text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
C� | (A� · nˆ), where ˆn =
A, � C). The latter identity can also be checked
� · (B� × C� ) = det( � B, �
C . So volume = A
C/
B� × � |B� × � |
directly using components. | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/649253ba60d11b0598cc58e9dcf58142_lec_week1.pdf |
INTRODUCTION
Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful
mathematical subject which has many applications, ranging from number theory and combinatorics
to geometry, probability theory, quantum mechanics and quantum field theory.
Representation theory was born in ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/64ac0a7f209d31929d393a80e3ab85bb_MIT18_712F10_intro.pdf |
mostly follow [FH], with the exception of the sections discussing quivers, which follow
[BGP]. We also recommend the comprehensive textbook [CR]. The notes should be accessible to
students with a strong background in linear algebra and a basic knowledge of abstract algebra.
Acknowledgements. The authors are grateful... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/64ac0a7f209d31929d393a80e3ab85bb_MIT18_712F10_intro.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
Physics 8.07: Electromagnetism II
Prof. Alan Guth
September 5, 2012
LECTURE NOTES 1
VECTOR ANALYSIS
DEFINITION: A vector is a quantity that has magnitude and direction.
Examples: displacement, velocity, acceleration, force, momentum, electric and magnet... | https://ocw.mit.edu/courses/8-07-electromagnetism-ii-fall-2012/64baa9cb5023c49bc9147bee5416b6ec_MIT8_07F12_ln1.pdf |
In fact, one can show that if AA · BA obeys rotational invariance
and the distributive law, then
A AA · B = const|AA||BA | cos θ .
We’ll come back to this later.
8.07 LECTURE NOTES 1, FALL 2012
p. 3
Cross product of two vectors:
AA × BA ≡ |AA||BA | sin θ ˆn ,
where ˆn is a unit vector perpendicular to AA a... | https://ocw.mit.edu/courses/8-07-electromagnetism-ii-fall-2012/64baa9cb5023c49bc9147bee5416b6ec_MIT8_07F12_ln1.pdf |
Ax + Bx)ˆex + (Ay + By )ˆey + (Az + Bz)ˆez .
8.07 LECTURE NOTES 1, FALL 2012
p. 4
Vector dot product:
AA · BA = (Axeˆx + Ay eˆy + Az eˆz ) · (Bxeˆx + By eˆy + Bzeˆz ) .
eˆx · eˆx = ˆey · eˆy = ˆez · eˆz = 1 ,
and ˆex · eˆy = 0, as does the dot product of any two basis vectors.
. A BA = AxBx + AyBy + AzBz
.. A... | https://ocw.mit.edu/courses/8-07-electromagnetism-ii-fall-2012/64baa9cb5023c49bc9147bee5416b6ec_MIT8_07F12_ln1.pdf |
product of any two distinct basis vectors must vanish,
so
AA · BA = const (AxBx + Ay By + Az Bz) .
This had better be equivalent to const|AA||BA | cos θ, but we can see it
explicitly by using the rotational invariance to orient AA along the pos
itive x-axis, so Ax = |AA|, and Ay = Az = 0. Then the above formula
g... | https://ocw.mit.edu/courses/8-07-electromagnetism-ii-fall-2012/64baa9cb5023c49bc9147bee5416b6ec_MIT8_07F12_ln1.pdf |
://ocw.mit.edu
8.07 Electromagnetism II
(cid:41)(cid:68)(cid:79)(cid:79)(cid:3)(cid:21)(cid:19)(cid:20)(cid:21)
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/8-07-electromagnetism-ii-fall-2012/64baa9cb5023c49bc9147bee5416b6ec_MIT8_07F12_ln1.pdf |
Type Classes and Subtyping
Armando Solar-Lezama
Computer Science and Artificial Intelligence Laboratory
MIT
October 5, 2015
With content by Arvind and Adam Chlipala.
Used with permission.
