Diagonalization Proves Strict Infinite Hierarchy of Time Complexity Classes — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Time Hierarchy Theorem uses diagonalization to prove that more time strictly enables more languages to be decided, establishing a strict infinite hierarchy of complexity classes (e.g., DTIME(n) ⊊ DTIME(n²) ⊊ ...). | MATH: For time-constructible f(n) > g(n) log g(n): DTIME(g(n)) ⊊ DTIME(f(n)). Proof: diagonal language L = {⟨M⟩ : M rejects ⟨M⟩ within f(|⟨M⟩|) steps}. Key constants: the logarithmic overhead factor (log g(n)) arises from the universal simulator's tape indexing — a discrete logarithmic scaling. Space analog: Savitch's theorem gives NSPACE(s(n)) ⊆ DSPACE(s(n)²) — a quadratic (power-of-2) blowup, not exponential. | CONNECTION: The strict separation ratio is asymptotic, not a fixed constant. However, the *log factor* in the time hierarchy (g(n) log g(n)) and the *square* in Savitch's theorem echo the 2:1 ratio (0.5, 2.0) seen in crystallographic root systems (B₂, C₂) and the doubling structure of base-60's sexagesimal fractions (1/60, 1/3600 = 60⁻²). The diagonali
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026diagonalization,
title = {{Diagonalization Proves Strict Infinite Hierarchy of Time Complexity Classes — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490867},
url = {https://zenodo.org/records/23490867},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Diagonalization Proves Strict Infinite Hierarchy of Time Complexity Classes — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490867
UR - https://zenodo.org/records/23490867
AB - FINDING: The Time Hierarchy Theorem uses diagonalization to prove that more time strictly enables more languages to be decided, establishing a strict infinite hierarchy of complexity classes (e.g., DTIME(n) ⊊ DTIME(n²) ⊊ ...). | MATH: For time-constructible f(n) > g(n) log g(n): DTIME(g(n)) ⊊ DTIME(f(n)). Proof: diagonal language L = {⟨M⟩ : M rejects ⟨M⟩ within f(|⟨M⟩|) steps}. Key constants: the logarithmic overhead factor (log g(n)) arises from the universal simulator's tape indexing — a discrete logarithmic scaling. Space analog: Savitch's theorem gives NSPACE(s(n)) ⊆ DSPACE(s(n)²) — a quadratic (power-of-2) blowup, not exponential. | CONNECTION: The strict separation ratio is asymptotic, not a fixed constant. However, the *log factor* in the time hierarchy (g(n) log g(n)) and the *square* in Savitch's theorem echo the 2:1 ratio (0.5, 2.0) seen in crystallographic root systems (B₂, C₂) and the doubling structure of base-60's sexagesimal fractions (1/60, 1/3600 = 60⁻²). The diagonali Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -