1. Circuit complexity of shuffle
Michael Soltys
July 10, 2013
Shuffle - Soltys IWOCA 2013 Title - 1/12

2. Shuffle
w is a shuffle of x and y: Shuffle(x, y, w)
x = 01101110
y = 10101000
w = 0110110011101000
Shuffle - Soltys IWOCA 2013 Definition - 2/12

3. Shuffle
w is a shuffle of x and y: Shuffle(x, y, w)
x = 01101110
y = 10101000
w = 0110110011101000
w is a shuffle of x and y provided:
x = x1x2 · · · xk and y = y1y2 · · · yk
and w obtained by “interleaving”:
w = x1y1x2y2 · · · xkyk.
Shuffle - Soltys IWOCA 2013 Definition - 2/12

4. Motivation
Modelling and verification of concurrent systems
Used in XML database systems for schema definitions
Plan recognition
Natural language processing
See http://bit.ly/17OD2md for more details
Shuffle - Soltys IWOCA 2013 Motivation - 3/12

5. Square Shuffle
w is a square provided it is equal to a shuffle of a x with itself, i.e.,
∃x s.t. Shuffle(x, x, w).
The string w = 0110110011101000 is a square:
w = 0110110011101000
and
x = 01101100 = 01101100
Shuffle - Soltys IWOCA 2013 Square Shuffle - 4/12

6. given an alphabet Σ, |Σ| ≥ 7,
Square = {w : ∃x Shuffle(x, x, w)}
is NP-complete; see http://arxiv.org/abs/1211.7161
What we leave open:
What about |Σ| = 2 (for |Σ| = 1, Square is just the set of
even length strings)
What about if |Σ| = ∞ but each symbol cannot occur more
often than, say, 6 times (if each symbol occurs at most 4
times, Square can be reduced to 2-Sat – see P. Austrin
Stack Exchange post http://bit.ly/WATco3)
Shuffle - Soltys IWOCA 2013 Square Shuffle - 5/12

7. Upper Bound
Shuffle(000, 111, 010101)
Shuffle(011, 011, 001111)
Based on Mansfield’s algorithm, Shuffle ∈ NL
Shuffle - Soltys IWOCA 2013 Upper bound - 6/12

8. Circuits
5.1. BASIC RESULTS AND DEFINITIONS 67
¬
OO
_
OO
^
OO
^
OO
GG
¬
??
¬
__
>> `` OO
66
OO
KK
Figure 1. Using de Morgan and replicating gates.
_
OO
_
OO
?? __GG WWShuffle - Soltys IWOCA 2013 Upper bound - 7/12

9. Suppose that we want a family of circuits that recognize
{1n : n ∈ N} ⊆ {0, 1}∗.
4x x x x x x x x x x1 21 1 12 23 3
I.e., it can be done with a ciruit family {Ci } where |Ci | ≤ 1 (and
depth 1), and hence in AC0
.
Shuffle - Soltys IWOCA 2013 Upper bound - 8/12

10. By results of Sudborough & Venkateswaran:
NL ⊆ SAC1
⊆ AC1
Shuffle - Soltys IWOCA 2013 Upper bound - 9/12

11. Lower Bound
#(x)s be the number of occurrences of a symbol s in the string x.
Shuffle(0#(x)0 , 1#(x)1 , x) is always true.
Parity(x) =
0 ≤ i ≤ |x|
i is odd
Shuffle(0|x|−i
, 1i
, x),
Shuffle - Soltys IWOCA 2013 Lower bound - 10/12

12. n−i
i=1 i=3 i=5 i=n
0 x 1 1 10 0 0x x x1
ii n−i i in−i n−i
Shuffle - Soltys IWOCA 2013 Lower bound - 11/12

13. By the famous result of Furst, Saxe, Sipser, Parity ∈ AC0
.
Therefore Shuffle ∈ AC0
.
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