Diagonalization Without Relativization A Closer Look at the Baker-Gill-Solovay Theorem

TL;DR

The paper investigates the P vs NP problem through semi-relativization, a weaker form of relativization that uses only an acceptance-problem oracle. By contrasting R and RE with $A_{TM}$ and extending to the polynomial-time analogue $A_{TM-PTIME}$, it shows that diagonalization can distinguish R from RE without full relativization, and that semi-relativization can be extended to the polynomial setting via reductions to $CIRCUIT-SAT$ and $3$-CNF-SAT. It argues that this approach avoids arithmetization and the algebrization barrier, as well as the natural proofs barrier, thereby offering a framework where $P eq NP$ can be demonstrated without relying on traditional relativization techniques. The work also provides a concrete polynomial-time analogue and an oracle construction $T^{A_{TM-PTIME}}$ to illustrate semi-relativized diagonalization in the polynomial realm, contributing to the broader understanding of barriers and potential avenues beyond them.

Abstract

We already know that several problems like the inequivalence of P and EXP as well as the undecidability of the acceptance problem and halting problem relativize. However, relativization is a limited tool which cannot separate other complexity classes. What has not been proven explicitly is whether the Turing-recognizability of the acceptance problem relativizes. We will consider an oracle for which R and RE are equivalent; RA = REA, where A is an oracle for the equivalence problem in the class ALL, but not in RE nor co-RE. We will then differentiate between relativization and what we will call "semi-relativization", i.e., separating classes using only the acceptance problem oracle. We argue the separation of R and RE is a fact that only "semi-relativization" proves. We will then "scale down" to the polynomial analog of R and RE, to evade the Baker-Gill-Solovay barrier using "semi-relativized" diagonalization, noting this subtle distinction between diagonalization and relativization. This "polynomial acceptance problem" is then reducible to CIRCUIT-SAT and 3-CNF-SAT proving that these problems are undecidable in polynomial time yet verifiable in polynomial time. "Semi-relativization" does not employ arithmetization to evade the relativization barrier, and so itself evades the algebrization barrier of Aaronson and Wigderson. Finally, since semi-relativization is a non-constructive technique, the natural proofs barrier of Razborov and Rudich is evaded. Thus the separation of R and RE as well as P and NP both do not relativize but do "semi-relativize", evading all three barriers.

Figures (2)

• Figure 1: Diagonalization for the acceptance problem. Now that we know, from the bottom-right cell, that not all Turing machines will halt, we can ask Turing's question of whether or not there exists a generalized algorithm to decide whether a Turing machine halts or never halts. As Turing proved, there is no such algorithm.
• Figure 2: A Cook-Levin Tableau representing the internal structure of a Turing machine running in polynomial time, with the 7-tuple represented. The "step circuit" has a constant number of logical gates---$AND$, $OR$, $NOT$---and is how we describe the transition function $\delta:Q \times \Gamma \times\{L,R\}$ of the Turing machine as a circuit, moving left $L$ or right $R$ or neither direction on the tape. Note that the top entries of a step circuit can include blanks, and not just 0's and 1's, so a tape can move out to the right and back to the left. Polynomial time and polynomial space with a constant-sized step-circuit gives us a polynomial number of logical gates. $CIRCUIT-SAT$ is not to be confused with with $CIRCUIT-VALUE$ problem which takes the input and in polynomial time accepts or rejects, placing $CIRCUIT-VALUE$ in $\textbf{P}$, which is the polynomial analog of $\textbf{R}$. $A_{TM-PTIME}$ is reducible to $CIRCUIT-SAT$ through the Cook-Levin Tableau, which itself is reducible to $3-CNF-SAT$ through Tseitin transformations.