Tight Bounds for the Space Complexity of Nonregular Language Recognition by Real-Time Machines

Abstract

We examine the minimum amount of memory for real-time, as opposed to one-way, computation accepting nonregular languages. We consider deterministic, nondeterministic and alternating machines working within strong, middle and weak space, and processing general or unary inputs. In most cases, we are able to show that the lower bounds for one-way machines remain tight in the real-time case. Memory lower bounds for nonregular acceptance on other devices are also addressed. It is shown that increasing the number of stacks of real-time pushdown automata can result in exponential improvement in the total amount of space usage for nonregular language recognition.

Uncountable realtime probabilistic classes

“…⊓ ⊔ Let L ⊆ Σ * be a language recognized by a one-way DTM, say D, and σ be a symbol not in Σ. We can execute D in realtime reading mode on the inputs defined on Σ ∪ {σ} as follows [9]: For each original "wait" move on a symbol from Σ, the machine expects to read symbol σ. If it reads something else or there is no more input symbol, then the input is rejected.…”

“…In [9], it was shown that realtime deterministic Turing machines (DTMs) can recognize unary nonregular languages in O(log n) space. By adopting the technique given there, we can show that bounded-error realtime PTMs can recognize uncountably many unary languages.…”

“…Proof. We can obtain a realtime algorithm from P ′ , say R c,I or shortly R, by using aforementioned technique borrowed from [9]. Let LOGLOG(I) ′ be the language recognized by R. Then, the language LOGLOG(I) ′ differs from the language LOGLOG(I) with the presence of symbols "3": for each "wait" move on "0", "1", "2" or "4" by P ′ , R expects to read one symbol of "3".…”

New Results on the Minimum Amount of Useful Space

“…It is known that (i) no weak o(log log n) space bounded alternating two-way Turing machine (TM) can recognize a nonregular language and (ii) there exists a unary nonregular language recognized by a deterministic two-way TM in strong O(log log n) space [1,17,30]. For oneway TMs, the bounds are given in Table 1, taken from a recent paper by Yakaryılmaz and Say [36], in which it was shown that all these bounds are tight for almost all realtime TMs. However, for realtime nondeterministic and alternating TMs accepting unary nonregular languages, it was left open whether the double logarithmic lower bounds are tight.…”

“…Another interesting direction is to identify the tight bounds for one-way/realtime multicounter/pushdown automata. Yakaryılmaz and Say [36] showed that realtime deterministic automata with k counters can recognize some nonregular languages in middle O(n 1/k ) space, where k > 1. The same result can be obtained by bounded-error probabilistic one-counter automata, but the error bound increases in k.…”