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(i) For levels k = 0..7 it extracts every 2-kernel element A(2^k t + r, 2^k x + s) on a 32 x 32 window and counts cumulative distinct elements. (ii) From the 2048 x 2048 sub-pattern it builds the automaton whose states are the distinct 64 x 64 kernel windows (levels k <= 5), with transition (k, r, s) -> (k+1, r + 2^k dt, s + 2^k dx) on digit pair (dt, dx), checks that no state has conflicting or undefined transitions, then runs the automaton on the least-significant-first binary digits of (t, x) for all 16,777,216 cells of the 4096 x 4096 array (12 digit pairs) and compares with A; it also stops the automaton after bitlen(t) digit pairs and compares on the 8,390,656 cells with x <= t. Controls 30, 110 (mirrored), 90 and 150 (right halves) are treated identically. Run time about 30 s." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 0 cells (of 2 x 16,777,216) where the digit automaton disagrees with direct evolution, rules 60 and 188 (n = 33554432, exact). Interpretation: Wolfram's statement (p. 744) is that the colour of a cell of rule 60 or rule 188 after t steps can be found by a cellular automaton acting on the base-2 digit sequences of the cell's coordinates (6 colours for rule 60, 12 for rule 188) in about log2(t) steps instead of t steps. The finite-automaton formulation used here is equivalent: a two-dimensional pattern computable from the binary digits of (t, x) by a finite-state device is a 2-automatic double sequence, whose characterization (Cobham, Salon) is finiteness of its 2-kernel; the automaton's step count is the number of digit pairs, floor(log2 t) + 1. The exact state counts Wolfram quotes (6 and 12) refer to his particular digit-CA encoding and are not tested; only the existence of a finite digit automaton and its logarithmic step count are. Rules 60 and 188 both grow only to the right from a single black cell at x = 0, so A(t, x) is defined for 0 <= x <= t and taken as 0 for x > t. For the controls, the right half (x >= 0) of rule 30, 90 and 150 and the mirror image of rule 110 are used. The clause 'nested patterns generally' is addressed only through rules 90 and 150 as further nested examples." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch12-17: supports, confidence 0.95" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch12-17/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-60-188-single-cell-4096-steps, eca-controls-30-110-90-150-single-cell-4096-steps" } ], "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: for rule 60 or rule 188 the kernel count keeps growing beyond level 5 (no saturation) or exceeds 64, or the automaton is inconsistent, or any cell of the 2^12 x 2^12 pattern disagrees with the automaton's output. Control rules 30 and 110 (not nested) are expected to fail (i) and (ii); if they passed, the test would be too weak and the verdict would be 'inconclusive' with reason 'not-operationalizable'." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "The automaton was built from the 2048 x 2048 pattern and reproduces all 16,777,216 cells of the 4096 x 4096 pattern, 12,582,912 of which were not used in its construction; agreement for all t is implied by 2-automaticity but was verified only up to t = 4095." }, { "@value": "The pre-registered saturation criterion (i) as written ('cumulative count over k <= 5 equals the count over k <= K - 6 with K = 11') is vacuous because K - 6 = 5; it was applied in the intended stronger form on the 4096 x 4096 pattern with a 32 x 32 window (levels up to 7): for rules 60 and 188 no new kernel element appears after level 1 and 2 respectively (2 and 7 distinct elements), while the controls 30 and 110 gain 4^k new elements at every level." }, { "@value": "Wolfram's exact state counts (6 colours for rule 60, 12 for rule 188) belong to his particular digit cellular automaton and were not reproduced; the kernel automata found here have 2 and 7 states. The claim's extension to 'nested patterns generally' was checked only on rules 90 and 150 (4 states each)." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered, for each of rules 60 and 188: (i) the number of distinct 2-kernel elements A(2^k t + r, 2^k x + s), compared on a 64 x 64 window, saturates: the cumulative count over levels k <= 5 equals the cumulative count over all levels k <= K - 6 of a 2^K x 2^K pattern with K = 11 (i.e. no new element appears beyond level 5), and that count is at most 64; (ii) the finite automaton whose states are those kernel elements and whose transitions on a digit pair (dt, dx) go from element (k, r, s) to (k+1, r + 2^k dt, s + 2^k dx) is consistent (every representative of a state yields the same target state for each digit pair) and, run on the least-significant-first binary digits of (t, x), reproduces 100% of the cells of an independently evolved 2^12 x 2^12 pattern (t < 4096), three quarters of which lie outside the K = 11 construction range; (iii) the number of automaton steps for a cell at time t equals the number of binary digits of t, i.e. floor(log2 t) + 1." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered: for rule 60 or rule 188 the kernel count keeps growing beyond level 5 (no saturation) or exceeds 64, or the automaton is inconsistent, or any cell of the 2^12 x 2^12 pattern disagrees with the automaton's output. Control rules 30 and 110 (not nested) are expected to fail (i) and (ii); if they passed, the test would be too weak and the verdict would be 'inconclusive' with reason 'not-operationalizable'." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.95", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-17--claim-test--dataset-1" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-17--claim-test--dataset-2" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "For elementary rules 60 and 188 started from a single black cell, the colour of the cell at position x after t steps is a 2-automatic function of (t, x): a fixed finite automaton reading the base-2 digits of t and x in parallel (one digit pair per step, hence about log2(t) steps) outputs the colour, and this automaton can be constructed from the pattern and reproduces it exactly, including at times not used in its construction." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py evolves each rule from a single black cell to a 4096 x 4096 array A(t, x). (i) For levels k = 0..7 it extracts every 2-kernel element A(2^k t + r, 2^k x + s) on a 32 x 32 window and counts cumulative distinct elements. (ii) From the 2048 x 2048 sub-pattern it builds the automaton whose states are the distinct 64 x 64 kernel windows (levels k <= 5), with transition (k, r, s) -> (k+1, r + 2^k dt, s + 2^k dx) on digit pair (dt, dx), checks that no state has conflicting or undefined transitions, then runs the automaton on the least-significant-first binary digits of (t, x) for all 16,777,216 cells of the 4096 x 4096 array (12 digit pairs) and compares with A; it also stops the automaton after bitlen(t) digit pairs and compares on the 8,390,656 cells with x <= t. Controls 30, 110 (mirrored), 90 and 150 (right halves) are treated identically. Run time about 30 s." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered, for each of rules 60 and 188: (i) the number of distinct 2-kernel elements A(2^k t + r, 2^k x + s), compared on a 64 x 64 window, saturates: the cumulative count over levels k <= 5 equals the cumulative count over all levels k <= K - 6 of a 2^K x 2^K pattern with K = 11 (i.e. no new element appears beyond level 5), and that count is at most 64; (ii) the finite automaton whose states are those kernel elements and whose transitions on a digit pair (dt, dx) go from element (k, r, s) to (k+1, r + 2^k dt, s + 2^k dx) is consistent (every representative of a state yields the same target state for each digit pair) and, run on the least-significant-first binary digits of (t, x), reproduces 100% of the cells of an independently evolved 2^12 x 2^12 pattern (t < 4096), three quarters of which lie outside the K = 11 construction range; (iii) the number of automaton steps for a cell at time t equals the number of binary digits of t, i.e. floor(log2 t) + 1." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-17--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "the two patterns named in the claim" } ], "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: Elementary rules 60 and 188 from a single black cell, 4096 steps, 4096 columns (generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-17--claim-test--dataset-2", "http://purl.org/dc/terms/description": [ { "@value": "negative controls (30, 110) and further nested examples (90, 150); not part of the verdict" } ], "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: Elementary rules 30, 110 (mirrored), 90 and 150 from a single black cell, 4096 steps, right half x >= 0 (generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch12-17", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-17--claim-test" } ] } ] }, { "@id": "https://w3id.org/np/RAR_SDYuDd_vRvMuobtRLfAKjIQfF5aEtdWlfvookQx5Q/Head", "@graph": [ { "@id": "https://w3id.org/np/RAR_SDYuDd_vRvMuobtRLfAKjIQfF5aEtdWlfvookQx5Q", "http://www.nanopub.org/nschema#hasAssertion": [ { "@id": "https://w3id.org/np/RAR_SDYuDd_vRvMuobtRLfAKjIQfF5aEtdWlfvookQx5Q/assertion" } ], "http://www.nanopub.org/nschema#hasProvenance": [ { "@id": "https://w3id.org/np/RAR_SDYuDd_vRvMuobtRLfAKjIQfF5aEtdWlfvookQx5Q/provenance" } ], "http://www.nanopub.org/nschema#hasPublicationInfo": [ { "@id": "https://w3id.org/np/RAR_SDYuDd_vRvMuobtRLfAKjIQfF5aEtdWlfvookQx5Q/pubinfo" } ], "@type": [ "http://www.nanopub.org/nschema#Nanopublication" ] } ] } ]