text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
satisfied: team success!
• Preconditions of all executed actions met
• No missed deadlines
Concurrent Plan Recognition & Execution for Human-Robot Teams | Steven J. Levine
Slide
40
Concurrent Plan Recognition & Execution for Human-Robot Teams | Steven J. Levine
Slide 41
Compile constraints with ATMS
• Compile const... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
)(cid:3)(cid:3)
(cid:14)(cid:15)(cid:16)(cid:3)(cid:5)(cid:17)(cid:8)(cid:2)(cid:3)(cid:3)
(cid:18)(cid:8)(cid:9)(cid:7)(cid:5)(cid:17)(cid:8)(cid:2)(cid:3)(cid:3)(cid:5)(cid:13)(cid:9)(cid:6)
(cid:18)(cid:8)(cid:9)(cid:7)(cid:5)(cid:17)(cid:8)(cid:2)(cid:3)(cid:3)(cid:5)(cid:13)(cid:9)(cid:6)
Concurrent Plan Recogniti... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
−3
100
Offline Compilation Time
101
102
Worst Online Commit Latency
103
• Randomly-generated TPNU’s with
randomly-generated causal link
structure (probably harder)
• Compilation time roughly
104
proportional to candidate subplans,
(large variance)
• Reactive online performance
101
103
102
Number of Candidate Subplans
10... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
Labeled all-pairs shortest path (APSP)
(cid:6)
[6, 11]
[2, 11]
if x = 1
if x = 2
• Causal link extraction: provides ordering
• In this case, producer guaranteed to precede consumer
• Labeled causal link extracted
Concurrent Plan Recognition & Execution for Human-Robot Teams | Steven J. Levine
Slide
55
Labeled Value ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
each pair of graph, representing
shortest distance as function of environment
• Generalization of Floyd Warshall algorithm that uses LVS
operations instead of standard + and <
• See (Conrad 2010) for details
Concurrent Plan Recognition & Execution for Human-Robot Teams | Steven J. Levine
Slide 58
Optimization: Causal ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
A3 = bagel}
•
Concurrent Plan Recognition & Execution for Human-Robot Teams | Steven J. Levine
Slide 66
Environments and subsumption
• Environment subsumes iff contains all
e2
e2
e1
assignments in
e1
•
ex., subsumes xR1 = j
}
{xR1 = juice
{
uice, xA3 = bagel}
• Intuitively, all subplans represented by also
e2
represe... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
pi
⇒
(cid:3)
(cid:2)
(cid:4)
∧ φec from epi to ec
end
foreach epi
∈ P do
foreach eti ∈ T do
∧ φec
then
∧ φeti
(cid:5)
(cid:5)
(cid:4)
φC
φC ← {aec,p = epi } ∧ φepi
(cid:5)
(cid:5)
if epi ≺ eti
φC
AddConstraint
and eti ≺ ec
(cid:2)
¬φC
and eti (cid:9) ec
φC
Add [(cid:4), ∞] : φC from ec to eti
(cid:5)
(cid:5)
and eti ≺ ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
8)(cid:11)(cid:9)
Concurrent Plan Recognition & Execution for Human-Robot Teams | Steven J. Levine
Slide 73
Labeled causal link extraction
• Uses an LVS, except:
• Values are TPNU events, rather than numbers
• Relation R is not < but rather succession (via labeled APSP):
(cid:6)
e R e =
a
b
if Qdea e (φa φb)
∪
True
→ ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
2 ∧ z = 1)
At least 1 causal link holds
Threat resolved
Concurrent Plan Recognition & Execution for Human-Robot Teams | Steven J. Levine
Slide 79
Causal link extraction in a nutshell*
• For each precondition of each consumer event:
• Find all producer provably before or during consumer
Concurrent Plan Recognition & Ex... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/497c66f8f0334208d69fc5a29ff2a956_MIT16_412JS16_L13.pdf |
Chapter 4 and 5 Wear Mechanisms
1
Delamination Wear Mechanisms
•
Four mechanisms of delamination wear
1. Plastic deformation of the surface
2. Crack nucleation at the sub-surface due to
plastic deformation
3. Crack propagation from these nucleated cracks
due to plastic deformation
4. Creation of loose wear s... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/49841547dcddc87b82263b982a60486e_ch5_wear_mech.pdf |
∂(σ )
xz
∂x
r +
r +
∂ ( σ xz ) r = 0
∂z
∂(σ zz ) r = 0
∂z
f 3 = c 1
f 2 = c 2
(σ ) = f ( z )
r
xx
r
(σ yy ) = νf ( z )
(σ ) = 0
r
(σ ) = 0
r
xz
zz
10
Plastic Deformation of a Semi-Infinite
Elastoplastic Solid
Residual Stress Calculation -- Equilibrium Equations
(σzz )' r
( ) = −
ε
zz r
(
γxz r
) = −
1− 2... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/49841547dcddc87b82263b982a60486e_ch5_wear_mech.pdf |
primary wear-rate controlling
mechanism.
