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

Abstract

The closure of deterministic context-free (dcf) 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 changing a memory device from pushdown stacks to access-controlled storage tapes, we introduce a computational model of deterministic depth-k storage automata (k-sda’s) whose tape cells are freely modified during the first k accesses and then erased and frozen 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 hierarchy of $\mathrm {LOG}k\mathrm {SDA}$ for all $k\ge 1$ therefore lies between $\mathrm {LOGDCFL}$ and $\mathrm {SC}$. As an immediate consequence, we obtain the same simulation bounds for Hibbard’s limited automata. We 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 machine are also shown to be as powerful as a polynomial-time two-way multi-head deterministic depth-k storage automata.