October 5, 2015
L08-1
Hindley-Milner gives us generic
functions
• Can generalize a type if the function
makes no assump... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/64d576d7d140cb9fbbd23364c2c2b453_MIT6_820F15_L08.pdf |
) :: a -> a -> a
fromInteger :: Integer -> a
}
October 5, 2015
L08-5
Generalized Functions w/ “class”
types
matrixMul :: Num a -> Mat a -> Mat a -> Mat a
dft :: Num a -> Vec a -> Vec a -> Vec a
• All of the numeric aspects of the type has been isolated to
the Num type
– For each type, we built a num... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/64d576d7d140cb9fbbd23364c2c2b453_MIT6_820F15_L08.pdf |
October 5, 2015
L08-8
Laws for a type class
• A type class often has laws associated
with it
– E.g., + in Num should be associate and
commutative
• These laws are not checked or ensured
by the compiler; the programmer has to
ensure that the implementation of each
instance correctly follows the law
more o... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/64d576d7d140cb9fbbd23364c2c2b453_MIT6_820F15_L08.pdf |
super.append(ls);
}
int howManyAppends() { return numAppends; }
}
L08-15
Subtyping as Graph Search
float
List
Arrow from A to B to indicate
that B is a “direct” subtype of A
int
Nil
Cons
LoggingCons
Q: How do we decide if A is a subtype of B?
A: Graph reachability! (Ea... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/64d576d7d140cb9fbbd23364c2c2b453_MIT6_820F15_L08.pdf |
��
L08-20
Sanity-Checking the Principle
Primitive rule:
int ≤ float
✔Any integer N can be treated as N.0, with no loss of meaning.
Primitive rule:
float ≤ int
✗E.g., “%” operator defined for int but not float.
Primitive rule:
int ≤ int → int
✗Can't call an int!
L08-21
From Nominal to Structural
A str... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/64d576d7d140cb9fbbd23364c2c2b453_MIT6_820F15_L08.pdf |
A : int, B : float} ≤ {A : float, B : float}
{A : float, B : float} ≤ {A : int, B : float}
?
{A : int, B : float} ≤ {A : float}
?
Depth:
∀i. 𝜏i ≤ 𝜏'i
{ai : 𝜏i} ≤ {ai : 𝜏'i}
Width:
∀j. ∃i. ai = a'j ∧ 𝜏i = 𝜏'j
{ai : 𝜏i} ≤ {a'j : 𝜏'j}
Yes!
No!
Yes!
L08-25
Function Types
Consider types 𝜏1 → 𝜏2. ... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/64d576d7d140cb9fbbd23364c2c2b453_MIT6_820F15_L08.pdf |
Read a 𝜏 from some index.
2. Write a 𝜏 to some index.
𝜏1 ≤ 𝜏2
𝜏1[] ≤ 𝜏2[]
Covariant rule:
Counterexample:
int[] x = new int[1];
float[] y = x; // Use subtyping here.
y[0] = 1.23;
int z = x[0]; // Not an int!
L08-28
Arrays
Consider types 𝜏[].
What operations must we support?
1. Read a 𝜏 f... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/64d576d7d140cb9fbbd23364c2c2b453_MIT6_820F15_L08.pdf |
information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/64d576d7d140cb9fbbd23364c2c2b453_MIT6_820F15_L08.pdf |
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
19. Sequential Transactions with Caching
There are many situations in which we want to make a ‘big’ action atomic, either with respect to
concurrent execution of other actions (everyone else sees that the big action has either not... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
more efficient code that allows for
caching data; this is usually necessary for decent performance. Unfortunately, it complicates
matters considerably. We give a rather abstract version of the caching code, and then sketch the
concrete specialization that is in common use. Finally we discuss some pragmatic issues.
... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
with Caching
2
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
Action
Do(x := x – 1);
Do(y := y + 1)
Commit
Crash before commit
X
5
5
5
4
5
Y
5
5
5
6
5
x
5
4
4
4
5
y
5
5
6
6
5
If we want to take account of the possibility that the server (specified ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
ph := run >>
% aborts any current trans.
APROC Do(a) -> V RAISES {crashed} = <<
IF ph = run => VAR v | (v, vs) := a(vs); RET v [*] RAISE crashed FI >>
Here is the previous example extended with the ph variable.
