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.