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Run time about 18 s." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 1 fraction of (rule, block) pairs whose eventual temporal period is <= k^n (n = 5507844, exact (exhaustive within each family; the range-2 family is a seeded sample of 500 of 2^32 rules)). Interpretation: 'Initial conditions which consist just of a fixed block of cells repeated forever' is a spatially n-periodic configuration on the infinite line. 'Simple repetitive behavior' is read as eventual time-periodicity of the configuration, and the stated bound 2^n is read, following Wolfram's own reasoning and the assessors, as k^n for k colours (2^n in the two-colour case the text has in mind). 'Any cellular automaton' cannot be enumerated, so three families are tested: all 256 elementary rules exhaustively (n = 1..12), all 2187 three-colour nearest-neighbour totalistic rules (n = 1..6, bound 3^n), and a fixed-seed random sample of 500 two-colour range-2 (5-neighbour) rules out of 2^32 (n = 1..10). The reduction 'each block evolves like a finite cyclic system' is tested separately by simulating the actual repeated-block line and comparing it with the ring." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch06-17: supports, confidence 0.98" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch06-17/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: eca-repeated-blocks-n1-12, totalistic-3colour-repeated-blocks-n1-6, range2-binary-sample-repeated-blocks-n1-10, line-vs-ring-equivalence-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: any (rule, block) pair whose eventual period exceeds k^n, or any pair for which the repeated-block line evolution differs from the ring evolution inside the region unaffected by the line's ends." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "The bound k^n is attained only in degenerate cases (n = 1, rules that cycle through every state); for n >= 2 the overwhelming majority of periods are below k^n / 10 (see the histogram), so the bound is loose but never violated." }, { "@value": "The claim quantifies over all cellular automata; three rule families (one exhaustive over all 256 elementary rules, one exhaustive over all 2187 three-colour totalistic rules, one a 500-rule sample of range-2 rules) were tested, and the general statement rests on the finite-state argument that the line-vs-ring check confirms empirically for the elementary rules." }, { "@value": "Wolfram writes 2^n for two colours; the three-colour family tests the natural k^n generalisation that the assessors and the book's own argument imply." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered: (1) across all (rule, block) pairs enumerated, 100% have eventual temporal period <= k^n (zero violations); (2) for every (rule, block) pair in the equivalence sample, the central block of a long line made of repeated copies of the block evolves identically to the n-cell ring for all simulated steps (zero mismatching cells)." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered: any (rule, block) pair whose eventual period exceeds k^n, or any pair for which the repeated-block line evolution differs from the ring evolution inside the region unaffected by the line's ends." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.98", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-17--claim-test--dataset-1" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-17--claim-test--dataset-2" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-17--claim-test--dataset-3" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-17--claim-test--dataset-4" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "For every cellular automaton tested and every block of n cells repeated forever as the initial condition, the evolution is eventually time-periodic with period at most k^n (k = number of colours; 2^n for two colours), and the infinite repeated-block configuration evolves exactly as a cyclic ring of n cells." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py treats a block of n cells repeated forever as an n-cell ring, builds the ring's transition map on all k^n states for each rule and computes the eventual period of every state's orbit from the functional graph (cycle detection by in-degree peeling). Families: all 256 ECA for n = 1..12 (2,096,640 rule-block pairs), all 2187 three-colour totalistic rules for n = 1..6 (2,388,204 pairs, bound 3^n), and 500 seeded random range-2 binary rules for n = 1..10 (1,023,000 pairs). It then checks the reduction itself: for all 256 ECA and all 32,256 blocks of n = 1..6, a line of repeated copies of the block is evolved for 2*2^n steps and its central copy compared cell by cell with the ring evolution (15,845,376 cells). 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