Action
Begin();
Do(x := x - 1);
Do(y := y + 1)
Commit
Crash before commit
X
5
5
5
5
4
5
Y
5... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
% volatile state
% stable log
% volatile log
% phase (volatile)
% recovering
APROC Commit() RAISES {crashed} =
<< IF ph = run => ss := vs; ph := idle [*] RAISE crashed FI >>
EXCEPTION crashed
APROC Abort () = << vs := ss; ph := idle >>
APROC Crash () = << vs := ss; ph := idle >>
% same as Crash
% ‘aborts’ the... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
the crash does
% the recovery action
vl := sl; Redo(); vs := ss; rec := false
PROC Redo() =
% sl = vl before this is called
% replay vl, then clear sl
DO vl # {} => << ss := ss + {vl.head} >>; << vl := vl.tail >> OD
<< sl := {} >>
Action
Begin();
Do(x := x - 1);
Do(y := y + 1)
Commit
Redo: apply x:=4
Redo:... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
l
k = {}
RET
Here is what this code does for the previous example, assuming for simplicity that A = U. You
may wish to apply the abstraction function to the state at each point and check that each action
simulates an action of the spec.
\/ (EXISTS k', l'|
k = k' + l' /\ l' # {} /\ l' <= l
/\ IsHiccups(k', l) )
be... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
block. Then you can tell whether a U has already been applied just by comparing its VN
with the VN of the block. For example, if the version number of the update is 23:
Original
Idempotent
The disk block x: Int
x: Int
vn: Int
The update
x := x + 1
IF
vn = 22 => x := x + 1;
[*]
SKIP FI
vn := 23
Note: vn = ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
=
<< vs := ss; vl := {}; vph := idle >>;
vph := sph; vl := (vph = idle => {} [*] sl);
Redo(); vs := ss
% what the crash does
% the recovery
% action
PROC Redo() =
% replay vl, then clear sl
DO vl # {} => << ss := ss + {vl.head} >>; << vl := vl.tail >> OD
DO sl # {} => << sl := sl.tail >> OD;
<< sph := idle; vph... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
8
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
Caching
We would like to have code for SequentialTr that can run fast. To this end it should:
1. Allow the volatile state vs to be cached so that the frequently used parts of it are fast to
access, but not require a complete ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
stable state, which no longer exists. So there’s an incentive to limit the amount by
which the background process runs behind.
Normally the volatile state consists of entries in the cache. Although the abstract code below
does not make this explicit, the cache usually contains the most recent values of variables, th... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
can abort a transaction after installing some parts of the cache to the stable
state, we have to follow the “write ahead log” or WAL rule, which says that before a cache entry
can be installed, all the actions that affected that entry (and therefore all their undo’s) must be in
the stable log.
Although we don’t wan... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
2002
6.826—Principles of Computer Systems
2002
UndoLog : sequence of undo actions (not updates)
The essential step is installing a cache update into the stable state. This is an internal action, so it
must not change the abstract stable or volatile state. As we shall see, there are many ways to
satisfy this requi... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
:= L{}
undoing
:= false
% Stable State
% Stable Log
% Stable Undo Log
% Cache (dirty part)
% Volatile Log
% Volatile Undo Log
% Volatile PHase
% Permanent Log
Note that there are two logs, called L and UL (for undo log). A L records groups of updates; the
difference between an update U and an action A is th... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
of the undo action to vl/sl. For
example, after un1, un2, and un3 have been processed, ul might be
un0 un1 un2 cancel un3 cancel cancel
= un0 un1 un2 cancel cancel
= un0 un1 cancel
= un0
Of course many other encodings are possible, but this one is simple and fits in well with the rest
of the code.
Examples of a... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
vl)–(sul+vul)
SequentialTr.ph = (~undoing => vph idle)
% INVARIANTS
[1] (ALL l1, l2 |
sl = l1 + {commit} + l2
/\ ~ commit IN l2
==> ss + l1 = ss + sl - sul )
% Stable undos cancel
% uncommitted tail of sl
[2] ss + sl = ss + sl + vl - vul
% Volatile undos cancel vl
[3] ~ undoing ==> ss + sl + vl = ss ++ c
% C... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
ss ++ {w} + sl
+ sl
= ss
% Any cache entry can be installed
VAR ul := sul + vul, i := 0 |
DO ul # {} => VAR un := ul.last |
ul := ul.reml;
IF un=cancel => i := i+1 [*] i>0 => i := i-1 [*] Apply(un, cancel) FI
OD; undoing := false
% Every entry in sul + vul has a cancel, and everything is undone in vs.
