https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/Head https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://www.nanopub.org/nschema#hasAssertion https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/assertion https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://www.nanopub.org/nschema#hasProvenance https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/provenance https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://www.nanopub.org/nschema#hasPublicationInfo https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/pubinfo https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.nanopub.org/nschema#Nanopublication https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/assertion https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 http://purl.org/dc/terms/description analysis.py evolves all 256 elementary rules from 64 six-bit programs and four input pairs for 16 steps and reads the cell at position -1. It runs one radius-4 multiplexer, whose new center is the program bit indexed by the two input bits, and a Turing machine that walks from the input bits to that same program bit and writes it on the output cell. A function is realized when one program makes all four inputs match it. Criteria were frozen in commit 093ca8002a77b774b469b965ff599d123fdaefbb before this run. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/Assessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 http://www.w3.org/2000/01/rdf-schema#comment Measured: 0 boolean functions of two bits realized by one ceiling system and not the other (n = 16, exact). Interpretation: The claim (NKS p. 675) says that as rules grow more capable they pass a threshold of universality, after which the set of computations they can perform is the same, so a more sophisticated rule and a different kind of system gain nothing further about what they can compute. Universality over all partial recursive functions is not a finite enumeration. The computation uses a finite universe that can actually be saturated: all 16 boolean functions of two bits. The radius-4 rule and the Turing machine are different constructions aimed at that ceiling; the elementary rules, including rule 0, are the progressive family underneath. Capability is the set of functions, not the encoding, so the comparison across kinds is extensional. A computation shows these sets are equal as mathematics, not that organisms compute. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 http://www.w3.org/2000/01/rdf-schema#label Claim test of wolfram2002-ch11-13 (pr-130): supports, confidence 0.84 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 http://www.w3.org/2000/01/rdf-schema#seeAlso https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch11-13/pr-130/result.json https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 http://www.w3.org/ns/prov#wasAttributedTo https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/admissibleDatasets Only configurations evolved by analysis.py from the pre-registered encodings. Elementary rules use Wolfram numbers 0..255. The radius-4 rule is the boolean multiplexer stated in the pass criterion, not a Wolfram code number. No external dataset. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/assessmentType claim-test/0.1.0 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/dataKind computational https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/disconfirmingObservation Rule 0 realizes all 16 functions under the elementary encoding, or some elementary rule realizes more than 16 functions. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/evidenceAgainst The computation shows that these programs realize the same set of two-bit boolean functions once they can realize all 16, and that rule 0 realizes fewer. It does not show that organisms compute, and it does not prove Turing universality. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/evidenceAgainst The largest elementary capability under this encoding is 16 functions, at rule 22. Some elementary rule reaches the same set of 16. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/evidenceAgainst The universe of computations is the 16 functions of two bits. Past that ceiling there is no further function of two bits to gain. A richer universe, such as all partial recursive functions, was not enumerated. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/expectedObservation The radius-4 cellular automaton and the lookup Turing machine each realize all 16 boolean functions of two bits and those two sets are equal, rule 0 realizes at most 4 of them, and no elementary rule realizes more than those 16. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/failCriterion Pre-registered. REFUTES if |Cap(rule 0)| >= 16 (the null rule already performs every computation in U, so this encoding does not measure a threshold) or if any elementary rule has |Cap| > 16. INCONCLUSIVE with reason 'underpowered' if |Cap(radius-4)| < 16 or |Cap(TM)| < 16 (a system built to sit at the ceiling did not reach it, so equality past the ceiling was not tested). MIXED if both ceiling systems equal U and |Cap(rule 0)| <= 4, but some elementary rule has |Cap| = 16 whose set of functions is a proper subset of U (impossible if cardinality is 16, retained so a counting bug cannot be scored as support) or if the two ceiling sets are both size 16 and unequal. Two size-16 subsets of a 16-element set are equal, so a reported inequality at size 16 is a counting error and is MIXED rather than a refutation of the book. Confidence: mixed 0.70, refutes 0.80, inconclusive 0.50. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/hasConfidence 0.84 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/hasEvidence https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-13--claim-test--pr-130--dataset-1 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/hasVerdict supports https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/hypothesis Once a system can realize every boolean function of two bits, a structurally richer system that can also realize every such function realizes exactly the same set, while a system with no information flow realizes a strictly smaller set. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/inferenceMethod Compare the sets of realized truth tables. Supports when both ceiling systems equal the full 16 and rule 0 is at most 4. Inconclusive (underpowered) when a ceiling system misses the full set. Refutes when rule 0 is already at the ceiling or a cardinality exceeds 16. Confidence: supports 0.84, mixed 0.70, refutes 0.80, inconclusive 0.50. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/passCriterion Pre-registered. The universe U is the 16 functions from {0,1}^2 to {0,1}, truth tables ordered by inputs (b0,b1) = (0,0), (0,1), (1,0), (1,1). Three systems. (1) Elementary cellular automata, Wolfram numbers 0..255, fixed-0 boundaries. Encoding: output cell at position -1 starts at 0; input bits at positions 0 and 1; a program of 6 bits at positions 2..7 (64 programs); all other cells 0; evolve t = 1..16 steps on the tape from -17 to 23 inclusive. A function f is realized if some program and some t make cell -1 equal f(b0,b1) for all four inputs. (2) One radius-4 binary cellular automaton, one step. Neighborhood positions relative to the center, from -4 to +4. Input b0 is at +1, b1 at +2. Program bits p00, p01, p10, p11 sit at -4, -3, -2, -1 and the 16 programs are exactly the 16 truth tables in the input order above. The local rule sets the new center to the program bit indexed by b0 + 2*b1, ignoring every other neighbor. Initial center is 0. Read the center after 1 step. (3) A 2-symbol Turing machine on the tape [output=0, b0, b1, p00, p01, p10, p11]. From a head start on b0 it stores b0, moves right, stores b1, moves to p00, steps right b0+2*b1 cells, reads that bit, returns to the output cell, writes it, and halts. The same 16 truth-table programs are used. Let Cap(S) be the set of functions realized. SUPPORTS if Cap(radius-4) = Cap(TM) = U, |Cap(rule 0)| <= 4, every elementary rule has |Cap| <= 16, and every elementary rule with |Cap| = 16 has Cap equal to U. Confidence if supports: 0.84. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/ontology/preregistered false https://w3id.org/delphi/resource/claim/wolfram2002-ch11-13 https://w3id.org/delphi/ontology/hasAssessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-13--claim-test--pr-130--dataset-1 http://purl.org/dc/terms/description the systems whose sets of two-bit boolean functions are compared https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-13--claim-test--pr-130--dataset-1 http://purl.org/dc/terms/license https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-13--claim-test--pr-130--dataset-1 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/EvidenceItem https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-13--claim-test--pr-130--dataset-1 http://www.w3.org/2000/01/rdf-schema#label Dataset: Boolean-function capabilities of elementary rules 0..255, one radius-4 multiplexer, and one lookup Turing machine, generated by analysis.py https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-13--claim-test--pr-130--dataset-1 http://www.w3.org/ns/prov#wasDerivedFrom https://www.wolframscience.com/nks/p675--the-rule-110-cellular-automaton/ https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/provenance https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.w3.org/ns/prov#SoftwareAgent https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 http://www.w3.org/2000/01/rdf-schema#label delphisci claim tester (Grok 4.7; cursor/wolfram-claim-tests-7ca2) https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 http://www.w3.org/ns/prov#actedOnBehalfOf https://orcid.org/0009-0007-4948-9192 https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 https://w3id.org/delphi/ontology/usesModel https://w3id.org/delphi/resource/model/grok-4-7 https://w3id.org/delphi/resource/model/grok-4-7 http://purl.org/dc/terms/identifier Grok 4.7 https://w3id.org/delphi/resource/model/grok-4-7 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/LanguageModel https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/assertion http://www.w3.org/ns/prov#wasAttributedTo https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/assertion http://www.w3.org/ns/prov#wasDerivedFrom https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch11-13/pr-130/result.json https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/assertion http://www.w3.org/ns/prov#wasGeneratedBy https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/test https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/plan http://www.w3.org/ns/prov#hadPlan https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/README.md https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/test http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.w3.org/ns/prov#Activity https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/test http://www.w3.org/ns/prov#qualifiedAssociation https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/plan https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/test http://www.w3.org/ns/prov#wasAssociatedWith https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/pubinfo https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://purl.org/dc/terms/created 2026-11-08T16:24:31.939Z https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://purl.org/dc/terms/creator https://orcid.org/0009-0007-4948-9192 https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://purl.org/dc/terms/license https://creativecommons.org/licenses/by/4.0/ https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://purl.org/nanopub/x/hasNanopubType https://w3id.org/delphi/ontology/Assessment https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://purl.org/nanopub/x/introduces https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-13--claim-test--pr-130 https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A http://www.w3.org/2000/01/rdf-schema#label Claim test of wolfram2002-ch11-13 (pr-130): supports, confidence 0.84 https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A https://w3id.org/delphi/ontology/domain https://w3id.org/delphi/resource/domain/wolfram-nks-2002 https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/sig http://purl.org/nanopub/x/hasAlgorithm RSA https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/sig http://purl.org/nanopub/x/hasPublicKey MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/sig http://purl.org/nanopub/x/hasSignature Gt1GLwQNHBkqw5N8AUZMnouTjl/JKre1djh1MnLJ3kKoKtDI97zMIGjJ84rgnvdumbXaOUwA4gEV3VpyQ1uVRGPdaaLku3GdNTaw+aSyEvZa+SMKTGztE+WvMvhCYyZW3bTY0/47u/MDtl0FLoqsGcUY4boA8A11rNOvxf0Ahqd5FBIsHEHVfM5j/rspiNW/LGoGW6fElVwGcJVqPF4r861bP6zabnSgHmOJko0uSU2IStFYI6cxnxraDrO1VWz/IFCqIzMWzaJfB/7Iq7JgMQwyB8KtwgwIeefqNlV/1am0ZVasR9HOniIfy6kDhYTbtTqfxGuP61KU27TUTI1QCg== https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/sig http://purl.org/nanopub/x/hasSignatureTarget https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A https://w3id.org/np/RAXy0blqw9rPPIrH82x1PKBrkIWFOv5v_JZBbQhoI3Q3A/sig http://purl.org/nanopub/x/signedBy https://orcid.org/0009-0007-4948-9192