Linear Model Measurements
Linear
Model Measurements
with
Application
to Bird Flocking
with Application to Bird Flocking
Scott A. Smolka
Scott A. Smolka
Stony Brook University
Joint work with Radu Grosu, Doron Peled, C.R.
Ramakrishnan, Scott Stoller, Junxing Yang
Congratulations Ed!
• I first met Ed at Harvard in 1980,
on occasion of visit by Amir Pnueli
• Closely followed Ed’s work
throughout his illustrious career
• Finally got to work with Ed in 2009 with launch of NSF
CMACS Expedition in Computing
• Ed is mentor, friend, colleague, and inspiration!
Talk Outline
• Flocking Model
• Neighborhood-based Measurements
• Path-based Measurements
• Application to Flocking
Goal: Model-measurement framework provides
fitness values for parameter-optimization framework
Flocking Model
Cucker’s model
n
vai (t) = åf ( xi (t) - x j (t) )(v j (t) - vi (t))
j=1
f (d) =
H
(1+ d 2 )b
Reynold’s model
Velocity Matching
• Measures how well the velocities are aligned:
1
2
1
VM (t) = ( å vi (t) - v j (t) ) 2
n i> j
• LTL property to be “measured”:
FG(VM £ q )
Neighborhood-Based Measurement
• State space is a tuple S, s0 , R
• S is finite set of states
•
•
s0 ÎS is initial state
R Í S ´ S is transition relation
• Components in each state
•
•
•
•
•
Tuple of measurement variables V
Well-founded value set W, <
Expressions E based on V that result in values from W
Constants
Update function f
Measurement Algorithm
With each clock tick, execute in each state
s
If E(s) is not minimal, then do
Send V(s) to all neighbors
Receive V q (s) from neighbors q Î{q1, q2 ,..., qn(s) }
q
Update V = f (V,V q1 ,...,V n ( s ) ) decreasing E(s)
Example Measurements
• Find maximal value in a graph
current maximal
(m, d) = (max(m, m1, m2 ,..., msucc(s) ), d -1)
counter, initialized to width of the structure
• Well-founded domain is the natural numbers with usual <
• Decreasing expression E assigned to each state is simply d
Example Measurements (contd.)
• CTL Model Checking
• Variable vh for each subformula h
• Two counters: phase-counter and down-counter
• For sub-formula e.g. (hEU r ) :
v(hEU r ) = vr Ú (vh Ù
Ú
v(qhEU r ) )
qÎN (s)
f := (VM £ q ).
Measure linear combination of how fast Gf becomes true in
FGf + average value of VM while Gf is true.
• LTL Model Checking: Let
Path Measurements
1. Paths may be infinite.
Assume measurements are affected mainly by a finite
prefix of sequences. Impose a limit on the length.
2. Multiple paths in the structure (possibly infinite).
Use generalized Monte Carlo measurements to
conclude that a large enough number of executions
has guaranteed some measurement threshold.
Generalized Monte-Carlo Measurements
• Obtain joint estimate
mZ = ( m B , m R )
of mean values of Boolean-real pairs
• Additive approximation (AAA algorithm):
• Multiplicative approximation (SRA and OAA algorithms):
Experimental Results
Table 1. Results obtained from OAA
Table 2. Results obtained from AAA
Runs
μR
N
Runs
μR
N
1
0.00644
66846
1
0.00681
84
2
0.00651
65592
2
0.00685
84
3
0.00646
66696
3
0.00589
84
4
0.00648
66173
4
0.00630
84
5
0.00646
66588
5
0.00585
84
Avg
0.00647
66379
Avg
0.00634
84
Std
2.7e-05
505.9
Std
0.00047
0
Figure 1. Birds’ positions and velocities after 50 steps of simulation
V-formation
• Model measurements as fitness in Genetic Algorithm
• To achieve V-formation, measure
clear view + upwash benefit
Figure 2. Birds’ positions and velocities after 500 steps of simulation