% Background... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
w} = ss => c := c - {w} >>
APROC Truncate() = << VAR l1, l2 |
% Move some of sl to pl
sl = l1 + l2 /\ ss + l2 = ss + sl => pl := pl + l1; sl := l2 >>
% Media recovery
The idea is to reconstruct the stable state ss from the permanent log pl by redoing all the
updates, starting with a fixed initial ss. Details are l... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
)
[] VAR ul1, un, ul2 | un # cancel /\ ul = ul1 + {un, cancel} + ul2 =>
RET UndoLog(s, ul1 + ul2)
FI
A cache is a set of commuting update functions. When we combine two caches c1 and c2, as we
do in apply, we want the total effect of c1 and c2, and all the updates still have to commute and
be atomic updates. The Com... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
if the existing systems don’t implement two-phase commit, you
can still build a multi-level system on them.
• Often you can get more concurrency by allowing lower level transactions to commit and
release their locks. For example, a B-tree typically holds locks on whole disk blocks to
maintain the integrity of the i... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
’s are also used to code the WAL guard in Install and the guard in Truncate. It’s OK
to install a W if the LSN of the last entry in sl is at least as big as the n of the W. It’s OK to drop a
U from the front of sl if every uninstalled W in the cache has a bigger LSN.
The simplest case is a block equal to a single di... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
u.ba ==> w
commutes with u, because u only looks at u.ba. Stated precisely:
(ALL s | (ALL ba | ba # u.ba ==> u(s)(ba) = s(ba)
/\ (ALL s' | s(u.ba) = s'(u.ba) ==> u(s)(u.ba) = u(s')(u.ba)) )
So the guard in Install testing whether w is already installed is just
(EXISTS u | u IN vl /\ u.ba = w.a)
because in Do we get... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
current one, or to issue a
stop payment order for a check that was printed erroneously. Or it might require manual
intervention, for instance, calling up the bank customer and asking what really happened in
yesterday’s ATM transaction. Usually compensation is not perfect.
Because compensation is complicated and imp... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/64d732d98d264f9a61e381bf1358c907_19.pdf |
18.465 notes, March 29, 2005, revised May 2
The spatial median
In one dimension, for any probability distribution function F with a finite first mo-
ment, the medians are exactly the values of m for which ∫ |x −m|dF (x) is minimized, using
a definition allowing an interval of medians on which the distribution function ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/64fb1ad786d5fc7157b0f30b430e7c9a_spatialmedian.pdf |
be called strictly convex if whenever x (cid:6)= y ∈ C and 0 < λ < 1 we have
f (λx + (1 − λ)y) < λf (x) + (1 − λ)f (y). It’s easily seen that a function f on an interval is
convex if its second derivative is nonnegative and strictly convex if its second derivative is
strictly positive. For example, on R1 , f (x) = x... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/64fb1ad786d5fc7157b0f30b430e7c9a_spatialmedian.pdf |
0) := |s − x| − |s0 − x| ≤ |s − s0|,
g(s) := M (s, P, s0) is always finite. Clearly, it’s continuous in s, and goes to ∞ as s → ∞
for fixed s0. Thus the infimum of M (s, P, s0) is attained, and a spatial median always
exists. Changing s0 only adds a constant to the integral, so the minimization doesn’t
depend on s0.
... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/64fb1ad786d5fc7157b0f30b430e7c9a_spatialmedian.pdf |
Notes. Haldane (1948) proved uniqueness of the spatial median in Rk, k ≥ 2. (In
Haldane’s proof, note that d2R/dx2 > 0 unless yr = 0 for all r, in which case all the
observations are on a line.) Haldane gives the proof in detail for a finite sample (empirical
measure).
The device of taking |s − x| − |s0 − x| in plac... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/64fb1ad786d5fc7157b0f30b430e7c9a_spatialmedian.pdf |
Engineering Risk Benefit Analysis
1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82,
ESD.72, ESD.721
DA 2. The Value of Perfect Information
George E. Apostolakis
Massachusetts Institute of Technology
Spring 2007
DA 2. The Value of Perfect Information
1
Recall the evaluation of the survey results
(Slide 14, DA... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/6515ae688d570abf0946ec1059c38644_da2.pdf |
/s)=0.706
P(L3/s)=0.294
P(L4/s)=0.000
1.000
P[L2 materializes/survey says L2] = 0.706, because the survey is not perfect.
