Between SC and LOGDCFL: families of languages accepted by logarithmic-space deterministic auxiliary depth-k storage automata
Tomoyuki Yamakami · International Journal of Computer Mathematics Computer Systems Theory · 2023
The closure of deterministic context-free languages under logarithmic-space many-one reductions (L-m-reductions), known as LOGDCFL, has been studied in depth from an aspect of parallel computability because it is nicely situated between L and AC1∩SC2. By replacing a memory device of pushdown automata with an access-controlled storage tape, 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 kSDA. Similarly to LOGDCFL, we study the closure LOGkSDA of all languages in kSDA under L-m-reductions. We demonstrate that DCFL⊆kSDA⊆SCk by significantly extending Cook's early result (1979) of DCFL⊆SC2. The entire hierarch of LOGkSDA for all k≥1 therefore lies between LOGDCFL and SC. As an immediate consequence, we obtain the same simulation bounds for Hibbard's limited automata. We further characterize LOGkSDA 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’ LOGkSDA-complete language under L-m-reductions by constructing a two-way universal simulator working for all k-sda's.