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It generates rule 30 from a single black cell with big-integer bit operations, taking the central 40001 cells of row 20000 and the first 65536 centre-column cells, and applies the same count. It also measures what share of the centre column's length-14 factors appear in any binary substitution word, and runs the one-cell-per-cell neighbour-dependent substitution systems for rules 90 and 30 against the direct CA for 1000 steps. Runtime about 19 s." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 157 maximum number of distinct length-14 factors over all 1924 neighbour-independent substitution words (threshold 225) (n = 1924, exact combinatorial counts). Interpretation: 'Repetitive' is operationalised as eventually periodic (bounded subword complexity, Morse-Hedlund), 'nested' as a self-similar word, which for fixed points of morphisms has subword complexity O(n^2) at most (Pansiot); the threshold (n+1)^2 at n = 14 is therefore a generous upper bound for repetitive-or-nested and about a tenth of what a random-like binary sequence of this length shows. 'Cannot emulate rule 30' is tested within the enumerated family by the complexity gap (a word with p(14) <= 225 cannot contain the rule 30 row, whose own p(14) is measured); it is not a proof over all encodings. The neighbour-dependent half is tested exactly as the book describes it (rules 'always yielding just one cell and corresponding quite directly to cellular automata', p. 666)." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch11-09: supports, confidence 0.85" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch11-09/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: substitution-systems-and-rule30-generator" } ], "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. FAIL (refutes) if any neighbour-independent rule in the family has p(14) >= 2000 (rule-30-like complexity), or if the neighbour-dependent construction does not reproduce rule 90 or rule 30. If some neighbour-independent rule has 225 < p(14) < 2000 the result is 'mixed' and the rule is reported with its complexity profile." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "The impossibility half is tested on the two complete enumerated families named in the book's own discussion (p. 83), not for arbitrarily long replacement blocks or alphabets; the complexity gap (at most 157 distinct length-14 factors versus 14922-16034 for rule 30) is a finite proxy for the theorem that fixed points of morphisms have polynomially bounded subword complexity." }, { "@value": "The length-14 factor share result (at most 0.5% of rule 30's centre-column factors occur in any binary substitution word) is reported as a direct check that none of the enumerated systems reproduces rule 30's sequence under the identity encoding; other encodings were not explored." }, { "@value": "The neighbour-dependent half is near-definitional: a substitution system whose single output cell depends on the cell and its two neighbours is a cellular automaton; the test confirms it reproduces rules 90 and 30 exactly for 1000 steps, which is what the book's figure shows." }, { "@value": "The subword-complexity criterion treats any word with p(14) <= 225 as repetitive-or-nested; the words with the largest complexity (e.g. 0 -> 0, 1 -> 011, p(n) = 2,4,6,9,13,...,81) are non-primitive morphic words whose complexity grows faster than linearly but far below that of rule 30, consistent with the known n log n / n^2 classes for such words." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered. For each of the 196 two-colour rules {0 -> w0, 1 -> w1} with |w0|, |w1| in {1,2,3} and each of the 1728 three-colour rules with block lengths in {1,2}, start from a single cell of colour 1 and iterate until the word has >= L = 131072 symbols (or 60 steps if it never does), truncate to L, and compute the number p(n) of distinct length-n factors for n = 1..14. For rule 30 from a single black cell compute p(14) of (i) the central 40001 cells of the row at step 20000 and (ii) the first 65536 cells of the centre column. Build the neighbour-dependent substitution system whose rule maps each cell to one new cell given by the elementary-CA table of its (left, self, right) neighbourhood, and run it for 1000 steps from a single black cell. PASS (supports) if (a) every neighbour-independent rule has p(14) <= 225 = (14+1)^2 (eventually periodic words are those with p(14) = p(13); the rest are counted as nested), (b) both rule 30 sequences have p(14) >= 2000, and (c) the neighbour-dependent systems for rules 90 and 30 reproduce the cellular-automaton rows exactly at every one of the 1000 steps." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered. FAIL (refutes) if any neighbour-independent rule in the family has p(14) >= 2000 (rule-30-like complexity), or if the neighbour-dependent construction does not reproduce rule 90 or rule 30. If some neighbour-independent rule has 225 < p(14) < 2000 the result is 'mixed' and the rule is reported with its complexity profile." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.85", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-09--claim-test--dataset-1" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "If the claim holds, every neighbour-independent substitution system in a complete enumerated family (2 colours with replacement blocks of length 1-3; 3 colours with blocks of length 1-2) produces, from a single cell, a sequence whose subword complexity stays bounded (repetitive) or grows at most polynomially slowly (nested), far below the near-exponential subword complexity of rule 30's rows and centre column, so none of them can reproduce rule 30; while a neighbour-dependent substitution system (one output cell per cell, chosen from the cell and its two neighbours) reproduces rules 90 and 30 exactly." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py enumerates all 196 two-colour (blocks 1-3) and 1728 three-colour (blocks 1-2) neighbour-independent substitution rules, iterates each from a single cell of colour 1 until the word has 131072 symbols (or 60 steps), and counts distinct factors of length 1..14 with a sorted rolling-code method. It generates rule 30 from a single black cell with big-integer bit operations, taking the central 40001 cells of row 20000 and the first 65536 centre-column cells, and applies the same count. It also measures what share of the centre column's length-14 factors appear in any binary substitution word, and runs the one-cell-per-cell neighbour-dependent substitution systems for rules 90 and 30 against the direct CA for 1000 steps. Runtime about 19 s." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered. For each of the 196 two-colour rules {0 -> w0, 1 -> w1} with |w0|, |w1| in {1,2,3} and each of the 1728 three-colour rules with block lengths in {1,2}, start from a single cell of colour 1 and iterate until the word has >= L = 131072 symbols (or 60 steps if it never does), truncate to L, and compute the number p(n) of distinct length-n factors for n = 1..14. For rule 30 from a single black cell compute p(14) of (i) the central 40001 cells of the row at step 20000 and (ii) the first 65536 cells of the centre column. Build the neighbour-dependent substitution system whose rule maps each cell to one new cell given by the elementary-CA table of its (left, self, right) neighbourhood, and run it for 1000 steps from a single black cell. PASS (supports) if (a) every neighbour-independent rule has p(14) <= 225 = (14+1)^2 (eventually periodic words are those with p(14) = p(13); the rest are counted as nested), (b) both rule 30 sequences have p(14) >= 2000, and (c) the neighbour-dependent systems for rules 90 and 30 reproduce the cellular-automaton rows exactly at every one of the 1000 steps." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-09--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "the computational objects the claim is about" } ], "http://purl.org/dc/terms/license": [ { "@value": "generated by this test" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: Enumerated neighbour-independent substitution systems (2 colours, blocks 1-3; 3 colours, blocks 1-2), rule 30 evolutions, and neighbour-dependent substitution emulations of rules 90 and 30, generated by analysis.py" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://www.wolframscience.com/nks/p666--emulating-cellular-automata-with-other-systems/" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch11-09", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-09--claim-test" } ] } ] } ]