Computational Complexity: Hierarchy Theorems, P vs. NP, and NP-Completeness — E8 Intelligence Research

FINDING: Computational complexity theory establishes hard lower bounds on algorithmic resource requirements, with P vs. NP as the central unresolved separation problem | MATH: Time hierarchy: DTIME(f(n)) ⊊ DTIME(g(n)) if f(n)·log f(n) = o(g(n)); Space hierarchy: DSPACE(f(n)) ⊊ DSPACE(g(n)) if f(n) = o(g(n)); P ⊆ NP ⊆ PSPACE ⊆ EXPTIME, with P ⊊ EXPTIME (strict, by diagonalization); NP-complete reductions via ≤_m^p (Karp) and ≤_T^p (Cook); SAT is NP-complete (Cook–Levin, 1971); circuit lower bounds: PARITY ∉ AC⁰ (Furst–Saxe–Sipser, Ajtai); Time-space tradeoffs: NTIME(n) ⊄ TISP(n^{1.2}, n^{0.2}) (Ryan Williams, 2011); #P-completeness of permanent (Valiant); BPP ⊆ P/poly (Adleman); IP = PSPACE (Shamir); PCP theorem: NP = PCP[O(log n), O(1)] | CONNECTION: Complexity classes form a partial order (lattice-like hierarchy) under inclusion; the polynomial hierarchy PH = ∪_k Σ_k^p mirrors a graded algebraic structure analogous to root system filtrations; circuit depth-width tradeoffs exhibit scal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com