A Clari cation Concerning the L Hierarchy



Eric Allender

Department of Computer Science, Rutgers University

P.O. Box 1179, Piscataway, NJ 08855-117 9,USA

e-mail: [email protected]

Octob er 29, 1997

Abstract

1 0

In [AO96], it is stated without proof  that if NC L = AC L, then the

L hierarchy col lapses to some level. This note provides a proof of that claim.

The reader is referredto[AO96] for al l background, de nitions, and motivation.

Al l of the circuit complexity classes mentioned in this note arelogtime-uniform

[BIS90 ], unless otherwise speci ed.

1 The Pro of

The pro of consists of establishing the following fact.

1

Fact 1.1 The set of languages NC DET has a complete set under logspace

many-one reducibility.

This fact is sucient to establish the claim made in [AO96]. To see this,

1 0

assume that NC DET = AC DET. By Claim 1.1, there is a complete set

:L

:

L

0 0 L

, this complete set is in for AC DET. Since AC DET is equal to L

0

some xed level of this hierarchy, and thus AC DET collapses to this level.

It remains for us to establish Fact 1.1.



Supp orted in part by NSF grant CCR-9509603. 1

First, let's come up with a de nition of a \canonical" way for a log-time

1 2

machine to sp ecify a circuit. Let M b e a clo cked 3-tap e Turing machine

running for c log n time for some c.

Let us say that M is k-good on length n if for all p of length  k log n, M n; p

is a string in the set

fb; ORACLE; 2; AND; 2; OR; 1; NOT; i; INPUTg

where jbj + jpj k log n, and i  n, and such that if jpj = k log n, then M n; p=

i; INPUT  for some i, and for each pre x q of p,ifM n; q = j; INPUT, then

i = j . Intuitively, p is an enco ding of a path from the output gate to a gate

1

g in the NC circuit, and M , on input n; p, is pro ducing as output the fan-in

of the g , and the typ e of gate that g is. The other conditions are merely for

technical convenience. The depth of the circuit is k log n.

Note that { for a suitable enco ding of clo cked Turing machines { the following

0

language is in uniform AC , for each c and k : fM; n : M is a clo cked Turing

machine running in c log n time, and M is k -go o d on length ng. This do es

dep end somewhat on the enco ding of Turing machines. However, note that

M n p i

for any reasonable enco ding of Turings machines M , the language f1 0 1 0 b

0

: the i-th output bit of M n; pisbg is in Dlogtime-uniform AC since it is

in Dlogtime. Checking if a circuit is 2-go o d can b e expressed as a rst-order

0 0

sentence over a uniform AC predicate, and thus it is in Dlogtime-uniform AC .

De ne the circuit C as follows: If M is a clo cked Turing machine

M ;c;k ;n

rnning in time c log n and M is k -go o d on length n, then this is the circuit

with gates having lab els of the form p, where the typ e and fan-in of gate p is

given by M n; p. If gate p has fan-in b, then for all strings q of length jbj that

lexicographically precede b, the gates that are input to p are the gates pq . If

M is not a clo cked Turing machine that is k -go o d on length n, then the circuit

is a circuit that trivially accepts the empty set.

I claim that the set C = fM; x : M is a clo cked Turing machine running in

5 log n time, and M is 2-go o d on length jxj, and the circuit C accepts

M ;c;k ;jxj

3

x where the oracle gates give the middle bit of the function DET applied to

1

their inputs, and jM j log log jxjg is complete for NC L.

1

We need to showthatCisinNC L, and that it is hard. Neither seems

completely trivial.

1

A clo cked TM running in time c log n is a machine that, on input n; p

1. computes c log n which is just c times jnj.

2. starts a counter that will allow it to execute only c log n steps.

Actually, steps 1 and 2 can b e b egun simultaneously; there are a numb er of programming

tricks with Turing machines that one can use. The p oint is, a there is some purely syntactic

part of the Turing machine description that we will call the \clo ck", b this \clo ck" enforces

a run-time on the Turing machine, and c every Turing machine is equivalent to a \clo cked"

Turing machine of comparable complexity.

2 0 1

Note that Dlogtime-uniform AC and NC have circuits that are Dlogtime-uniform even

with this additional restriction that the uniformity machine have three tap es [BIS90].

