Constant-depth Circuits Research Articles

Pseudorandom bits for constant depth circuits

For every integerd we explicitly construct a family of functions (pseudo-random bit generators) that convert a polylogarithmic number of truly random bits ton bits that appear random to any family of circuits of polynomial size and depthd. The functions we construct are computable by a uniform family of circuits of polynomial size and constant depth. This allows us to simulate randomized constant depth polynomial size circuits inDSPACE(polylog) and inDTIME(2 polylog ). As a corollary we show that the complexity class AM is equal to the class of languages recognizable in NP with a random oracle. Our technique may be applied in order to get pseudo random generators for other complexity classes as well; a further paper [16] explores these issues.

An oracle separating ΘP from PP PH

We prove the existence of an oracle A such that ΘP A is not contained in PP PH A . This separation follows in a straightforward manner from a circuit complexity result, which is also proved here: To compute the parity of n inputs, any constant depth circuit consisting of a single threshold gate on top of and's and or's requires exponential size in n.

Some Results on Uniform Arithmetic Circuit Complexity

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fields F_(2)n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fields F_(2)n may be identified with an <em>n</em>-input and an n-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME) then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

Non-uniform automata over groups

A new model, non-uniform deterministic finite automata (NUDFA's) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC 1 , NUDFA characterizations of several important subclasses of NC 1 , and a new proof of the old result that the dot-dephth hierarchy is infinite, using M. Sipser's (1983, in “Proceedings, 15th ACM Symposium on the Theory of Computing,” Association for Computing Machinery, New York, pp. 61–69) work on constant depth circuits. Here we extend this theory to NUDFA's over solvable groups (NUDFA's over non-solvable groups have the maximum possible computing power). We characterize the power of NUDFA's over nilpotent groups and prove some optimal lower bounds for NUDFA's over certain groups which are solvable but not nilpotent. Most of these results appeared in preliminary form in ( D. A. Barrington and D. Thérien, 1987 , in “Automata, Languages, and Programming: 14th International Colloquium,” Springer-Verlag, Berlin, pp. 163–173).

The complexity of ranking simple languages

Ranking is the problem of computing for an input string its lexicographic index in a given (fixed) language. This paper concerns the complexity of ranking. We show that ranking languages accepted by 1-way unambiguous auxiliary pushdown automata operating in polynomial time is inNC (2). We also prove negative results about ranking for several classes of simple languages.C is rankable in deterministic polynomial time iffP=P #P , whereC is any of the following six classes of languages: (1) languages accepted by logtime-bounded nondeterministic Turing machines, (2) languages accepted by (uniform) families of unbounded fan-in circuits of constant depth and polynomial size, (3) languages accepted by 2-way deterministic pushdown automata, (4) languages accepted by multihead deterministic finite automata, (5) languages accepted by 1-way nondeterministic logspace-bounded Turing machines, and (6) finitely ambiguous linear context-free languages.

Lower bounds for recognizing small cliques on CRCW PRAM's

We show that any CRCW PRAM which recognizes k-cliques in n-node graphs in time T requires at least n Ω( K T 2) processors independent of its memory size. As a corollary we obtain essentially the same trade-off for unbounded fan-in circuits. We also demonstrate a similar but weaker trade-off for the memory size of CRCW PRAM's solving this problem independent of the number of processors. These bounds also answer an open question posed in [13], i.e., they show that constant-depth circuits for recognizing k-cliques in n-node graphs require size n ⊖(k) .

Lower bounds for constant-depth circuits in the presence of help bits

We consider the problem of how many extra bits of “help” a constant-depth Boolean circuit needs in order to compute m functions of the same input. Each help bit can be an arbitrary Boolean function of the input. We prove an exponential lower bound on the size of the circuit computing m parity functions in the presence of m - 1 help bits. The proof is carried out using the algebraic machinery of Razborov and Smolensky. A by-product of the proof is that the same bound holds for circuits with Mod p gates for a fixed prime p > 2. The lower bound implies a random oracle separation for PH and PSPACE, which is optimal in a technical sense.

Characterization of Associative Operations with Prefix Circuits of Constant Depth and Linear Size

The prefix problem consits of computing all the products $x_{0}x_{1}\cdots x_{j}(j = 0,\cdots, N-1)$, given a sequence ${\bf x} = (x_{0}, x_{1},\cdots , x_{N-1})$ of elements in a semigroup. It is shown that there are unbounded fan-in and fan-out Boolean circuits for the prefix problem with constant depth and linear size if and only if the Cayley graph of the semigroup does not contain a special type of cycle called monoidal cycle.

One-way functions and circuit complexity

A finite function f is a mapping of {0, 1}n into {0, 1}m⌣{#}, where “#” is a symbol to be thought of as “undefined.” A family of finite functions is said to be one-way (in a circuit complexity sense) if it can be computed with polynomial-size circuits, but every family of inverses of these functions cannot. In this paper we show that, provided functions that are not one-to-one are allowed, one-way functions exist if and only if the satisfiability problem SAT does not have polynomial-size circuits. A family of functions fi(x) can be checked if some family of polynomial-size circuits with inputs x and y can determine if fi(x) = y. A family of functions fi(x) can be evaluated if some family of polynomial-size circuits with input x can compute fi(x). Can all families of total functions that can be checked also be evaluated? We show that this is true if and only if the nonuniform versions of the complexity classes P and UP ⋔ co-UP are equal. A family of functions fi is one-way for constant depth circuits if fi can be computed with unbounded famin circuits of polynomial size and constant depth, but every family of inverses fi−1 cannot. We give two provably one-way functions (in fact permutaions) for constant-depth circuits. The second example has the stronger property that no bit of its inverse can be computed in polynomial size and constant depth.