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```									Wild Circuits
Investigating the Limits of MIN/MAX/AVG Circuits Brendan Juba

Definitions: MIN/MAX/AVG Circuits
unsatisfied

 We are given a circuit, C, with feedback, operating on real numbers from the closed interval [0,1].  C contains
MIN, MAX, or AVG gates with two inputs “Inputs” to the circuit that are hard-wired to either 0 or 1.

satisfied

1 0

0

MIN

AVG

 |C| denotes the number of gates of C
Here, |C| = 3

0
MAX

 When the output of a gate is the appropriate function of its inputs, we say that the gate is satisfied

0 satisfied

Definitions: MIN/MAX/AVG Circuits
 Settings of the gate outputs from the interval [0,1] are value vectors
 A value vector for C, v  [0,1]|C|  The ith entry, vi, is the output of the ith gate.
 This is an implicit ordering of the gates of C
1 0

MIN

 We may also consider an update function, F: [0,1]|C|  [0,1]|C|
 A single-gate update function replaces the output of a single designated gate with the correct output value.  We will call iterating over the single gate update functions “gate-by-gate update”

AVG

MAX

Definition: Stable Circuit Problem
 A vector v is stable iff every gate is satisfied. (F(v) = v)  Gate-by-gate update from the vector 0 obtains a stable vector in the limit. This is the minimum stable solution
 We wish to find the minimum stable solution 1 0 1 0

stable
MIN

unstable
0

MIN

0 1/2
AVG

1/2

AVG

MAX

MAX

1/2

0

Definition: STABLE CIRCUIT (Decision Problem)
 We are given a circuit C, and some designated ith gate. In the minimum stable solution of C, s, “is si ≥ 1/2?”  If we can efficiently solve this decision problem, we can efficiently solve the function problem: we can find 2|C| bits of any si, which may be shown to be sufficient.
 Inductively suppose we know the first k-1 bits of si to be v  Modify C:
 (1-1/2k-v) requires k gates
(1-1/2k-v)

ith gate

AVG

 In the minimum stable solution, this new AVG gate’s output is above 1/2 iff the kth bit of si is a 1, so the decision problem tells us the kth bit of si
 Ex: Suppose v = .011010, si = .0110101… (k = 7) then AVG(si,1-1/2k-v) = (.0110101… + .1001011)/2 = .10000000…  If si = .0110100… then AVG(si,1-1/2k-v) = (.0110100… + .1001011)/2 = .01111111…

Previously, on STABLE CIRCUIT
STABLE CIRCUIT is in NPco-NP (Condon, 1992)
We can modify our circuits to have a unique solution that is identical to the minimum stable solution up to the 2|C|th bit This unique solution can be guessed and checked

STABLE CIRCUIT is P-hard
MONOTONE CIRCUIT is a special case

Observations and Motivations
 Our original motivation was to show STABLE CIRCUIT was hard for some class beyond P  If we apply gate-by-gate update to arbitrary starting value vectors, we can obtain “interesting” circuits
 We do not necessarily obtain stable configurations of our circuits -- this is not Stable Circuit

 If we apply gate-by-gate update to the value vector 0, can we still obtain “interesting” circuits?
 If so, the minimum stable solution is the configuration of the device after an unbounded amount of time!

Can we obtain “interesting” circuits starting from 0?

YES

“Leapfrog” circuits
 We assign each wire a “threshold” wire and interpret its value relative to that threshold
 Above threshold: T  Below threshold: F

 It is already clear that we still have AND and OR  There is also a construction for NOT (next slide)
 If there are W wires which we wish to interpret relative to the same threshold, this gadget takes Θ(W) gates

 NB: The circuits are still monotone!
 As we update, a value may seem to rise or fall, as we follow it across different wires through the circuit  The value on any particular wire only rises as the gates of the circuit are updated

th
AVG

x0

x1

x2

MAX AVG MIN MIN MIN

MAX

MAX

th

~x0

x1

x2

th

x0

x1

x2

Caveats
 Assumptions:
1. All values above [below] threshold are equal 2. th has a value distinct from all other inputs 3. We may specify the update order for the gates of the circuit



Take each in turn:
1. Everything starts from zero and the property is preserved by our AND, OR, and NOT gates 2. We can push th above zero by means of an AVG gate
 With feedback, we must also pass the other wires through AVG gates to preserve relative values

3. Update order doesn’t change the solution we approach

Two-bit Counter Circuit
1
x0
NOT NOT
AVG

1

x1

th

MIN

MIN

MAX

1
AVG AVG

1

0
x0 x1 th

Two-bit Counter Circuit
1
x0
NOT NOT
AVG

17/32

x1

th

MIN

MIN

MAX

1
AVG AVG

1

1/2 x0 x1 th

Two-bit Counter Circuit
1
x0
NOT NOT
AVG

x1

781/ 1024

th

MIN

MIN

MAX

1
AVG AVG

1

195/ 256

x0

x1

th

Two-bit Counter Circuit
1
x0
NOT NOT
AVG

7217/ 8192

x1

th

MIN

MIN

MAX

1
AVG AVG

1

28867/ 32768

x0

x1

th

Serving Suggestions
 The counter generalizes to n bits easily
 The n-bit counter takes Θ(n2) gates, due to the size of the NOT gadgets
carry-in

xi

NOT
NOT
MIN MIN MIN

 We now have our counter  We next investigate the power of Leapfrog circuits, using the counter…  First, we will need to make precise what we mean by “Leapfrog circuits”

MAX

carryout

xi

Definition: LEAPFROG
 Let LEAPFROG be the following problem: Given a circuit C and designated gates i and th, consider the sequence of vectors v1, v2, … obtained during gate-by-gate update of C from 0 in the order of the gate indices of C: “Is there an index t such that vti > vtth?”

 LEAPFROG captures our notion of what Leapfrog circuits “compute”

LEAPFROG vs. STABLE CIRCUIT
NB: Not the same problem!!
But, STABLE CIRCUIT obviously reduces to LEAPFROG (include a gate that outputs constant 1/2-1/22|C|…)

Is LEAPFROG hard?
YES -- we will see in a moment

Does LEAPFROG reduce to STABLE CIRCUIT?
If “yes,” then STABLE CIRCUIT is also hard.

LEAPFROG is hard! (NP-hard)
Let any boolean formula be given… Ex: (x1~x2x3)  (~x1~x2x3) Since we have AND, OR, and NOT gates, formulas easily translate into circuits. If we attach xi to the ith bit of the counter, we try all possible assignments, allowing us to reduce SAT to LEAPFROG. The number of gates in these SAT circuits is quadratic in the length of the formula.
MAX

x1 NOT NOT

x2 x3 th, etc.

MAX

MAX

MAX

MIN

(x1~x2x3)(~x1~x2x3)

LEAPFROG is really hard! (PSPACE-hard)
 We can still do better: using the counter, we will decide whether quantified boolean formulas are valid (Reducing TQBF to LEAPFROG)  Assume WLOG that the quantifiers alternate: odd variables are universal, even ones are existential   Leaves in this tree correspond to assignments
 The counter walks along the leaves, left to right



x1

x0

x0

 At the bottom we evaluate the quantifier-free part 00 01 10 of the formula on the specified assignment.  Each  level of the tree has one bit of memory for the left branch
 Set it to T when the branch is T, reset it to F when leaving that subtree.

11

 Pass T up the tree when we see
 T at either branch at an  level  T at the right branch of a  level with the left branch bit already set to T.

 T is passed up from the top of the tree iff we have a TQBF.

Quantifier Circuit: xi (xi-1 A)
xi Carry-out: xi v i0 A

NOT
MIN

• IH: the wire A will be T iff the shorter formula with alternating quantifiers, A, is satisfied by the assignment to xn,…,xi-1 from the counter • vi0 is our bit of memory storing the value of (A|xi = F) (the left branch) under the fixed assignment to xn,…,xi+1 • When there is a carry out of xi, xi+1 has altered, so we reset vi0 to F

NOT

MIN MIN

• If vi0 = (A|xi = F) = T and (A|xi = T) = T (on the right branch), then the wire labeled xixi-1 A is set to T. Otherwise, the wire remains F. • Notice we try both settings of xi-1 for each branch. The wire xi xi-1 A is T iff xi xi-1 A is satisfied by the assignment to xn,…,xi+1, so the Inductive Hypothesis is satisfied

MAX

MIN

xi

xi xi-1 A

v i0

End of the Line: Thwarted by PSPACE
 Recall: finding values in the limit (the minimum stable solution) is known to be in NPco-NP  Answers to PSPACE-hard problems (TQBF) may be encoded on the wires as we update
Since circuits of AND/OR/NOT gates can be evaluated in PSPACE, we would need to drastically alter our model to solve anything harder

 Hence, unless NP = PSPACE, LEAPFROG does not reduce to STABLE CIRCUIT  Thus, in general, Leapfrog circuits (specifically, our counter) cannot be “stopped”

Open problems
How hard is STABLE CIRCUIT?
We had also succeeded in placing the function version in PLS, but still no hardness results Is Stable Circuit PLS-complete? Is STABLE CIRCUIT in P?

How hard is LEAPFROG, actually?
Trivially RE, but this says rather little Is LEAPFROG decidable?

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