Functional Characterizations of Uniform Log-depth and Polylog-depth Circuit Families
نویسنده
چکیده
We characterize the classes of functions computable by uniform log-depth (NC) and polylog-depth circuit families as closures of a set of base functions. (The former is equivalent to ALOGTIME, the latter to polylogarithmic space.) The closures involve the \safe" composition of Bellantoni and Cook as well as a safe \divide and conquer" recursion; a simple change to the de nition of the latter distinguishes between log and polylog depth. The proofs proceed, in one direction, by showing that safe composition and divideand-conquer recursion preserve growth rate and circuit depth bounds, and in the other, by simulating alternating Turing machines with divide-and-conquer recursion.
منابع مشابه
The Size and Depth of Boolean Circuits: A Dissertation Proposal
In this thesis, we study the relationship between size and depth for Boolean circuits. Over four decades, very few results were obtained for either special or general Boolean circuits since Spira gave the first related result. Spira showed in 1971 that any Boolean formula of size s can be simulated in depth O(log s). (A Boolean formula is a tree-like circuit, that is the fan-out of every gate i...
متن کاملA Generalization of Spira's Theorem and Circuits with Small Segregators or Separators
Spira [36] showed that any Boolean formula of size s can be simulated in depth O(log s). We generalize Spira’s theorem and show that any Boolean circuit of size s with segregators of size f(s) can be simulated in depth O(f(s) log s). If the segregator size is at least sε for some constant ε > 0, then we can obtain a simulation of depth O(f(s)). This improves and generalizes a simulation of poly...
متن کاملOn Uniformity Within
1 Abstract In order to study circuit complexity classes within NC 1 in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC 1 uniformity Ru81,Co85], have appeared in recent research: Immerman's families of circuits deened by rst-order formulas Im87a,Im87b] and a uniformity corresponding to Buss' deterministic...
متن کاملCircuits with composite moduli
This work studies the power of ACC circuits. These are circuits that have modular gates, in addition to the usual AND,OR,NOT. We are particularly interested in circuits where the modulus is a composite number. Our main result is that every ACC circuit of polynomial size and depth d can be reduced to a depth-2 circuit SYM◦AND of size 2(logn)O(d) . This improves exponentially the previously best-...
متن کاملThe Size and Depth of Layered Boolean Circuits
We consider the relationship between size and depth for layered Boolean circuits and synchronous circuits. We show that every layered Boolean circuit of size s can be simulated by a layered Boolean circuit of depth O( √ s log s). For synchronous circuits of size s, we obtain simulations of depth O( √ s). The best known result so far was by Paterson and Valiant [17], and Dymond and Tompa [6], wh...
متن کامل