text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
:17)
(cid:30) (cid:7)
(cid:0)
(cid:1)
which
allows
to
treat
them
as
coordinate
and
k
k
kk
momentum(cid:5)
In
terms
of
the
operators
p(cid:17)
(cid:1)
q(cid:17)
the
hamiltonian
is
represented
as
a
sum
of
k
k
independent
harmonic
oscillators(cid:5)
Indeed(cid:1)
since
a
a
(cid:8)
a
a
(cid:7)
p(cid:17)
p(cid:17)
(cid:8)... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
:7)
k
k
k
(0)
(cid:12)
q(cid:17)
(cid:7)
e
q(cid:17) (cid:2)
p(cid:17)
(cid:7)
e
p(cid:17) (cid:2)
e
(cid:7)
(cid:3)(cid:11)(cid:22)(cid:4)
k
k
(cid:1)
(cid:0)
(cid:1)
k
k
(0)
(cid:12)
(cid:8) (cid:9)
(cid:11)n
k
which
acts
on
the
operators
a
(cid:1)
a
as
k
k
+
a
(cid:7)
cosh
(cid:13)
b
sinh
(cid:13)
b
(cid:2)
a
(cid... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
(cid:7)
(cid:11)n
V
(cid:8)
E
b
b
(cid:3)(cid:11)(cid:24)(cid:4)
k
k
k
k
k
k
k
k
H
(cid:9)
(cid:9)
(cid:0)
(cid:0)
(
(cid:6)
)
k
k
X
(cid:0)
(cid:0)
(cid:1)
k
�0
X
describing
a
gas
of
Bogoliubov
quasiparticles(cid:1)
the
noninteracting
bosons
created
and
an(cid:0)
nihilated
by
the
operators
b
(cid:1)
b
(cid:1)
having
e... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
9)(cid:15)(cid:4)
k
k
k
k
k
+ +
(cid:5)
(cid:7)
(cid:0)
(cid:0)
(cid:0)
k
�0
X
(cid:0)
(cid:1)
(cid:6)
A
(cid:13)
(cid:4)
(cid:4)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
1)
the
dispersion
k
takes
the
form
of
a
usual
free(cid:0)particle
expression
E
(cid:7) (cid:10)
h
(cid:7)(cid:9)m
(cid:8)
(cid:11)n(cid:5)
k
2
2
k
E
(cid:7)
hc
(cid:10)
(cid:2)
c
(cid:7)
(cid:11)n(cid:7)m
(cid:3)(cid:9)(cid:11)(cid:4)
k
k
j
j
q
Remarkably(cid:1)
both
the
collective
modes(cid:1)
sound
waves(cid:1)
and
t... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
:11)n
(cid:3)(cid:11)n(cid:4)
the
same
as
Eq(cid:5)(cid:3)(cid:11)(cid:28)(cid:4)(cid:5)
In
other
words(cid:1)
one
can
con(cid:0)
(0)
2
2
k
(cid:6)
(cid:0)
r
(cid:0)
(cid:1)
sider
condensate
with
(cid:26)uctuating
amplitude
and
phase(cid:1)
and
show
that
these
(cid:26)uctuations
propagate
in
just
the
same
way
as
the
co... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
in
magnitude(cid:1)
which
means
that
the
oscillation
follows
a
small
circle
in
the
complex
(cid:10)
plane(cid:1)
i(cid:5)e(cid:5)
the
phase
and
the
modulus
(cid:0)
of
(cid:10)
participate
in
the
collective
oscillations
roughly
equally(cid:5)
We
can
use
the
above
results
to
estimate
the
e(cid:6)ect
of
condensate
depleti... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
:1)
3(cid:1)2
Estimating
the
sum
as
O(cid:3)(cid:11)
(cid:4)(cid:1)
we
(cid:16)nd
that
the
condensate
depletion
is
a
small
e(cid:6)ect(cid:5)
In
contrast(cid:1)
in
superfuid
He
only
few
percent
of
the
helium
atoms
are
in
the
single(cid:0)particle
4
ground
state(cid:5)
(cid:19)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
Routing
Second main application of Chernoff: analysis of load balancing.
• Already saw balls in bins example
• synchronous message passing
• bidirectional links, one message per step
• queues on links
• permutation routing
• oblivious algorithms only consider self packet.
• Theorem Any deterministic oblivious perm... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
�
– Adversary doesn’t know our routing so cannot plan worst permu
tation
• What if don’t wait for next phase?
– FIFO queuing
– total time is length plus delay
– Expected delay ≤ E[ T (el)] = n/2.
– Chernoff bound? no. dependence of T (ei).
�
• High prob. bound:
– consider paths sharing i’s fixed route (e0, . . . ... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
, then l at ej +1 next time)
�
∗
∗ charge one delay to w.