14
Crack Nucleation
Imaginary
sphere
A
B
(a)
(b)
Figure 4.36
Deformed sphere
of the Imaginary
sphere
15
Criteria for Crack Nucleation
Energy Criterion and Strength Criterion
Energy criterion
e
,
n
i
a
r
t
S
Strength criterion
d*
Particle size, d
Figure 4.37... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/49841547dcddc87b82263b982a60486e_ch5_wear_mech.pdf |
=6k
Graph removed for copyright reasons.
See Figure 4.44 in [Suh 1986].
23
Crack Propagation
•
In fracture mechanics, crack propagation is
classified in terms of three modes.
1. The load is applied perpendicular to the crack.
2. The load is applied parallel to the crack direction.
3. The load is applied tran... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/49841547dcddc87b82263b982a60486e_ch5_wear_mech.pdf |
, H-C. “Surface Fraction and
Crack Propagation in Delamination Wear.”
Ph.D. Thesis, MIT, 1981.
31
Shear strain as a function of distance from the
left tip for different depths of crack location
Source: Sin, H-C. “Surface Fraction and
Crack Propagation in Delamination Wear.”
Ph.D. Thesis, MIT, 1981.
32
Void ... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/49841547dcddc87b82263b982a60486e_ch5_wear_mech.pdf |
39
Fretting Wear
Graph removed for copyright reasons.
See Figure 5.10 in [Suh 1986].
40
Source: Sin, H-C. “Surface Fraction and
Crack Propagation in Delamination Wear.”
Ph.D. Thesis, MIT, 1981.
41
Sequence of graphs and photos
removed for copyright reasons.
See Figures 5.12-5.23 in [Suh 1986].
42 | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/49841547dcddc87b82263b982a60486e_ch5_wear_mech.pdf |
Fast Fourier Transform:
Theory and Algorithms
Lecture 8
Vladimir Stojanović
6.973 Communication System Design – Spring 2006
Massachusetts Institute of Technology
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts I... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
Divide and conquer
Divide and conquer always has less computations
Suppose all Il sets have same number
of elements N1 so, N=N1*N2, r=N2
Each inner-m... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
and conquer
Different types balance mapping with subproblem
cost
E.g. in radix-2
subproblems are trivial (only sum and differences)
Mapping requires twiddle factors (large number of
multiplies)
E.g. in prime-factor algorithm
Subproblems are DFTs with coprime lengths (costly)
Mapping trivial (no ar... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
: N multiplications with twiddle factors
Step 3: Evaluate N2 DFTs of length N1
Vector xi mapped to matrix xn1,n2 (N1xN2)
Compute N1 DFTs of length N2 on each row
Point-to-point multiply with twiddle factors
Compute N2 DFTs of length N1 on the columns
6.973 Communication System Design
9
Cite as: Vladim... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
1
x2
x3
x4
x5 x10
x6 x11
x7 x12
x8 x13
x
x
14
9
Not just transpose
Figure by MIT OpenCourseWare.
6.973 Communication System Design
10
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
X7
DFT
N = 4
{ }x2i
DFT
N = 4
{ }
x2i+1
w1
8
w2
8
w3
8
Division
into even and
odd numbered
sequences
DFT of
N / 2
Multiplication
by twiddle
factors
DFT
N = 2
DFT
N = 2
DFT
N = 2
DFT
N = 2
DFT of
2
X0
X4
X1
X5
X2
X6
X3
X7
Which type is this implementation?
Figure by MIT OpenCourseWare.
6.973 Communication System Desig... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
}x2k+1
X0
X2
X4
X6
X1
X3
X5
X7
DFT of
2
Multiplication
by twiddle
factors
DFT of
N / 2
6.973 Communication System Design
13
Figure by MIT OpenCourseWare.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
by MIT OpenCourseW. are.
Which one is DIT (DIF)?
How can we get one from another?
6.973 Communication System Design
14
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded o... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
8
X0
X14
X1
X15
DFT
4
DFT
4
DFT
4
DFT
4
X0
X12
X1
X13
X2
X14
X3
X15
Reduces the number of stages to log4N
Figure by MIT OpenCourseWare.
Radix-8 can reduce number of operations even more
6.973 Communication System Design
1
6
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Sp... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
4
X1
X13
X3
X15
R2
R4
S-R
Figure by MIT OpenCourseWare.
6.973 Communication System Design
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
17
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
Split-radix ... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
N / 4)],
Even samples X2k in DIF should
be computed separately from
other samples
With same algorithm (recursively) as
the original sequence
No general rule for odd samples
Radix-4 is more efficient than radix-2
Higher radices are inefficient
DFT of
2
Multiplication
by twiddle
factors
DFT of
N / 2
X4k... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
conquer requirements
N-long DFT computed from DFTs with lengths that are
factors of N (allows the inner sum to be a DFT)
Provided that subsets Il guarantee periodic xi
When N factors into co-prime factors N=N1*N2
Starting from any xi form subset with compatible periodicity (the
periodicity of the subset d... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
N1 and N2 are coprime
All congruences modulo N1 obtained
For a given congruence modulo N2 and vice versa
Figure by MIT OpenCourseW. are.
6.973 Communication System Design
21
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
3
8
4
14
Reversing N1 and N2
Results in transposed mapping
6.973
Figure by MIT OpenCourseW. are.
22
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
0
n1k1
n2k2
Figure by MIT OCW.
x12
x9
x6
DFT
3
x3
x0
x5
x10
DFT
5
X9
X4
X14
X8
X2
X11
X5
Figure by MIT OpenCourseWare.
True bidimensional transform!
(no extra twiddle factors)
6.973 Communication System Design
23
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT Op... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
4
X8
X2
X11
X5
Efficient small DFTs are a key to the
feasibility of this algorithm
Figure by MIT OpenCourseWare.
6.973 Communication System Design
26
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49a17bd0078f7723100ac1395a476595_lecture_8.pdf |
Fast Fourier Transform:
Practical aspects and Basic Architectures
Lecture 9
Vladimir Stojanović
6.973 Communication System Design – Spring 2006
Massachusetts Institute of Technology
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
1100
68
136
276
632
2524
1572
5804
3548
17660
9492
N
Radix 2
Radix 4
SRFFT
PFA
Winograd
152
408
1032
2504
5896
13566
30728
68616
148
976
5488
28336
148
388
964
2308
5380
12292
27652
61444
30
60
120
240
504
1008
2520
384
888
2076
4812
384
888
2076
5016
13388
14540
29548
34668
84076
99628
16
32
64
128
256
512
1024
2048
1... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
(cid:137) In-place computation
(cid:137) Regularity
(cid:132) Computation
(cid:132) Interconnect
(cid:137) Parallelism and pipelining
(cid:137) Quantization noise
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts In... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
(cid:137) Cooley-Tukey and SRFFT are most compatible
with longer size FFTs
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Desig... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
) Four error sources per butterfly (variance 2-2B/12)
(cid:132) Total variance per butterfly 2-2B/3
(cid:132) Each output node receives signals from a total of N-1 butterflies in the flow graph
(N/2 from the first stage, N/4 in the second, …)
(cid:132) Total variance for each output ~ N/3*2-2B
(cid:132) Assuming input... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
.973 Communication System Design
8
Particular cases
(cid:137) DFT algorithms for real data sequence xk
(cid:132) Xk has Hermitian symmetry (XN-k=Xk
*)
(cid:132) X0 is real, and when N even, XN/2 real as well
(cid:132) N input values map to
(cid:132) 2 real and N/2-1 complex conjugate values when N even
(cid:132) 1 re... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
) Can be obtained by simply pruning the FFT flow graph
(cid:132) Alternately, looks just like a recursive 1-tap filter for each
tone
x(n)
X(k)
z-1
-k
WN
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute o... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
973 Communication System Design
12
Related transforms: DCT
(cid:137) Lots of applications in image and video processing
(cid:137) Scale factor of 1/sqrt(2) for X0 left out
(cid:132) Formula above appears as a sub-problem in length-4N
real DFT
(cid:132) Multiplicative complexity can be related to real DFT
(cid:132) Pr... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
-2
+ (3.2n-2 - 4) multiplications + (2n +3.2n-2 -n) additions
1 real DFT 2n-1 + 2 DCTs 2n-2
+ 3.2n-1 -2 additions
1 real DFT 2n
+ (3.2n-1 -2) multiplications + (3.2n-1 -3) additions
1 odd DFT 2n-1+ 1 complex DFT 2n-1
+2n+1 additions
2 complex DFT's 2n-2
+ 2(3.2n-2-4) multiplications + (2n+3.2n-1 -8) additions
1 DHT 2n
... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
s
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Design
15
Implementation on general purpose computers
(cid:137) FFT algorithms... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Design
17
Vector and multi-processors
(cid:137) Must deal with two interconnected problems
(cid:132) The vector size of the data that can be processed at the ... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
2) Communication cost very hard to estimate
(cid:137) Dedicated arithmetic units
(cid:132) Butterfly unit
(cid:132) CORDIC unit
(cid:137) Still, many heuristics and local tricks to reduce complexity and improve
communication
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
UTTERFLY
DATA OUT
FFT
RAM
Figure by MIT OpenCourseWare.
N
CONTROL
(cid:137) For large FFTs storage of intermediate results is a problem
(cid:132) N-long FFT requires
(cid:132) N/r*logrN, radix-r butterfly operations
(cid:132) 2N*logrN read or write RAM accesses
(cid:132) E.g. to do the 8K FFT in 1ms, need to access int... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
of
2
Multiplication
by twiddle
factors
DFT of
N / 2
Figure by MIT OpenCourseWare.
(cid:132) Produces the output values in bit-reversed order
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
73 Communication System Design
24
FFT network
(cid:137) Inputs are in “bit-shuffled” order (decimated)
(cid:137) Outputs are in “bit-reversed” order
(cid:132) Minimizes the amount of interconnects
X0X4
X1X5
X2X6
X3X7
(cid:137) General scheme for interconnections
(cid:132) Number the cells naturally
(cid:132) 0 to N/2-... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
X2,X3
X4,X5
X6,X7
Figure by MIT OpenCourseWare.
(cid:137) Each multiply-add cell associated with xk and xk+1 (k- even
number between 0 and N-1)
(cid:137) A connection from cell with xk to cell with xj when j=2k mod
N-1 (this mapping is one-to-one)
(cid:132) Represents “circular left shift” of the logN-bit binary repr... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
137) View as time-multiplexed version of the FFT
network
(cid:132) In each step, N/2 nodes take the role of N/2 cells
Figure by MIT OpenCourseWare.
in FFT network
(cid:132) Other half routes the data other nodes
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT Ope... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Design
29
Readings
(cid:137) [1] C.D. Thompson "Fourier Transforms in
VLSI," no. UCB/CSD-82-105, 1982.
(cid... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/49aa5e11da2190ceb1dcadbf5abf7393_lecture_9.pdf |
Lecture 4
van Emde Boas
6.046J
Spring 2015
Lecture 4: Divide and Conquer:
van Emde Boas Trees
• Series of Improved Data Structures
• Insert, Successor
• Delete
• Space
This lecture is based on personal communication with Michael Bender, 2001.
Goal
We want to maintain n elements in the range {0, 1, 2, . .... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/49c8fa24dffce58052c90d46ac800387_MIT6_046JS15_lec04.pdf |
3
0
4
0
5
0
6
0
7
0
8
0
9
1
10
1
11
0
12
0
13
0
14
0
15
1
Figure 1: Bit vector for u = 16. THe current set is {1, 9, 10, 15}.
Split Universe into Clusters
√
We can improve performance by splitting up the range {0, 1, 2, . . . , u − 1} into u
clusters of size u. If x = i u + j, then V[x] = V.Clust... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/49c8fa24dffce58052c90d46ac800387_MIT6_046JS15_lec04.pdf |
)
√
O( u)
√
O( u)
√
Total = O( u)
The three operations in Successor are also Successor calls to vectors of size u. We
can use recursion to speed things up.
• V.cluster[i] is a size- u van Emde Boas structure (∀ 0 ≤ i < u)
√
√
√
• V.summary is a size- u van Emde Boas structure
√
• V.summary[i] indicates... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/49c8fa24dffce58052c90d46ac800387_MIT6_046JS15_lec04.pdf |
each structure. This gives an O(1)
time overhead for each Insert operation.
SUCCESSOR(V, x)
i = high(x)
1
if low(x) < V.cluster[i].max
2
j = Successor(V.cluster[i], low(x))
3
4 else i = Successor(V.summary, high(x))
5
6 return index(i, j)
j = V.cluster[i].min
√
T(u) = T( u) + O(1)
=⇒ T(u) = O(log ... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/49c8fa24dffce58052c90d46ac800387_MIT6_046JS15_lec04.pdf |
I Now we update V.max
11
12
13 else
14
15
if x == V.max
if V.summary.max = None
i = V.summary.max
V.max = index(i, V.cluster[i].max)
I O(1) time
I Unstore new min
I First Call
I Second Call
If the second call is executed, the first call only takes O(1) time. So
√
T(u) = T( u) + O(1)
=⇒ T(u) = O(log log ... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/49c8fa24dffce58052c90d46ac800387_MIT6_046JS15_lec04.pdf |
Details: Split/Merge small structures
· log log u) = O(n) space for large
6
MIT OpenCourseWare
http://ocw.mit.edu
6.046J / 18.410J Design and Analysis of Algorithms
Spring 2015
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/49c8fa24dffce58052c90d46ac800387_MIT6_046JS15_lec04.pdf |
6.897: Selected Topics in Cryptography
Lectures 7 and 8
Lecturer: Ran Canetti
Highlights of past lectures
• Presented a basic framework for analyzing the
security of protocols for multi-party function
evaluation.
• Presented the notion of modular composition.
• Stated and proved the non-concurrent.
composition theo... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
parties.
• Only function evaluation.
• Environment interacts with the computation only at the
beginning and the end.
• Only non-concurrent composition.
Wish-list for a more general framework
• Deal with more “real-life” settings such as:
– Asynchronous communication
– Unreliable and unauthenticated communication ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
value for the security
parameter.
In each activation an ITM may request to write to at most one
tape of another ITM. A request includes:
•
– Identity of the requesting ITM
– Identity of the target ITM and tape, code for the target ITM.
– Contents
If the control function C allows the tuple (source id, target id,... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
output tape of Z.
– All other ITMs can write to the incoming comm. tape of A, can invoke
new ITMs, and can write to the subroutine output tape of their invoker
and the input tapes of their subroutines.
– Modeling corruptions: A can write a “corrupt” message on incoming
comm. Tape of ITM M. Then:
• M writes “Corru... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
• Copy all outputs from F to the subroutine output tape of its
invoker.
The ideal process
The ideal process for evaluating functionality F with environment Z
and adversary S is the following system of interacting ITMs:
Initial ITM:
– Environment Z (the initial ITM, with fixed ID)
•
• Control function:
– Z can ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
The ensemble {IDEALf
S,Z (k,z)} (k in N, z in {0,1}*)
Definition of security:
Protocol P emulates the ideal process for F if
for any adversary A there exists an adversary S
such that for all Z we have:
IDEALF
S,Z ~ EXECP,A,Z .
In this case we say that protocol P securely realizes F.
Note: There is no parameterizati... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
that for all Z we have
IDEALF
S,Z ~ EXECP,Ad,Z .
• Claim: P realizes F w.r.t. the dummy adversary iff it realizes F.
Proof:
•
• Assume P realizes F then w.r.t. dummy adversaries. That is,
If P realizes F then it also realizes F w.r.t the dummy adversary.
Sd,Z ~ EXECP,Ad,Z . Now, let A be an arbitrary adversary.
t... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
• State the UC theorem
• Discuss some implications
• Prove the theorem
Modular composition:
The basic idea for a single copy of f
Q
Q
Q
P
Q
P
Q
Q
Î
Q
P
Q
P
F
The basic idea for multiple calls to F:
Q
Q
Q
Q
Î
Q
P
P
PP
PPP
P
Q
P
P
F
F
F
Q
P
P
P
P
P
P
Q
P
P
P
PPP
PPP
The “hybrid model” fo... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
• M writes “Corrupted” on subr. output tape of Z
• From now on, in each activation M sends its entire state to Z
• A assumes all write privileges of M.
• Corruption messages to copies of F are treated as in the ideal
process (I.e., up to the discretion of F), with the exception that the
“corrupted” outputs are wri... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
ALF
Q,H,Z ~ EXECQp,A,Z .
S,Z ~ EXECQ,H,Z .
The corollary follows.
Implications of the UC theorem
1. Can design and analyze protocols in a modular way:
– Partition a given task T to simpler sub-tasks T1…Tk
– Construct protocols for realizing T1…Tk.
– Construct a protocol for T assuming ideal access to
T1…Tk.
– U... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
exactly like Ad.)
– Messages sent to and from each instance of P are treated
as follows:
•
•
For each instance of P, H keeps a simulated copy of S.
All messages from Z to parties of P are forwarded to the
corresponding instance of S.
• Messages generated by each instance of S are forwarded to Z.
• Messages fro... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
notifies S
that it received input from (id). When receiving “ok” from S, the shell
forwards the input to the main program.
– Whenever the main program wishes to write an output to party (id),
the shell tells S that it wants to give output to (id). When receiving
“ok” from S, the shell forwards the output to (id). ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
Send (sid,pida,pidb) to S
•
• Halt.
Example:
The key-exchange functionality FKE (II)
Wait to receive:
(sid,pida,“exchange”,pidb) from party (sid,pida)
(sid,pidb,“exchange”,pida) from party (sid,pidb)
•
•
Then:
•
If one of the parties is corrupted then obtain a value
a from S. Else, choose a Å{0,1}k
• Outpu... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
,pid)
where pid is in T, do:
– Output r to the party, plus all the messages addressed to it
that were not yet delivered.
– Obtain from the party a list of messages to be delivered in the
next round. Send this list to S.
3. Once all the parties in T have sent their messages for
this round, increment r Å r+1 and r... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/4a4bd69a1e3c2fe6208174e2d38a9146_lecture7_8.pdf |
=
T
η
i
η
t
cos
θ
t
cos
θ
i
L10-3
SOLVE TE BOUNDARY EQUATIONS
We found:
1+ Γ = T
and
Solving yields:
1− Γ = T
ηi cos θt
ηt cos θi
) =
Γ θ(
i
t
η cos θ − η cos θ
t =
η cos θ + η cos θ
t
i
t
i
i
i
η −' 1
n
'
η + 1
n
where η ' (cid:22)
n
t
η cos θ
i
η cos θ
t
i
Normal incidence: θ = 0, ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/4a6f6970935940d8e2b391f9dbadf598_MIT6_013S09_lec10.pdf |
TE solution:
E
→
H
E
H
→ −
ε ↔ μ
Our original
equations :
Become these:
E
∇ × = −μ
H
∂
t
∂
E
∂
t
∂
E 0
H
∇ × = ε
∇ ε =
i
H
E
∂
→ ∇ × = ε
t
∂
H
∂
→ − ∇ × = μ
t
∂
E
→ ∇ μ =
H 0
i
∇ μ =
H 0
i
→
∇ ε =
E 0
i
Are they valid?
Under what circumstances does the solution to the resulti... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/4a6f6970935940d8e2b391f9dbadf598_MIT6_013S09_lec10.pdf |
L10-7
TE AND TM WAVE REFLECTIONS
TE
TM
Γ
-1
0
1
|Γ|2
1
TE
|Γ|2
1
TE
Critical
angle
θ
90o
0
TM
90o
θ
0
TM
θ
90o
Brewster’s angle θB
Brewster’s angle θB
High power
laser beam
Brewster’s angle
laser window
Horizontally
polarized glasses
cut glare
Ocean wave
No reflection
at θB
L10-8
M... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/4a6f6970935940d8e2b391f9dbadf598_MIT6_013S09_lec10.pdf |
Massachusetts Institute of Technology
6.270 Autonomous LEGO Robot Competition
IAP 2005: Attack of the Drones
Workshop 1 — Basic LEGO Structure and Bracing
Monday, January 3, and Tuesday, January 4, 2005
1
Items to Bring
• All of your LEGOs
2 Reading
• Section 6.3 of course notes
3 LEGO Measurements
• A Funda... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/4aa718211a3d9142118d5dd0f138e54b_1_basiclego.pdf |
work if the number of nubs (FLUs) in the beam conform to
Pythagoras’ theorem. Using Pythagoras’ theorem not only leads to stronger bracing, but also saves beams. For
the purposes of LEGO, this means that the only useful combinations are 3-4-5, 5-12-13, and 6-8-10 (Figure 4).
• Other Bracing. Try experimenting with u... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/4aa718211a3d9142118d5dd0f138e54b_1_basiclego.pdf |
a cube at least 10 FLUs on a side that can survive a 6 droptest.
′
For more practice, try constructing the motor jig as shown in Section 4.4. You will also see some creative uses of
bracing when you complete Assignment 1.
6 A few things for the future
• Drop Test Blues Doing the drop test can really scary when it i... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/4aa718211a3d9142118d5dd0f138e54b_1_basiclego.pdf |
Effective Innovation
Don Clausing
ESD 33, MIT
July 2004
Three types of innovations
• Launch
• Growth
• Library
July 2004
MIT
2
Enterprise Processes
INTEGRATION & DIRECTION
July 2004
MIT
3
Product acquisition process
GLOBAL ECONOMY
I
S
E
G
O
L
O
N
H
C
E
T
E
L
B
A
L
I
A
V
A
L
L
A
PRODUCT
PORTFOLIO
ARCHITECTURE
BUSIN... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
t
-
t
i
f
e
n
e
B
γ
β
n
ptimizatio
O
“Renaissance”
Decline
Saturation
Present system
Next-generation system
Leapfrog
α
Infancy
Rapid growth
Maturity
Typical steps of evolution of technological systems can be illustrated by an S-shaped curve that reflects changes of the system's benefit-to cost ratio with time since the... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
is relatively
fixed, then the performance alone can be used. If the performance is relatively fixed, then the reciprocal of cost can be used alone.
July 2004
MIT
8
Evolution of inventive activity
S CURVE
NUMBER
OF INVENTIONS
LEVEL
OF INVENTIONS
PROFITABILITY
OF INVENTIONS
July 2004
MIT
9
TIME
Levels of invention
• 1... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
Law of Increasing Degree of
Ideality.
Degree of Ideality is defined as a ratio of Index of Functionality to Index of Cost, where cost can be expressed in dollars, or units of size or weight, etc. It is,
essentially, the Benefit-to-Cost ratio. A truly ideal system in most cases is a virtual reality, it exists only in... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
approach
REQUIRED
TRADE-OFF
Compromise doesn’t satisfy either requirement.
July 2004
MIT
16
Separation of physical conflicts
• Separation of opposite properties in time
• Separation of opposite properties in space
• Separation of opposite properties between
the whole and its parts
These simple ideas lead to many inve... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
• Apply the first three (above) for a specific
purpose; e.g., mitigate a harmful effect
July 2004
MIT
25
Structural change in Sufield
CONTACT
FORCE
PAPER
ROLL
SHAFT
FIELD
July 2004
MIT
26
Changes to fields
• Change from one type of field to another
• Intensify
• Concentrate in a smaller region
• Vary strength of fie... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
250 MM/SEC
VELOCITY 300 MM/SEC
PAPER
STACK
STACK FORCE
0.7 LB
GUIDE:
ANGLE 45
MOUTH OPENING 7 MM
FRICTION 1.0
RETARD:
RADIUS 25 MM
FRICTION 1.5
Critical parameters guide the detailed design: assure robustness
Fig. 5.18
July 2004
MIT
32
Technology readiness
SELECTION
PRODUCT N
PRODUCT DEVELOPMENT
AND COMMERCIALIZATI... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
of
xerographic copiers
July 2004
MIT
38
5 problems of EI management
• Innovation done ineffectively; EI process
not followed
• EI not well integrated into PA
• EI not well integrated with other enterprise
processes
• Spending on EI is at wrong level
• EI had wrong people
July 2004
MIT
39
Management for success
• R... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
Statistics as a Catalyst p1.pdf
Box_Statistics as a Catalyst p2.pdf
Frey_One Factor at a Time.pdf
DeWeck_Isoperforamnce.pdf
Senin_Wallace_Distributed Modeling.pdf
Hazelrigg_Role and Use of Models.pdf
Taguchi_Clausing_Robust Quality.pdf
Ulrich_Eppinger_Product Design and Dev ch13.pdf
Beck_Extreme Programming.pdf... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
MIT
#1
#2
#3
#4
#5
#1
#2
#3
#4
#6
#5
#6
#7
#8
#7
#8
42
Next Steps
• Due date changed for HW #4
– Due 8:30AM Thurs 8 July
• Reading assignment
– Crevelling_Critical Paramter Management.pdf
– Frey_Error Budgeting.pdf
• See you at the ne... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/4ab81884f461adfcb659b04d7e62349e_s8_efctv_invtn.pdf |
18.417 Introduction to Computational Molecular Biology
Lecture 5: September 23, 2004
Lecturer: Ross Lippert
Scribe: Tony Scelfo
Editor: Athicha Muthitacharoen
Local/Multi Alignments
Introduction
Last Time
• Global Alignment
This Time
• Local Alignment Method of aligning two sequences that share a highly commo... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4acde14888af72996f60ab92a9a430bf_lecture_05.pdf |
(ascii)
• asn - another format that isn’t as popular
Homeobox Genes
Homeobox Genes are good to study when looking at local alignments. The reason
for this is that the region that codes for the Homeobox Gene has been identified in
many species and biologists can look for the Homeobox Gene in new species or test
the ... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4acde14888af72996f60ab92a9a430bf_lecture_05.pdf |
�, j�) in the edit graph.
Figure 5.1: Figure showing Local Alignment region
Problem Statement
Input: S1, S2, �
Output: maxi� �i,j��j GlobalScore(S1[i� · · · i], S2[j� · · · j])
Time: O(n4 · n2) = O(n6)
cell(i,j) = GlobalAlignmentScore(S1 [1 · · · i], S2[1 · · · j])
• reduces local alignment to O(n4).
By using the ... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4acde14888af72996f60ab92a9a430bf_lecture_05.pdf |
3, 2004
5-5
Hi,j = max
⎨
Di,j = max
⎨
Vi,j = max
⎨
Hi,j−1 − λ
D
i,j−1 − � − �
Vi,j , Hi,j
Di−1,j−1 + �(S1(i), S2(j))
Vi−1,j − �
D
i−1,j − � − �
Multiple Alignment
S1, S2, S3, · · · , Sd are the sequences to be aligned.
�(S1, S2, · · ·) d-way score measures the distance for all possible pairwise alignments
a... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4acde14888af72996f60ab92a9a430bf_lecture_05.pdf |
ai, aj )
⎩
5-6
Lecture 5: September 23, 2004
When aligning k sequences, the running time is O(2knd)
Progressive alignment
• e.g. clustal (based on aligning to an alignment)
• does not use ( �
2 ) pairwise alignments, instead aligns first pair and then aligns
next sequence to the ... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4acde14888af72996f60ab92a9a430bf_lecture_05.pdf |
Computational Ocean
Acoustics
• Ray Tracing
• Wavenumber Integration
• Normal Modes
• Parabolic Equation
13.853
COMPUTATIONAL OCEAN ACOUSTICS
Lecture 19
Parabolic Equation
• Mathematical Derivation (6.2)
– Phase Errors and Angular Limitations (6.2.4)
• Starting Fields (6.4)
– Modal starter
– PE Self Starter
– Analyti... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/4afbf5c07b1676c30dbea0bb4ef2dda1_lect_19.pdf |
ICS
Lecture 19
[See Fig 6.8, 6.10-6.12 in Jensen, Kuperman,
Porter and Schmidt. Computational Ocean
Acoustics. New York: Springer-Verlag, 2000.]
Upslope: Energy loss
Downslope: Energy Gain
Accounts for density variation
Energy Conserving
13.853
COMPUTATIONAL OCEAN ACOUSTICS
Lecture 19
Student Demos
Wavenumber Integ... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/4afbf5c07b1676c30dbea0bb4ef2dda1_lect_19.pdf |
18.405J/6.841J:
Advanced Complexity Theory
Spring 2016
Prof. Dana Moshkovitz
Lecture 5: Toda's Theorem
Scribe: Ilya Razenshteyn
Scribe Date: Fall 2012
1 Overview
In the last lecture we covered Valiant–Vazirani reduction and investigated the complexity of ap-
proximate counting.
In this lecture we prove Toda’s Theorem:... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/4b169a1c4e9ce5a4e0e1ba2ace2b2eba_MIT18_405JS16_Todas.pdf |
quantified Boolean
formula ϕ(x) with n variables (x stands for the set of free variables), and a parameter m and
outputs a formula (cid:76)
y ψ(x, y) such that for every x
Pr[ϕ(x) =
(cid:77)
y
ψ(x, y)]
≥ 1 − 2−m.
The running time of the reduction is (nm)Oc(1).
1
To prove this theorem we first need to establish some prop... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/4b169a1c4e9ce5a4e0e1ba2ace2b2eba_MIT18_405JS16_Todas.pdf |
)] ≥ 1
− 2−(m+n+10).
(cid:76)
y τ (x1, x, y) such that for every x1, x
y
Now by union bound we have that with probability at least 1 − 2−(m+10) we have
ψ(x1, x) for every x, x1.
remove the outer quantifier. For this we use Valiant–Vazirani
somewhat “abstract” setting.
In this case we have ϕ(x) = ∃x1
y τ (x1, x, y) =
y τ... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/4b169a1c4e9ce5a4e0e1ba2ace2b2eba_MIT18_405JS16_Todas.pdf |
we see that with
probability at least (1 − 1/8n)N (1) is equal to ϕ(x). So, if we choose N = O(nm) we can make this
probability at least 1 − 2−(m+10). Thus, the total probability of failure is at most 2−(m+9) (cid:28) 2−m.
It is left to observe that we can (and should!) transform (1) to a ⊕SAT instance using tricks fro... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/4b169a1c4e9ce5a4e0e1ba2ace2b2eba_MIT18_405JS16_Todas.pdf |
of possible values of this
formula are disjoint. So, we can compute this sum using an oracle from #P (since T and f are
polynomial-time deterministic reductions), and, thus, decide ϕ.
So, it is left to prove the Lemma. For l = 1 there is nothing to prove: we can take the reduction to
be the identity. By the inductive h... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/4b169a1c4e9ce5a4e0e1ba2ace2b2eba_MIT18_405JS16_Todas.pdf |
Lecture Notes on Wave Optics (04/07/14)
2.71/2.710 Introduction to Optics –Nick Fang
Outline:
A. Imaging with coherent light
B. Optical Spatial Filtering
C. The significance of PSF and ATF, and effect of coherence
D. Phase Contrast Imaging: Zernike and Schlieren methods
A. Imaging with Coherent Light
Recap: a c... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
0 Introduction to Optics –Nick Fang
By cascading two lenses together, we can reveal Abbe’s theory of imaging process:
Ideally, applying two forward Fourier transforms recovers the original function of
the object field, with a reversal in the coordinates:
𝐸𝑖𝑚𝑎𝑔𝑒(𝑥", 𝑦") ≈ ∬ 𝐸(𝑥′, 𝑦′)exp {
} 𝑑𝑥′𝑑𝑦′
... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
aveilluminationobject:decomposed intoHuygens waveletsimageplaneplanewaveilluminationobject:decomposed intospatial frequenciesimageplaneFourier (pupil)planediffraction ordercomes to focus
Lecture Notes on Wave Optics (04/07/14)
2.71/2.710 Introduction to Optics –Nick Fang
𝐸𝑖𝑚𝑎𝑔... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.