Hausdorff Reductions and the Exponential Hierarchies

TL;DR

The paper develops a universal Hausdorff-reductions framework to analyze generalized exponential hierarchies, showing that intermediate levels are precisely Hausdorff classes defined by the main level and the length of Hausdorff descriptions. This yields a structural explanation for Hemachandra’s SEH collapse by equating SEH levels with early EH steps, and it provides certificate- and alternating-machine characterizations that align with the Hausdorff perspective. By defining complete problems via Quantified Boolean Second-Order Formulas (QBSFs) and extending hardness results to Datalog$ ext{-} ext{±}$ knowledge bases, the work unifies several Oracle-class relationships and clarifies which kinds of reductions truly differentiate hierarchy levels. The results reveal a deep, uniform pattern across EHs, where the power of a level corresponds to the length of the Hausdorff description and the nature of the oracle used, with SEH occupying the first EH step rather than a standalone entity. The framework thus offers a principled explanation for the SEH collapse and a blueprint for extending hardness and completeness results to higher-order exponential hierarchies, impacting both theory and related areas in logic and knowledge representation.

Abstract

The Strong Exponential Hierarchy $SEH$ was shown to collapse to $P^{NExp}$ by Hemachandra by proving $P^{NExp} = NP^{NExp}$ via a census argument. Nonetheless, Hemachandra also asked for certificate-based and alternating Turing machine characterizations of the $SEH$ levels, in the hope that these might have revealed deeper structural reasons behind the collapse. These open questions have thus far remained unanswered. To close them, by building upon the notion of Hausdorff reductions, we investigate a natural normal form for the intermediate levels of the (generalized) exponential hierarchies, i.e., the single-, the double-Exponential Hierarchy, and so on. Although the two characterizations asked for derive from our Hausdorff characterization, it is nevertheless from the latter that a surprising structural reason behind the collapse of $SEH$ is uncovered as a consequence of a very general result: the intermediate levels of the exponential hierarchies are precisely characterized by specific "Hausdorff classes", which define these levels without resorting to oracle machines. By this, contrarily to oracle classes, which may have different shapes for a same class (e.g., $P^{NP}_{||} = P^{NP[Log]} = LogSpace^{NP}$), hierarchy intermediate levels are univocally identified by Hausdorff classes (under the hypothesis of no hierarchy collapse). In fact, we show that the rather simple reason behind many equivalences of oracle classes is that they just refer to different ways of deciding the languages of a same Hausdorff class, and this happens also for $P^{NExp}$ and $NP^{NExp}$. In addition, via Hausdorff classes, we define complete problems for various intermediate levels of the exponential hierarchies. Through these, we obtain matching lower-bounds for problems known to be in $P^{NExp[Log]}$, but whose hardness was left open due to the lack of known $P^{NExp[Log]}$-complete problems.

Figures (5)

• Figure 4.1: Some of the hierarchies of the Iterated Exponentials Meta-Hierarchy. The two on the left are the Polynomial and the (Weak) Exponential Hierarchy ($\textnormal{EH}\xspace$), respectively. Observe how the Strong Exponential Hierarchy ($\textnormal{SEH}\xspace$) is a portion of the first step of $\textnormal{EH}\xspace$. The other two hierarchies depicted are the ${2}$- Exponential Hierarchy and the ${i}$- Exponential Hierarchy. The complexity class $\mathbf{B}$ is a placeholder for the first main level of the exponential hierarchy considered.
• Figure 5.1: Inclusion relationships between oracle classes. In the table, $i,j,c \geq 0$ and $0 \leq k \leq i+j$. *In this row, $j \geq 1$; for $j = 0$, $\textnormal{${\textnormal{N{}$i$Exp}\xspace}^{{\textnormal{${\textnormal{$j$Exp}\xspace}^{{\textnormal{$\Sigma^\mathrm{P}_{c}$}\xspace}}{}$}\xspace}}{}$}\xspace = \textnormal{${\textnormal{N{}$i$Exp}\xspace}^{{\textnormal{$\Sigma^\mathrm{P}_{c}$}\xspace}}{}$}\xspace$.
• Figure 5.2: Equivalence relationships between oracle classes. In the table, $i,i',j,j',c \geq 0$ and $i+j = \ell = i'+j'$. *In this row, $j,j' \geq 1$.
• Figure 5.3: Example for the "generalized" binary search used in the proof of \ref{['theo_exp_nexp_containment']}, where $r(n) = 3$, and $s(n) = 3$. We have a space of $(3+1)^3 = 64$ points. Symbols '✓' and '✗' mean that the predicates $\mathcal{D}(w,z)$ at those positions are true and false, respectively (we haven't reported the symbols for all the points, only to avoid a too cumbersome drawing); $39$ is the maximum index $z$ at which $\mathcal{D}(w,z) = 1$. The three levels relate, from top to bottom, with the three rounds of parallel queries. The arrows individuate the "samples" taken at each round (notice the additional query in the first round). The greyed-out areas in the bottom two levels are the portions of the sequence that have been recognized as not relevant. In the bottom level, the longer arrow is the sample individuating the maximum index $z$ at which $\mathcal{D}(w,z)$ is true.
• Figure 6.1: Exemplification for the proof of \ref{['theo_complexity_LexMaxFunc']} on how the index $z$ is "positioned" on the input string, represented via the function $\mathit{str}$, starting from the unfolding of the function $f$. In the example, we assume that the representation size of $z$ is $4$, and hence its symbols can be indexed via integers from $0$ to $3$. In the figure we show that $\bar{v} + \bar{v}' = 3$, that the symbol '$\#$' is located at position $| w |$, and that the positions $\bar{u}$ of $z$'s symbols are computed via the relation $\bar{u} = | w | + 1 + \bar{v}'$.

Theorems & Definitions (110)

• Definition 1.0
• Theorem 1.1
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