Summary:
• 2 key roles for chernoff
• sampling
• load balancing
• “high probability” results at log n means.
3
The Probabilistic Method—Value of Random Answers
Idea: to show an object with certain properties exists
• generate a random object
• pro... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
sets T
sets S
–
–
–
Pr[] ≤
�
��
n
s
n
�
cs
(cs/n)ds
≤ (en/s)s(en/cs)cs(cs/n)ds
=
≤
≤
[(s/n)d−c−1 e
[(1/3)d−c−1 e
[(c/3)d(3e)c+1]
c d−c]
s
c d−c]
s
s
c+1
c+1
– Take c = 2, d = 18, get [(2/3)18(3e)3]<2−s
– sum over s, get < 1
Existence proof
• No known construction this good.
• N P -hard to verify
•... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
, value ˆ
w.
• rounding is Poisson vars, mean ˆw.
6
• Pr[≥ (1 + δ) ˆ
w] ≤ e−δ2 ˆ
w/4
• need 2n boundaries, so aim for prob. bound 1/2n2 .
• solve, δ =
�
(4 ln 2n2)/ ˆw.
√
•
So absolute error
8 ˆw ln n
– Good (o(1)-error) if ˆw � 8 ln n
– Bad (O(ln n) error) is ˆw = 2
– General rule: randomized rounding g... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
1-approx for k = 1
LP good for small clauses, random for large.
• Better: try both methods.
• n1, n2 number in both methods
• Show (n1 + n2)/2 ≥ (3/4)
�
zˆj
• n1 ≥
Cj ∈Sk (1 − 2−k )ˆzj
�
• n2 ≥
βk zˆj
�
• n1 + n2 ≥
(1 − 2−k + βk )ˆ
zj ≥
�
3 ˆ
2 zj
�
8 | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.265/15.070J
Lecture 1
Fall 2013
9/4/2013
Metric spaces and topology
Content. Metric spaces and topology. Polish Space. Arzel´a-Ascoli Theo
rem. Convergence of mappings. Skorohod metric and Skorohod space.
1
Metric spaces. Open, closed and compact sets
When we discuss p... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
∞ essentially corresponds to Ix − yI∞.
1≤j≤d
1
Problem 1. Show that Lp is not a metric when 0 < p < 1.
Another important example is S = C[0, T ] – the space of continuous func
tions x : [0, T ] → Rd and ρ(x, y) = ρT = sup0≤t≤T Ix(t) − y(t)I, where I · I
can be taken as any of Lp or L∞. We will usually concentrate ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
of E is a pair of nodes (u, v), u, v ∈ V . For every two nodes u
and v, which are not necessarily connected by an edge, let ρ(u, v) be the length
of a shortest path connecting u with v. Then it is easy to see that ρ is a metric
on the finite set V .
Definition 2. A sequence xn ∈ S is said to converge to a limit x ∈ S... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
T ] of C[0, ∞), namely are these spaces Polish as well? The answer is
yes, but we will get to this later.
Problem 3. Given a set S, consider the metric ρ defined by ρ(x, x) = 0, ρ(x, y) =
1 for x = y. Show that (S, ρ) is a metric space. Suppose S is uncountable. Show
that S is not separable.
Given x ∈ S and r > 0 d... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
A \ Ao . Examples of open sets are open balls Bo(x, r) = {y ∈ S :
ρ(x, y) < r} ⊂ B(x, r) (check this). A set K ⊂ S is defined to be compact
if every sequence xn ∈ K contains a converging subsequence xnk → x and
x ∈ K. It can be shown that K ⊂ Rd is compact if and only if K is closed and
bounded (namely supx∈K IxI < ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
to x ∈ S it is also true that f (xn) converges to f (x).
A mapping f is defined to be continuous if it is continuous in every x ∈ S1.
A mapping is uniformly continuous if for every E > 0 there exists δ > 0 such
that ρ1(x, y) < δ implies ρ2(f (x), f (y)) < E.
Problem 5. Show that f is a continuous mapping if and only... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
�1(y, z) < δ/2.
We just showed that there exists i, 1 ≤ i ≤ k such that ρ1(xi, y) ≤ δ(xi)/2.
By triangle inequality ρ1(xi, z) < δ(xi)/2 + δ/2 ≤ δ(xi). Namely both y
and z belong to Bo(xi, δ(xi)). Then f (y), f (z) ∈ Bo(f (xi), E). By triangle
inequality we have If (y) − f (z)I ≤ If (y) − f (xi)I + If (z) − f (xi)I ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
n If (xn)I = supx∈K If (x)I.
Since K is compact there exists a converging subsequence xnk → x0. Again us
ing continuity of f we have f (xnk ) → f (x0). But If (xnk )I → supx∈K If (x)I.
We conclude f (x0) = supx∈K If (x)I.
