CSci 5403 Lecture VII

1
CSci 5403
COMPLEXITY THEORY
LECTURE VII CIRCUITOUS COMPUTING
2
A CIRCUIT
Is a directed acyclic graph, where
x1
x2
Each node has a label 2 x1,,xn, Æ,Ç,
x0

Each node has in-degree either 0 (inputs), 1 (),
or m (Æ , Ç)
Nodes with out-degree 0 are the outputs of the
circuit.
?
A circuit C with n inputs and m outputs computes
a function C 0,1n ! 0,1m.
3
Definition. A circuit family is a sequence of
circuits C1, C2, C3, C4, where for each n, Cn
has n inputs.
We write the family as Cnn 2 N.
The language of a circuit family is the set L,
where x 2 L , Cx(x) 1.
The size of a circuit, C, is the number of
gates.
The size of a circuit family is the function s(n)
Cn
4
Definition. A language is in SIZE(t(n)) if it
is decided by a circuit family of size O(t(n)).
P/poly ?c 2 N SIZE(nc)
Theorem. P µ P/poly.
5
Given TM M and time bound t, we create a
circuit that takes n input bits and runs up to t
steps of M.
The circuit will have t rows, where the ith
row represents the configuration of M after i
steps
a tableau
t
t
6
Rows are made up of cells. Each cell has a
light for every state and every tape symbol.
Each light has a circuit that turns it on or off
based on the previous row.
7
EXAMPLE
1?1, R
1??, R
0?0, R
q1
q0
qa
???, L
0??, R
8
The lights in the first row are connected to the
circuit inputs and the tape head is hardwired in
The circuit should output 1 iff M ends in qaccept.
9
CIRCUIT-SAT (C) C is a satisfiable circuit
CNF-SAT ? ? is a satisfiable CNF formula
Proposition. CIRCUIT-SAT P CNF-SAT.
Proof.
Given a circuit C, we will output a CNF formula ?
that that is satisfiable iff C is.
(x1?x2)
(x1? x2)
((x1?x2) ? (x1? x2))
((x1?x2) ? (x1? x2))
((x1 ? x2) ? (x1? x2)) ? ((x1? x2) ? (x1 ? x2))
10
For every gate in the circuit C, we introduce
a new variable gi and force gi to satisfy the
gate.
g1 ? (x1 ? x2)
g1 (x1 ? x2)
(x1 ? x2) ? g1
g2 ? (x1 ? x2)
g2 (x1 ? x2)
(x1 ? x2) ? g2
g3 ? (g1 ? g2)
g3 (g1 ? g2)
(g1 ? g2) ?g3
g4 ? (g1 ? g2)
g4 (g1 ? g2)
(g1 ? g2) ?g4
g5 ? (g3 ? g4)
g5 (g3 ? g4)
(g3 ? g4) ?g5
g5
11
Theorem. Actually, P ( P/poly.
Proof. P/poly contains undecidable languages,
e.g. UATM 1n n ?M,x? and M accepts x
No one can decide which circuit to use for Cn
Definition. A circuit family C Cnn 2 N is
uniform if there is a TM T that outputs Cn on
input 1n.
C is logspace uniform if T uses space O(log n).
12
Theorem. L 2 P iff L is decided by a logspace
uniform polynomial size circuit family.
Proof.
()) The previous construction is logspace
computable.
(() There is an algorithm to evaluate a
circuit of size S in time O(S).
13
Definition. A circuit family is called
DC-uniform if there is a polytime algorithm that
computes bit i of Cn on input (n,i).
Theorem. L 2 PH iff it is decided by a
DC-uniform circuit family of size 2poly(n) and
depth O(1).
Proof.
()) Let L 2 ?i x 9Y1 8Y2 QiYi .
R(x,Y1Yi) We build a circuit with
depth i4
14
Given x, Y1, Y2,,Yi, we can build a 3CNF
formula ?(x,Y1Yi,Z) to be satisfiable iff
R(x,Y1Yi) accepts.
15
Theorem. L 2 PH iff it is decided by a
DC-uniform circuit family of size 2poly(n) and
depth O(1).
Proof.
(() We build a ?dATM to evaluate a depth-d
circuit
Set gate output. While (gate ? x1,xn) set
type TYPE(gate) if type Æ switch to
universal state. if type Ç switch to
existential state. if type flip current
state. set gate guess 2 INPUTS(gate) Output
value of gate xi.
16
NONUNIFORM ALGORITHMS
A Turing Machine with advice gets an
additional input ?n for each input length n.
M decides L with a(n) advice if there is an
advice sequence ?nn 2 N with ?n a(n) such
that 8 n 8 x2 0,1n M(x,?n) 1 , x 2 L.
Definition. L 2 DTIME(t(n))/a(n) if there is a
time O(t(n)) TM that decides L with advice
O(a(n)).
Example UATM 2 DTIME(n)/1 .
17
Theorem. L 2 P/poly iff there is a TM that
decides L in time O(nc) with advice O(nd)
P/poly c,d DTIME(nc)/nd
Proof.
()) The advice string ?n is the circuit Cn.
(() The circuit Cn is the Cook-Levin circuit for
M with ?n hardwired in.
18
CS5403.info