Rudimentary relations and Turing machines with linear alternation

Abstract

Smullyan [61] introduced the class R of rudimentary relations as the smallest class which contains the concatenation relation and which is closed under the boolean operations, explicit transformations and linearly bounded quantification. RUD, the class of rudimentary languages, consists of the sequential encodings of rudimentary relations. Wrathall [75] has shown that RUD can be described as the union LH of a linear time analogue of the polynomial time hierarchy of PH of Meyer,Stockmeyer [72].

Chandra,Kozen and Stockmeyer [76] introduced the concept of alternating Turing machines (=ATM). An ATM is a nondeterministic Turing machine with two disjoint sets of states,existential and universal states,which play dual roles in the definition of acceptance. A language L belongs to the alternation class STA(s,t,a) if there exists an ATM M such that for each word w in L there exists a finite accepting computation tree of M for w of depth ≤t(n), alternation depth ≤a(n) and space ≤s(n), where n=|w|.

There is a close connection between quantification and linear alternation. Chandra,Kozen and Stockmeyer noted that PH may be described as the union of a hierarchy of bounded alternation. An analogous result will be shown for RUD = LH:

• (1) RUD = U<STA (−,O(n) , k) :k ε N> ⊑ ATIME (O(n)) = STA(−,O(n),O(n)) Extending a result of Nepomnjascii [70] for NLOGSPACE we are able to prove that the alternating LOGSPACE hierarchy of Chandra, Kozen and Stockmeyer is contained in RUD:

• (2) U<STA (O (n^α), O^β),k):k ε N> ⊑ RUD for α<1≤β, and hence

• (3) U<STA(log(n),−,k):k ∈N>⊑RUD However, the question whether the inclusion RUD⊑ATIME (O(n)) is proper remains open. A negative answer would solve important open problems in complexity theory.