3

Any bit of the DETERMINANT can b e reduced to the middle bit. If I want the lower-

k

n

order bit of fx, this is the middle-bit of some function GapL function gx de ned as f x2

for some k . 2

1

First, let's show that it's in NC L. Let m b e xed. We'll describ e the

circuit accepting C for inputs of length m. First, given input M; x, where

0

jxj = n, the circuit will evaluate the AC predicate to check that M runs for

5 log n time and is 2-go o d on length jxj;thus let's assume that M is go o d, and

let's concentrate on length n.For each string p, there will b e circuitry evaluating

M n; p. The output of the circuit our circuit accepting C , of course, is the value

of gate  in circuit C on input x. Recall that  the empty string is the

M ;c;k ;n

name of the output gate of C . Here is the circuitry that will evaluate

M ;c;k ;n

0 0

any given gate p. With NC circuitry, with \free" calls to the AC predicates

that are computed only once, given M n; p we can compute the \typ e" of gate

p. If the gate is of typ e INPUT, then a sub circuit of depth log n can compute

the input bit i to which gate p is connected. If the gate is of any other typ e,

then a sub circuit of depth log b can compute the fan-in b of gate p. That is,

near the \top" of this circuit, there will b e gates checking if the fan-in is 2, and

attempting to compute the AND of the inputs of gate p; near the b ottom of

4

the circuit, there will b e gates checking if the fan-in is n , and attempting to

4

compute DETthe values of the input gates pq , etc. The total circuit depth

required to evaluate a gate of fan-in d is O log d+ depth of its inputs. This is

1

all that's required in order to show that this is in NC L.

Now, let's show hardness.

1

For this, we need to show that anything accepted byNC L circuits is

accepted by circuits of the form C for some k -good Turing machine M ,

M ;c;k ;n

for some c and k . If wehave this, then a standard \padding" reduction will

show that our set C is complete. What we need is that a log-time machine,

given a path p, can compute the typ e and fan-in of the gate that is reached by

following path p from the output gate. By \following a path p", I mean that

at each oracle gate of fan-in b, log b bits are used to determine the input wire

from the oracle gate that is followed.

1

Let A b e accepted by logspace-uniform NC L circuits C . Note that there

n

1

is a very uniform NC L family of circuits recognizing the language fn; p; t; i

: g is the gate reached by following path p in C , and either i is 0 and the gate

n

has typ e t,ori  logb and t is the i-th bit of the binary representation of the

fan-in of gate g .g That is, the L oracle can b e used to determine the name

4

Actually sp ecifying the lab eling M uses will b e a bit messy. The circuit has a very regular

structure:

 Use OR gates to guess the typ e of the gate g .

 Use ANDs to

{ check that the guess is correct, and

{ simulate the gate.

The circuits of typ e 1 are all similar. To do part 2, we a Use OR gates to guess

the next bit of the fan-in b of g b use AND gates to check that this bit is correct

Once wehave the entire fan-in, wehave a gate that actually simulates the gate of the

original circuit, and then we rep eat the pro cess for the inputs to gate g.

Lo oking at a path name, it is not to o hard to compute what typ e of gate one is at, and

what the fan-in needs to b e. However, it will b e a bit messy to describ e. 3

of the gate reached by following a given path, and to determine the output of

the uniformity machine for C .Now it is not hard to use oracle gates of this

n

form to build a new circuit family recognizing A, and having the prop erty that

the circuits are sp eci ed bya k -go o d machine M . The details are left to the

interested reader.

This completes the sketch of the pro of.

The language C we build is complete under pro jections which is more re-

strictive than b eing complete under many-one reductions.

The same construction can b e carried out for other functions f in place of

1

DET. However, if f is not at least hard for NC under pro jections, then the

1

language C we build won't necessarily b e hard for NC f . Furthermore, if we

1

want to simulate logspace-uniform NC f , then this construction will require

that f b e hard for logspace.

Acknowledgment: Thanks to Meena Maha jan and V. Vinay for p ointing

out that the claim made in [AO96] requires pro of.

References

[AO96] Allender and M. Ogihara. Relationships among PL, L, and the deter-

minant. RAIRO - Theoretical Information and Application, 30 1996,

1{21.

[BIS90] D. A. Mix Barrington, N. Immerman, and H. Straubing. On unifor-

1

mity within NC . Journal of Computer and System Sciences, 41:274{

306, 1990. 4