allowed; and every system with no such long valid word has no periodic arrangement either. Zero violations in both directions." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "exhaustive" } ], "http://purl.org/dc/terms/license": [ { "@value": "" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: All 16 allowed-word sets, 2 colours, window 2 (generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-2", "http://purl.org/dc/terms/description": [ { "@value": "exhaustive" } ], "http://purl.org/dc/terms/license": [ { "@value": "" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: All 256 allowed-word sets, 2 colours, window 3 (generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-3", "http://purl.org/dc/terms/description": [ { "@value": "exhaustive" } ], "http://purl.org/dc/terms/license": [ { "@value": "" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: All 65536 allowed-word sets, 2 colours, window 4 (generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-4", "http://purl.org/dc/terms/description": [ { "@value": "exhaustive" } ], "http://purl.org/dc/terms/license": [ { "@value": "" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: All 512 allowed-word sets, 3 colours, window 2 (generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-5", "http://purl.org/dc/terms/description": [ { "@value": "exhaustive" } ], "http://purl.org/dc/terms/license": [ { "@value": "" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: All 65536 allowed-word sets, 4 colours, window 2 (generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-15--claim-test--dataset-6", "http://purl.org/dc/terms/description": [ { "@value": "sample" } ], "http://purl.org/dc/terms/license": [ { "@value": "" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: 2000 allowed-word sets sampled uniformly from the 2^32 sets, 2 colours, window 5 (numpy default_rng seed 20261031; 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