NondeterministicNC1Computation

Abstract

We define the counting classes *NC 1 , GapNC 1 , PNC 1 , and C = NC 1 . We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that *NC 1 *L, that PNC 1 L, and that C = NC 1 L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separations of ACC 0 from MOD-PH and that of TC 0 from the counting hierarchy. Moreover, we obtain that if dlogtimeuniformity and logspace-uniformity for AC 0 coincide then the polynomial time hierarchy equals PSPACE. ]

Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

“…Because SC 1 ⊆ AC 0 (regardless of non-uniformity) [9], this improves Corollary 18 at least when k = 1. Nevertheless, note the usual uniformity condition for AC 0 is not Lbut the more restrictive DLOGTIME-uniformity [28], and there is good evidence that the two classes are distinct [5]. Using methods from [1], Corollary 20 may be rephrased for AC 0 in terms of TIME(poly(log n))-or even TIME((log n) 2 )-uniformity (since DELTA C is computable in quadratic time by a Turing machine), but the DLOGTIME-uniformity case remains unclear.…”

Groups with ALOGTIME-hard word problems and PSPACE-complete compressed word problems

“…A DLOGTIME-reduction is a DLOGTIME-computable many-one reduction. We say that a DLOGTIME-machine strongly computes a function f : Σ * → Γ * with |f (x)| ≤ C log(|x|) for all x ∈ Σ * and for some constant C if it computes the function value by writing it sequentially on a separate output tape (be aware of the subtle difference and that strong DLOGTIME-computability is not a standard terminology, but is coincides with FDLOGTIME in [13]. )…”

“…Note that the output length 2 d(n) is polynomial in n. Restricting the output length to a power of two (instead of an arbitrary polynomial) is convenient for our purpose but in no way crucial. Our definition of a projection is the same as in [13] except for our restriction on the output length. Moreover, in [13] projections were defined for arbitrary alphabets.…”

“…Our definition of a projection is the same as in [13] except for our restriction on the output length. Moreover, in [13] projections were defined for arbitrary alphabets. Let…”

Leaf languages and string compression

“…Among other results, a fixed deterministic context-free language K with PSPACE = LEAF L a (K) was presented. In [8], it was shown that in fact a fixed deterministic one-counter language K as well as a fixed linear deterministic context-free language [15] suffices in order to obtain PSPACE. Here "linear" means that the pushdown automaton makes only one turn.…”