Between SC and LOGDCFL: Families of Languages Accepted by Logarithmic-Space Deterministic Auxiliary Depth-k Storage Automata

TL;DR

This work extends the landscape of parallel language theory by introducing depth-k storage automata (k-SDA) that replace pushdown stacks with access-controlled, depth-bounded storage tapes, and by analyzing the LOG-closure LOGkSDA under L-m-reductions. It establishes DCFL ⊆ kSDA ⊆ SC^k, placing the entire LOGkSDA hierarchy between LOGDCFL and SC, and provides two machine characterizations (aux-k-SDA and k-SDA_2(ℓ)) that avoid reductions. A generic LOGkSDA-hard language MEMB_k and a depth-immune universal simulator are constructed, linking k-SDA to a universal encoding framework and proving L-m-hardness results. Finally, the paper shows k-SDA_imm ⊆ SC^k, extending Cook’s DCFL ⊆ SC^2 bound to depth-immune models and outlining several avenues for further refinement and natural hard problems within LOGkSDA.

Abstract

The closure of deterministic context-free languages under logarithmic-space many-one reductions ($\mathrm{L}$-m-reductions), known as LOGDCFL, has been studied in depth from an aspect of parallel computability because it is nicely situated between $\mathrm{L}$ and $\mathrm{AC}^{1}\cap\mathrm{SC}^2$. By replacing a memory device from pushdown stacks with access-controlled storage tapes, we introduce a computational model of one-way deterministic depth-$k$ storage automata ($k$-sda's) whose tape cells are freely modified during the first $k$ accesses and then become blank forever. These $k$-sda's naturally induce the language family $k\mathrm{SDA}$. Similarly to $\mathrm{LOGDCFL}$, we study the closure $\mathrm{LOG}k\mathrm{SDA}$ of all languages in $k\mathrm{SDA}$ under $\mathrm{L}$-m-reductions. We demonstrate that $\mathrm{DCFL}\subseteq k\mathrm{SDA}\subseteq \mathrm{SC}^k$ by significantly extending Cook's early result (1979) of $\mathrm{DCFL}\subseteq \mathrm{SC}^2$. The entire hierarch of $\mathrm{LOG}k\mathrm{SDA}$ for all $k\geq1$ therefore lies between $\mathrm{LOGDCFL}$ and $\mathrm{SC}$. As an immediate consequence, we obtain the same simulation bounds for Hibbard's limited automata. We further characterize $\mathrm{LOG}k\mathrm{SDA}$ in terms of a new machine model, called logarithmic-space deterministic auxiliary depth-$k$ storage automata that run in polynomial time. These machines are as powerful as a polynomial-time two-way multi-head deterministic depth-$k$ storage automata. We also provide a ``generic'' $\mathrm{LOG}k\mathrm{SDA}$-complete language under $\mathrm{L}$-m-reductions by constructing a two-way universal simulator working for all $k$-sda's.

Figures (6)

• Figure 1: A simulation of the movement of the storage-tape head and the counter head of $M$. (1) After moving from $\gamma_1$ to $\gamma_2$, an input-tape-head of $M$ changes $\gamma_2$ to $\gamma_{new}$ and moves in direction $d=+1$. (2) The counting head starts at cell $0$, travels through $i$ cells, and returns to cell $0$. (3) The storage-tape head of $N$ moves as depicted in this figure to simulate (1) and (2).
• Figure 2: Blocks on the storage tape and the tape head positions at steps $r-1$, $r$, and $r+1$, where $h_1,h_3,s_1,s_3\in[m_2]$ and $e+2<k$. The tape head moves to the right in (1)--(2) and to the left in (3)--(4) by making a left turn in (3).
• Figure 3: [left] A history of consecutive moves of a storage-tape head for $k=3$. The leftmost three vertical dashed lines indicate $d$-sections for $d=3,4,5$. The other vertical dashed lines indicate the tape cell numbers from $0$ to $10$ and the horizontal dotted lines show section time from $0$ to $32$. All storage-stationary-moves are suppressed into black circles and boxes. [right] A contingency tree, in which each node (except for the root) is a contingency list linked to a marker in the parent node.
• Figure 4: A storage content history. Here, we assume that an underlying $k$-sda makes no storage-stationary-move. Thus, runtime matches section time.
• Figure 5: The content of the principal contingency list at time $t=8,9,10,11$. The notation $C_i$ denotes a marker at (section) time $i$ given by Figure \ref{['fig:storage-hisotry']}. The parameters $a$ and $b$ satisfy $a\in\{l(eft),r(ight)\}$ and $b\in\{last,cur\}$ in $L^{(a)}_{b}(d,l_2,t)$.

Theorems & Definitions (16)

• Lemma 2.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Theorem 4.1
• Lemma 4.2
• Theorem 5.1