Consequences of polylogarithmic membership tools for solving SAT problems
Consequences of Polylogarithmic Membership Comparability for SAT
Computational Complexity
Summary
This paper studies special tools called membership comparators that help decide if certain configurations belong to a specific set, with applications to the SAT problem, a fundamental computational puzzle. The authors show that if these comparators work efficiently and reliably for certain problems, it leads to surprising implications for complexity theory, including how different classes of problems relate and how complex their solutions must be. They also demonstrate limits on improving algorithms for SAT under these assumptions, and explore various conditions that influence the difficulty of verification and problem solving.
What this means in practice
- •For complexity theorists: Guide design of SAT algorithms by clarifying complexity class boundaries and limitations under membership comparator assumptions.
- •For cryptography engineers: Inform cryptographic hardness assumptions through new insights on SAT and class collapses impacting security proofs.
A theory result. No direct application yet.
Authors
Sebastian Ben Daniel
Abstract
We study the consequences of membership comparators that exclude one possible membership vector, deterministically or with a relative advantage over uniform guessing. For every polynomially bounded arity, a randomized polynomial-time comparator of error at most $(1-1/poly(n))2^{-t}$ gives $ NP/ poly\cap coNP/ poly$ recognition with common advice. The proof uses limited independence, polynomial occurrence certificates, and a self-contained positive-relation advice transfer. For SAT at arity $O((\log n)^d)$, both this relative-gap hypothesis and deterministic comparability imply $PH=S^{NP}$, the uniform bound $PH\subseteq BPTIME(2^{O((\log n)^{d^2})})$, and symmetric verification with polynomial-length certificates and an oracle-free deterministic $2^{O((\log n)^d)}$ predicate. Polynomial-advice deterministic decoding has the same exponent $d$. Applying the randomized simulation to an unconditional diagonal language yields, for every fixed $\varepsilon>0$, $\mathrm{BPP}\subsetneq BPTIME(2^{O((\log n)^{d^2+\varepsilon})})$, without advice. The larger clock remains subexponential under every fixed number of self-compositions. A layered oracle satisfies deterministic comparability and $NP^O=coNP^O$ but excludes randomized NP algorithms with smaller logarithmic power, establishing a relativized limit on the SAT exponent $d$. This expanded version also develops the full weak-advantage regime, where saving $2^{-O((log n)^d)}$ gives randomized SAT exponent $d$ and PH exponent $d^k$ at fixed level $k$; the quasipolynomial and exponential hierarchy consequences; binary-comparator advice bounds; and the certificate-length boundary between the randomized regimes. Under deterministic comparability, uniform deterministic promise-unique search additionally gives $UEXP=EXP$. The ordinary second-level collapse $PH=Σ_2^p$ for $d>1$ remains unproved.