We mentioned that the sets in Rd which are compact are exactly bounded closed
sets. What abou... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
x(t)| + |x(t) − x(s)| + |x(s) − y(s)| ≤ |x(t) − x(s)| + 2Ix − yI.
Similarly we show that |x(t) − x(s)| ≤ |y(t) − y(s)| + 2Ix − yI. Therefore for
every δ > 0.
|wx(δ) − wy(δ)| < 2Ix − yI.
We now show (2). Check that (2) is equivalent to
lim sup wx( ) = 0.
n
x∈A
1
n
(3)
(4)
Suppose A is compact but (4) does not... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
. A sequence fn is defined to converge to f uniformly
if
lim sup ρ2(fn(x), f (x)) = 0.
n x∈S1
Also given K ⊂ S1, sequence fn is said to converge to f uniformly on K if the
restriction of fn, f onto K gives a uniform convergence. A sequence fn is said
to converge to f uniformly on compact sets u.o.c if fn converges... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
all n > n0,
sup ρ2(fn(z), f (z)) < E/3. Fix any such n > n0. Since, by assumption fn
is continuous, then there exists δ > 0 such that ρ2(fn(x), fn(y)) < E/3 for all
y ∈ Bo(x, δ). Then for any such y we have
ρ2(f (x), f (y)) ≤ ρ2(f (x), fn(x)) + ρ2(fn(x), fn(y)) + ρ2(fn(y), f (x)) < 3E/3 = E.
This proves continuity ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
think about a process describing the number of customers
in a branch of a bank. This process is described as a piece-wise constant func
tion. We adopt a convention that at a moment when a customer arrives/departs,
the number of customers is identified with the number of customers right af
ter arrival/departure. This... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
1−τ (t − τ − δ),
1−τ −δ
t ∈ [0, τ + δ];
t ∈ [τ + δ, T ].
7
We see that x(λ(t)) = y(t). In other words, we rescaled the axis [0, T ] by
a small amount and made y close to (in fact identical to) x. This motivates the
following definition. From here on we use the following notations: x ∧ y stands
for min(x, y) and x ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
orohod metric. Suppose now ρs(xn, x) → 0. We
need to show Ixn − xI → 0.
Consider any sequence λn ∈ Λ such that Iλn−II → 0 and Ix(λn)−xnI →
0. Such a sequence exists since ρs(xn, x) → 0 (check). We have
Ix − xnI ≤ Ix − xλnI + Ixλn − xnI.
The second summand in the right-hand side converges to zero by the choice of
... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS
by A. Megretski
Lecture 2: Differential Equations As System Models1
Ordinary differential requations (ODE) are the most frequently used tool for modeling
continuous-time ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
all t ⊂ T , and
x˙ (t) = a(x(t), t)
x(t2) − x(t1) =
t2
�
t1
a(x(t), t)dt � t1, t2 ⊂ T.
1Version of September 10, 2003
(2.1)
(2.2)
2
The variable t is usually referred to as the “time”.
Note the use of an integral form in the formal definition (2.2): it assumes that the
function t ∈� a(x(t), t) is integrab... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
(2.3) have the form
x(t) = max{c − t, 0} or x(t) = min{t − c, 0},
where c is an arbitrary real constant. These solutions are not differentiable at the critical
“stopping moment” t = c.
2.1.2 Standard ODE system models
Ordinary differential equations can be used in many ways for modeling of dynamical
systems. The no... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
mentioned before, not all ODE models are adequate for design and analysis
purposes. The notion of well-posedness introduces some typical constraints aimed at
insuring their applicability.
Definition A standard ODE model ODE(f, g) is called well posed if for every signal
v(t) ⊂ V and for every solution x1 : [0, t1] ∈... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
to dry friction and external force input v), the model is not well-posed.
To prove this, consider the input v(t) = 0.5 = const. It is sufficient to show that no
solution of the ODE
x˙ (t) = 0.5 − sgn(x(t))
satisfying x(0) = 0 exists on a time interval [0, tf ] for tf > 0. Indeed, let x = x(t) be such
solution. As an... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
solutions for continuous ODE
This section contains fundamental results establishing existence of solutions of differential
equations with a continuous right side.
2.2.1 Local existence of solutions for continuous ODE
In this subsection we study solutions x :
[t0, tf ] ∈� Rn of the standard ODE
(same as (2.1)), subj... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
.3 The ODE
x˙ (t) = c0 + c1 cos(t) + x(t)2 ,
where c0, c1 are given constants, belongs to the class of Riccati equations, which play a
prominent role in the linear system theory. According to Theorem 2.1, for any initial
condition x(0) = x0 there exists a solution of the Riccati equation, defined on some time
inter... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
which T is a proper subset of T , and x(t) = x(t) for all t ⊂ T . In
¯
particular, well-posedness of standard ODE system models contains the requirement that
all maximal solutions must be defined on the whole time-line t ⊂ [0, →).
¯
¯
The following theorem gives a useful characterization of maximal solutions.
Theore... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
lim |x(t)| = →.
t�tf
In Example 2.2.1 with c0 = 1, c1 = 0, one maximal ODE solution is x(t) = tan(t),
defined for t ⊂ (−�/2, �/2). It cannot be extended on either side because |x(t)| � → as
t � �/2 or t � −�/2.
2.2.3 Discontinuous dependence on time
The ODE describing systems dynamics are frequently discontinuous ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
t ⊂ [t0, t0 + r]}
is a subset of Z;
(b) the function t ∈� a(x(t), t) is integrable on [t0, t0 + r] for every continuous function
x :
[t0, t0 + r] ∈� Rn satisfying |x(t) − x0| ∀ r for all t ⊂ [t0, t0 + r];
(c) for every π > 0 there exists � > 0 such that
t0+r
�
t0
|a(x1(t), t) − a(x2(t), t)|dt < π
whenever x1, x... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
∀
t1
�
0
t−1/3dt max |x1(t) − x2(t)|
t�[0,t1]
holds.
On the contrary, the differential equation
x˙ (t) =
⎩
t−1x(t), t > 0
t = 0,
0,
x(0) = x0
does not have a solution on [0, →) for every x0 ∞= 0. Indeed, if x : [0, t1] ∈� R is a solution
for some t1 > 0 then
d
dt
⎧
x(t)
t
⎨
= 0
for all t ∞= 0. Hence x(t)... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
fining discontinuous ODE to guarantee existence of solutions.
It turns out that differential inclusion (2.9) subject to fixed initial condition x(t0) = x0
has a solution on a sufficiently small interval T = [t0, t1] whenever the set-valued function
� is compact convex set-valued and semicontinuous with respect to its arg... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
) = x0. Moreover, any
The discontinuous differential equation
x˙ (t) = −sgn(x(t)) + c,
8
where c is a fixed constant, can be re-defined as a continuous differential inclusion (2.9)
by introducing
�(y) =
{c − 1},
y > 0,
[c − 1, c + 1], y = 0,
y < 0.
{c + 1},
�
�
�
The newly obtained differential inclusion has t... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
6.825 Techniques in Artificial Intelligence
Logic
Lecture 3 • 1
Today we're going to start talking about logic. Now, my guess is that almost
everybody's been exposed to basic propositional logic in the context of
machine architecture or something like that. But, it turns out that that
exposure to logic was just a... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
6.825 Techniques in Artificial Intelligence
Logic
• When we have too many states, we want a
convenient way of dealing with sets of states.
• The sentence “It’s raining” stands for all the states
of the world in which it is raining.
Lecture 3 • 3
What if I say "It's raining."? One way to think about what it means... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
5
What is a logic?
• A formal language
Lecture 3 • 6
So, what is a logic? Well, a logic is a formal language. And what does that
mean? It has a syntax and a semantics, and a way of manipulating
expressions in the language. We’ll talk about each of these.
6
What is a logic?
• A formal language
• Syntax – what ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
So, why do we want to do proofs? There are lots of situations.
Lecture 3 • 10
10
What is a logic?
• A formal language
• Syntax – what expressions are legal
• Semantics – what legal expressions mean
• Proof system – a way of manipulating syntactic
expressions to get other syntactic expressions
(which will tell ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
things together and infer something about the next state
of the world. So these are two kinds of inferences that an agent might want
to do. We could come up with a lot of other ones, but those are two good
examples to keep in mind.
12
Propositional Logic Syntax
Lecture 3 • 13
In the book they start by talking ab... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
really, but it's syntactically well-formed. It's got the nouns,
the verbs, and the adjectives in the right place. If you scrambled the words
up, you wouldn’t get a sentence, right? You’d just get a string of words that
didn’t obey the rules of syntax. So, "furiously ideas green sleep colorless" is
not OK.
15
Prop... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
that a computer is going to
read. And so we don't have to be absolutely rigorous about what characters
are allowed in the name of a variable. But there are going to be things called
variables, and we'll just use uppercase letters for them. Those are
sentences. It's OK to say "P" -- that's well-formed.
18
Proposit... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
: well formed formulas)
• true and false are sentences
• Propositional variables are sentences: P,Q,R,Z
• If φ and ψ are sentences, then so are
(φ),
φ, φ Æψ, φ Çψ, φ
ψ, φ
→
• Nothing else is a sentence
¬
↔
ψ
Lecture 3 • 20
And there's one more part of the definition, which says nothing else is a
sentence. OK. ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
, which is negation. For every negation, you’d add an
open paren in front of the negation sign and a close parenthesis after the
next whole expression. This is exactly how minus behaves in arithmetic.
The next highest operator is wedge, which behaves like multiplication in
arithmetic.
Next is vee, which behaves li... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
So, in our study of logic, we’re not going to assign particular
values or meanings to the variables; rather, we’re going to study the general
properties of symbols and their potential meanings.
23
Semantics
• Meaning of a sentence is truth value {t, f}
Lecture 3 • 24
Ultimately, the meaning of every sentence, in... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
two
horizontal bars coming out to the right) to mean “sentence \Phi is true in
interpretation I”. The turnstile symbol is not part of our language. It's part of
the way logicians write things on the board when they're talking about what
they're doing.
This is a really important distinction. If you can think of our... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
Phi is true in interpretation I.
I'm going to write the semantics down in a way that's parallel to the way we
specified the syntax.
27
Semantics
• Meaning of a sentence is truth value {t, f}
• Interpretation is an assignment of truth values to
the propositional variables
[
• ² i φ Sentence φ is t in interpretat... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
i [the sentence false has truth value f in all
if and only if 2i φ
Now, let’s think about the negation sign. When is \negation \Phi true in an
interpretation I? Whenever \Phi is false in that interpretation.
Lecture 3 • 30
30
Semantics
• Meaning of a sentence is truth value {t, f}
• Interpretation is an assignm... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
if and only if 2i φ
if and only if ² i φ and ² i ψ [conjunction]
if and only if ² i φ or ² i ψ [disjunction]
Lecture 3 • 32
When is Phi vee Psi true in an interpretation I? Whenever either Phi or Psi
is true in I. This is called “disjuction”, and we’ll call the vee symbol “or”. It is
not an exclusive or; so that ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
33
Some important shorthand
Lecture 3 • 34
It seems like we left out the arrows in the semantic definitions of the previous
slide. But the arrows are not strictly necessary; that is, it’s going to turn out
that you can say anything you want to without them, but they’re a convenient
shorthand. (In fact, you can al... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
→
¬
antecedent
≡
• φ
ψ
↔
≡
→
(φ
→
consequent
ψ) Æ (ψ
φ) [biconditional, equivalence]
→
Truth Tables
P
P Æ Q
P Ç Q
P
f
f
t
t
Q
f
t
f
t
¬
t
t
f
f
f
f
f
t
f
t
t
t
Q
Q
P
→
t
t
f
t
P
→
t
f
t
t
P
Q
↔
t
f
f
t
Lecture 3 • 37
Just so you can see how all of these operators work, here are the t... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
t
t
f
f
f
f
f
t
f
t
t
t
Q
Q
P
→
t
t
f
t
P
→
t
f
t
t
P
Q
↔
t
f
f
t
Q is t
Note that implication is not “causality”, if P is f then P
→
Lecture 3 • 38
Most of them are fairly obvious, but it’s worth studying the truth table for
implication fairly closely. In particular, note that (P implies Q) is tru... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
• A sentence is satisfiable iff its truth value is t in at
¬
least one interpretation
Satisfiable sentences: P, true,
¬
P
Lecture 3 • 41
A sentence is satisfiable if and only if it's true in at least one interpretation.
The sentence P is satisfiable. The sentence True is satisfiable. Not P is
satisfiable.
41
... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
Write down
all the interpretations, figure out the value of the sentence in each
interpretation, and if they're all true, it's valid. If they're all false, it's
unsatisfiable. If it's somewhere in between, it's satisfiable. So there's a way;
there's just a completely dopey, tedious, mechanical way to figure out if ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
Lecture 3 • 46
46
Models and Entailment
• An interpretation i is a model
of a sentence φ iff ² i φ
• A set of sentences KB entails φ
iff every model of KB is also a
model of φ
Sentences
Sentences
Lecture 3 • 47
So, here’s the picture. If we consider two groups of sentences, we might like
to say that one set... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
m
e
s
s
c
i
t
n
a
m
e
s
Sentences
Sentences
Interpretations
Interpretations
subset
Now, we can ask whether the first set of interpretations is a subset of the
second set.
Lecture 3 • 50
50
Models and Entailment
• An interpretation i is a model
of a sentence φ iff ² i φ
• A set of sentences KB entails φ ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
m
e
s
s
c
i
t
n
a
m
e
s
Sentences
entails
Sentences
Interpretations
Interpretations
subset
KB = A Æ B
φ = B
U
Lecture 3 • 53
Now, we can use a Venn diagram to think about the interpretations. Let U
be the set of all possible interpretations.
53
Models and Entailment
• An interpretation i is a model
o... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
Models and Entailment
• An interpretation i is a model
of a sentence φ iff ² i φ
• A set of sentences KB entails φ
iff every model of KB is also a
model of φ
s
c
i
t
n
a
m
e
s
s
c
i
t
n
a
m
e
s
Sentences
entails
Sentences
Interpretations
Interpretations
subset
KB = A Æ B
φ = B
A Æ B ² B
A Æ B
B
U
Lect... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
if we have a way of deciding whether
sentences are valid, then we have a way of checking whether one set of
sentences entails another. That is, whether the truth of one set of sentences
semantically requires the truth of another. This is going to lead into
techniques for proof, which is the process of testing for v... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
→
f)
¬
Examples
Interpretation that make
sentence’s truth value = f
Valid?
valid
satisfiable,
not valid
satisfiable,
not valid
smoke = t, fire = f
s = f, f = t
s
f = t,
→
¬
s
→
¬
f = f
Lecture 3 • 61
Here is a form of reasoning that you hear people do a lot, but the question is,
is it OK? “Smoke im... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
fire, then there's no smoke.
62
smoke
Sentence
smoke
→
smoke Ç
smoke
¬
smoke
fire
→
(s
→
f)
→
(
¬
s
→
f)
¬
(s
f)
→
contrapositive
(
f
→
¬
→
¬
s)
b Ç d Ç (b
b Ç d Ç
d)
→
b Ç d
¬
Examples
Interpretation that make
sentence’s truth value = f
Valid?
valid
satisfiable,
not valid ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
Programs with Flexible Time
When?
Contributions:
Brian Williams
Patrick Conrad
Simon Fang
Paul Morris
Nicola Muscettola
Pedro Santana
Julie Shah
John Stedl
Andrew Wang
courtesy of JPL
Steve Levine
Tuesday, Feb 16th
(cid:2)
Assignments
Problems Sets:
• Pset 1 due tomorrow (Wednesday) at 11:59pm
• Pset 2 released tomorr... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
YX
Y X
Y
X
Y-
Y-
BF G(cid:29)CD
[0,inf]
X+
[0,0]
BF <(cid:29)CD
X+
CD G(cid:29)BF and X- < Y+
CD G(cid:29)BD and X+ < Y+
BD <(cid:29)CD and X+ < Y+
BD G(cid:29)CD and X+ = Y+
BD <(cid:29)CD and X+ = Y+
BF G(cid:29)CD or Y+ < X-
(cid:2)(cid:5)
[Villain & Kautz; Simmons]
Temporal Relations Described by a
Simple Tempora... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
:13)(cid:27) (cid:21)(cid:27)(cid:8)
(cid:2)(
Consistency of an STN
Input: STN <X, C> where Cj = < <Xk, Xi>, <aj, bj> >
[1,10]
B
A
[0,9]
C
[1,1]
D
[2,2]
STN is consistent iff there exists an assignment to times X
satisfying C.
(cid:2)-
Schedule of an STN
Input: STN <X, C> where Cj = < <Xk, Xi>, <aj, bj> >
[1,10]
B
A
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
xi’s neighbors & update their windows
(cid:3)(cid:4)
Naïve (and wrong) scheduling
• (Board)
(cid:3)(cid:5)
Propagating to neighbors
Tighten neighbor’s execution windows:
- outgoing edges to neighbor: u’ = min(u, ti + wu)
- w )
- incoming edges from neighbor: l’ = max(l, ti
l
[u, l] (cid:2)tightened [u’, l’]
xi = ti
w... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
(cid:4)(cid:2)
Summary
• To schedule, want a simple, local-propagation
algorithm
– Requires exposing implicit constraints
• All-pairs shortest path (APSP) exposes all
implicit constraints
– Puts network in dispatchable form
• Negative cycle in APSP: inconsistent.
(cid:4)(cid:3)
To Execute a Temporal Plan
(cid:25)(c... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
1
0
D -2 -1 -2
1
2
0
-1
9
1
-1
-1
2
1
-2
11
-2
10
0
(cid:4)(
Computing a schedule
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
(cid:4)-
Computing a schedule
[-∞, ∞]
10
0
[-∞, ∞]
-1
9
1
-1
-1
2
1
-2
[-∞, ∞]
11
[-∞, ∞]
-2
Initialize execution windows for each event in the plan
(cid:4).
t = 0
Computing a schedule
[-∞, ∞]
-1
9
1
-1
-... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
to neighbors
(cid:5)(cid:3)
t = 0
Computing a schedule
[1, 10]
-1
9
1
-1
-1
2
1
-2
10
0
[0, 9]
11
-2
[-∞, ∞]
Propagate updated time bounds to neighbors
(cid:5)(cid:4)
t = 0
Computing a schedule
[1, 10]
-1
9
1
-1
-1
2
1
-2
10
0
[0, 9]
11
-2
[2, 11]
Propagate updated time bounds to neighbors
(cid:5)(cid:5)
t = 0
Compu... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
)-
[4, 4]
t = 0
Computing a schedule
10
0
t = 3
-1
9
1
-1
-1
2
1
-2
t = 2
11
-2
Propagate to neighbors
(cid:5).
[4, 4]
t = 0
Computing a schedule
10
0
t = 3
-1
9
1
-1
-1
2
1
-2
t = 2
11
-2
Assign the final event
(cid:10)$
t = 4
Pre-computed schedules not robust
against fluctuations
• We’ve just computed a schedule:... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
cid:8)
(cid:13)(cid:27) (cid:21)(cid:27)(cid:8)
(cid:10)(cid:3)
How do we schedule online?
• First, consider naive (incorrect!) approach.
• Similar to offline schedule algorithm, but now online:
– Wait until current time in execution window (“active”)
• (Still a problem though as we’ll see shortly)
(cid:10)(cid:4)
Na... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
we constrain dispatcher to do this?
• Solution: determine “enablement conditions” by
analyzing negative edges.
– Allows us to infer if some edges must precede other edges
(cid:10)-
Enablement conditions dictate the
ordering of dispatched events
• Negative edges from APSP dictate ordering constraints
10
0
-1
9
1
-1
-... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
to E any now-enabled events
E = {A, C}
S = {}
[-∞, ∞]
[-∞, ∞]
10
0
-1
9
1
-1
-1
2
[-∞, ∞]
1
-2
A, C initially in E – have no
negative, outgoing edges
[-∞, ∞]
11
-2
(cid:11)3
Running online dispatcher
Compute dispatchable form (i.e., APSP)
Initialize execution windows to [-∞, ∞]
E (cid:3) {events with no predecessors... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
windows to [-∞, ∞]
E (cid:3) {events with no predecessors}
S (cid:3) {}
while unexecuted events:
Wait until some event xi in E is active
ti = now
Propagate to xi’s neighbors
Add xi to S
Add to E any now-enabled events
t = 0
10
0
B, D not enabled! But C still is.
E = {C}
S = {A}
[1, 10]
-1
9
1
-1
-1
2
[0, 9]
11
-2
[2, ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
1
-2
B is now enabled (but still not D).
t = 2
11
-2
68
Running online dispatcher
Compute dispatchable form (i.e., APSP)
Initialize execution windows to [-∞, ∞]
E (cid:3) {events with no predecessors}
S (cid:3) {}
while unexecuted events:
Wait until some event xi in E is active
ti = now
Propagate to xi’s neighbors
Ad... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
}
t = 3
t = 0
10
0
-1
9
1
-1
-1
2
Finish up by dispatching D!
(1
t = 2
11
-2
t = 4
1
-2
73
Online dispatching algorithm remarks
• By considering predecessors, we guarantee that events
assigned monotonically increasing times online.
• Capable of responding to fluctuations that do not affect
overall temporal feasibili... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
xi’s neighbors
Add xi to S
Add to E any now-enabled events
(5
Online dispatcher efficiency
• Consider an STN with n edges.
• How many edges in APSP distance graph? n2.
• How many neighbors to propagate to each step? n.
Compute dispatchable form (i.e., APSP)
Initialize execution windows to [-∞, ∞]
E (cid:3) {events wi... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
)(cid:27) (cid:21)(cid:27)(cid:8)
7-
79
You don’t need all those edges!
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
-0
You don’t need all those edges!
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
1
-1
1
9
-1
0
Equivalent minimal dispatchable network
-1
You don’t need all those edges!
Let’s consider a specific triangle
of edges.
10
0
-1
9
1
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
1
Can we remove rope AD without changing behavior?
10
11
1
Yes! Same possible positions for A, B,
D.
8.
Rope analogy
10
1
11
Can we remove ropes AB, BD without changing
behavior?
10
11
1
No. AD still constrained, but B could slide freely! Not the
same behavior. Collectively, AB and BD entail AD (but AD
does not en... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
10
0
-1
9
1
-1
-1
2
1
-2
11
-2 Upper dominated!
98
Dominance example
10
0
-1
9
1
-1
-1
2
1
-2
11
-2 Upper dominated!
99
Dominance example
Lower dominated!
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
(cid:2)$0
Dominance example
Lower dominated!
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
(cid:2)$1
Dominance example
10
0
-1
9
1
-1
-1
2
1
-2
1... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
1]
[0,10]
[2, 2]
Original STN
1
-1
... now in minimal
dispatchable form!
1
9
-1
0
(cid:2)$8
FilteringAlgorithm(G)
Input: A dispatchable APSP-graph G
Output: A minimal dispatchable graph
1 for each pair of intersecting edges in G
2 if both dominate each other
3 if neither is marked
4 arbitrarily mark one for eliminati... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
position.
4. Execute Plan
(cid:13))) (cid:21)(cid:27)(cid:8)
(cid:13)(cid:27) (cid:21)(cid:27)(cid:8)
(cid:2)(cid:2)2
[Dechter, Meiri, Pearl 91]
To Execute a Temporal Plan
(cid:25)(cid:26)*(cid:8)+" (cid:8)(cid:29),)) (cid:21)(cid:27)(cid:8)
1. Describe Temporal Plan
2. Test Consistency
3. Schedule Plan
4. Execute Pla... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
16)(cid:8)(cid:7)(cid:15)
A
[0,9]
t=0
A
[0,9]
[1,1]
[1,1]
D
B
C
t=3
B
[1,1]
[1,1]
t=4
D
C
t=2
(cid:13))) (cid:21)(cid:27)(cid:8)
(cid:13)(cid:27) (cid:21)(cid:27)(cid:8)
(cid:2)(cid:2)5
MIT OpenCourseWare
https://ocw.mit.edu
16.412J / 6.834J Cognitive Robotics
Spring 2016
For information about citing these materials o... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.005 Elements of Software Construction
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
6.005
elements of
software
construction
basics of mutable types
Daniel Jackson
heap semantics of Java
pop quiz
what ha... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
of object type
‣ but does not denote an object
‣ cannot call method on null, or get/set field
© Daniel Jackson 2008
6
the operator ==
the operator ==
‣ returns true when its arguments denote the same object
(or both evaluate to null)
for mutable objects
‣ if x == y is false, objects x and y are observably di... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
}
}
© Daniel Jackson 2008
9
mutable datatypes
mutable vs. immutable
String is an immutable datatype
‣ computation creates new objects with producers
class String {
String concat (String s);
...}
... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
08
15
equivalence
can define your own equality notion
‣ but is any spec reasonable?
reasonable equality predicates
‣ define objects to be equal when they represent the same abstract value
a simple theorem
‣ if we define a ≈ b when f(a) = f(b) for some function f
‣ then the predicate ≈ will be an equivalence
an... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
public class Duration {
private final int hours;
private final int mins;
public Duration(int h, int h) {hours = h; mins = m;}
public boolean equals (Duration d) {
return d.getMins() == this.getMins();
}
}
Duration d1 = new Duration(1,2);
Duration d2 = new Duration(1,2);
System.out.println(d1.equals(d2));... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
08
23
bug #3
an attempt at writing equals for subclass
@Override
public boolean equals(Object o) {
if (! (o instanceof ShortDuration))
return false;
ShortDuration d = (ShortDuration) o;
return d.getSecs () == this.getSecs();
}
will this work?
‣ no, now it’s not symmetric!
Duration d1 = new ShortDurat... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
K key; V val; Entry<K,V> next; ... }
© Daniel Jackson 2008
28
Entryk1: Kv1: VEntrynextkeyvalk2: Kv2: Vkeyval01234HashMaptable hash map operations
operations
‣ put(k,v): to associate value v with key k
compute index i = hash(k)
hash(k) = k.hashCode & table.length-1 (eg)
if find entry in table[i] with key equal t... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
happens
if equals and hashCode depend on key’s fields
‣ then value of hashCode can change
‣ rep invariant of hash map is violated
‣ lookup may fail to find key, even if one exists
problem is example of ‘abstract aliasing’
‣ hash map and key are aliased
© Daniel Jackson 2008
31
example
what does this print?
... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
consistently return false, provided no information used in equals comparisons on the object is modified
© Daniel Jackson 2008
33
non-determinism
to iterate over elements of a hash set
‣ use HashSet.iterator()
‣ elements yielded in unspecified order
what determines order?
‣ code iterates over table indices
‣ so... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
operator ξ S†ΓµW hv which is not equivalent to (10.6) because S and W do not commute. This apparent
problem is solved by considering the remaining two diagrams of the same order as this one
n,p
11 SCETII APPLICATIONS
Diagrams.
These diagrams both yield the current
Fig() = Fig() =
2
if abcT c
g
2
ν
µ
n¯
n
n · qs... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5a73e8bb8f7f637a9b3eb58c5e733e6b_MIT8_851S13_SCETIIApplicat.pdf |
only present in currents
JI = (ξ
(0)
n
W (0))ΓµY †hv
3. Matching SCETI onto SCETII by taking Yn → Sn.
JII = (ξ
(0)
n
W (0))ΓµS†hv
(10.10)
(10.11)
(10.12)
11 SCETII Applications
(ROUGH) In this section we will apply the SCETII formalism developed in previous sections to various
processes to illustrate the formalis... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5a73e8bb8f7f637a9b3eb58c5e733e6b_MIT8_851S13_SCETIIApplicat.pdf |
= [cγµPLT ab][dγµPLT a u].
(11.1)
(11.2)
(11.3)
We want to factorize the matrix element (Dπ| O0,8 |B). We can represent this factorization diagrammat
ically as (INSERT FIG) where there are no gluons between π quarks and B/D quarks. For this process
we expect a B → D form factor (Isgur-Wise form factor) and a pion... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5a73e8bb8f7f637a9b3eb58c5e733e6b_MIT8_851S13_SCETIIApplicat.pdf |
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