Oracle Separations for RPH

TL;DR

The paper investigates the power of computations over real numbers by defining a real Turing machine model that extends the binary Turing machine with a real tape, thereby forming the real polynomial hierarchy $\mathbb{R}$PH, whose first level aligns with $\text{ER}$ and satisfies $\text{NP} \subseteq \text{ER} \subseteq \text{PSPACE}$ and $\text{PH} \subseteq \mathbb{R}$PH$\subseteq \text{PSPACE}$. It develops a technique to translate oracle separations from the binary world to the real world and applies it to derive several oracle separations: $RPH^O \subsetneq PSPACE^O$ for some oracle, $\Sigma_{k+1}^O \nsubseteq \Sigma_k\mathbb{R}^O$ for all $k$, $\Sigma_k\mathbb{R}^O \subsetneq \Sigma_{k+1}\mathbb{R}^O$, and $BQP^O \nsubseteq RPH^O$. The work also provides a constant-depth circuit analogue for $\mathbb{R}$PH, establishing finite-witness reductions that yield circuit upper bounds and enabling the transfer of lower bounds to separations. Overall, the results suggest that real-number computation, while conceptually richer, does not dramatically extend the power of binary computational models, though it does expose meaningful separations between levels and between quantum and real models in oracle settings.

Abstract

While theoretical computer science primarily works with discrete models of computation, like the Turing machine and the wordRAM, there are many scenarios in which introducing real computation models is more adequate. We want to compare real models of computation with discrete models of computation. We do this by means of oracle separation results. We define the notion of a real Turing machine as an extension of the (binary) Turing machine by adding a real tape. Using those machines, we define and study the real polynomial hierarchy RPH. We are interested in RPH as the first level of the hierarchy corresponds to the well-known complexity class ER. It is known that $NP \subseteq ER \subseteq PSPACE$ and furthermore $PH \subseteq RPH \subseteq PSPACE$. We are interested to know if any of those inclusions are tight. In the absence of unconditional separations of complexity classes, we turn to oracle separation. We develop a technique that allows us to transform oracle separation results from the binary world to the real world. As applications, we show there are oracles such that: - $RPH^O$ proper subset of $PSPACE^O$, - $Σ_{k+1}^O$ not contained in $Σ_kR^O$, for all $k\geq 0$, - $Σ_kR^O$ proper subset of $Σ_{k+1}R^O$, for all $k\geq 0$, - $BQP^O$ not contained in $RPH^O$. Our results hint that ER is strictly contained in PSPACE and that there is a separation between the different levels of the real polynomial hierarchy. We also bound the power of real computations by showing that NP-hard problems are unlikely to be solvable using polynomial time on a realRAM. Furthermore, our oracle separations hint that polynomial-time quantum computing cannot be simulated on an efficient real Turing machine.

Figures (3)

• Figure 1: Illustration of a circuit with infinite fan-in corresponding to $\forall u \exists v : M(u,v,x) = 1$.
• Figure 2: Illustration of the circuit construction mentioned in the proof of \ref{['thm:CircuitLowerBoundsImplyNoMembershipInRPH']}. In the first step of the construction, each input node in the old circuit gets replaced by depth-$2$ circuits which have a fan-in of at most $T(n)$ for the gates that are the parents of the input nodes and have a fan-in of at most $2^{T(n)}$ for the gates that are the grandparents of the input nodes; here $T(n)$ is a polynomial. Furthermore, its input nodes are outputs of $M'^{O}$. The second step of the construction is to contract consecutive edges with the same labels. The final depth of the new circuit is $1$ more than the depth of the original circuit.
• Figure 3: A sketch of the circuit a $\textsc{PSPACE}\xspace$-machine uses to solve $\mathbb{R}\textsc{PH}\xspace^O$. The top levels of the circuit represent the different variable assignments. Each level represents a variable, with each variable getting one of $2^{\texttt{poly}(n)}$ possible assignments. Each variable assignment is encoded by a polynomial-sized $\texttt{Logic-}\mathbb{R}\textsc{PH}\xspace$ formula. As there are at most $\texttt{poly}(n)$ variables, the depth is $\texttt{poly}(n)$. At the bottom layers, all variables have been assigned a value. These then handle the different possible oracle queries the machine may attempt to ask. There are at most $2^{\texttt{poly}(n)}$ possible different oracle queries of polynomial length the machine might ask, so the fan-in is at most $2^{\texttt{poly}(n)}$. Furthermore, the depth is $\texttt{poly}(n)$, as the machine may ask at most $\texttt{poly}(n)$ different queries, as it can ask at most one per time step.

Theorems & Definitions (73)

• theorem 1
• theorem 2
• corollary 3
• corollary 4
• theorem 5
• definition 1: Real Oracle Turing machine equivalence
• lemma 6
• proof
• lemma 7
• remark 1