DA 2. The Value of Perfect Information
4
Bayes’ Theorem for the Clairvoyant
P[L2 materializes/ CV says L2] =
CVsaysL
L/
2
2
materializ
L(xP)es
materializ
)es
2
=
(P
CVsaysL
L/
i
2
materializ
L(xP)... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/6515ae688d570abf0946ec1059c38644_da2.pdf |
.0
α=$195K
L2 $300K
L3 $100K
L4
-$100K
L1 $150K
DA 2. The Value of Perfect Information
8
The value of alpha
α:
EMV, if the terminal decision is to be
made with perfect information at no cost.
α = 0.3x300 + 0.5x150 + 0.2x150 = $195K
DA 2. The Value of Perfect Information
9
The value of beta
• What is the EMV without ... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/6515ae688d570abf0946ec1059c38644_da2.pdf |
considering whether to
actually do it.
DA 2. The Value of Perfect Information
13
Summary and Observations
• We have developed single-attribute, multi-stage
sequential Decision Trees.
• The model is useful to a single decision maker.
• Decision Criterion: Maximize the EMV.
• Maximizing the EMV is not the best decisi... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/6515ae688d570abf0946ec1059c38644_da2.pdf |
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 3-1
Lecture 3 - Semiconductor Physics (II)
Carrier Transport
September 15, 2005
Contents:
1. Thermal motion
2. Carrier drift
3. Carrier diffusion
Reading assignment:
Howe and Sodini, Ch. 2, §§2.4-2.6
6.012 - Microelectronic Devices and Circuits ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/652b369381b25855a652d351077fe3b5_lecture3anotat.pdf |
devices
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 3-5
2. Carrier Drift
Apply electric field to semiconductor:
E ≡ electric field [V /cm]
⇒ net force on carrier
F = ±qE
E
-
Between collisions, carriers accelerate in direction of field:
v(t) = at = − qE t
mn
for electrons
v(t) = qE t
mp
... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/652b369381b25855a652d351077fe3b5_lecture3anotat.pdf |
• for medium and high doping level, µ limited by colli-
sions with ionized impurities
• holes ”heavier” than electrons:
→ for same doping level, µn > µp
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 3-8
Drift current
Net velocity of charged particles ⇒ electric current:
Drift current densi... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/652b369381b25855a652d351077fe3b5_lecture3anotat.pdf |
1E-3
1E-4
p-Si
n-Si
1E+12 1E+13 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21
Doping (cm-3)
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 3-11
Numerical example:
• Si with Nd = 3 × 1016 cm−3 at 300 K
µn (cid:2) 1000 cm2/V · s
ρn (cid:2) 0.21 Ω · cm
• apply |E| = 1 kV /cm
|vdn| (cid... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/652b369381b25855a652d351077fe3b5_lecture3anotat.pdf |
Fn = −Dn dx
dp
Fp = −Dp dx
Dn ≡ electron diffusion coefficient [cm2/s]
Dp ≡ hole diffusion coefficient [cm2/s]
D measures the ease of carrier diffusion in response to a
concentration gradient: D ↑ ⇒ F dif f ↑.
D limited by vibrating lattice atoms and ionized dopants
6.012 - Microelectronic Devices and Circuits - Fall 2... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/652b369381b25855a652d351077fe3b5_lecture3anotat.pdf |
current can flow by drift and diffusion sepa-
rately. Total current:
Jn = J drif t + J dif f = qnµnE + qDn
n
n
dn
dx
Jp = J drif t + J dif f
p
p
= qpµpE − qDp
dp
dx
And
Jtotal = Jn + Jp
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 3-17
Summary: relationship between v, F , and J
In semicon... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/652b369381b25855a652d351077fe3b5_lecture3anotat.pdf |
Clocking
Figure by MIT OCW.
6.884 - Spring 2005
2/18/05
L06 – Clocks 1
Why Clocks and Storage Elements?
Inputs
Combinational Logic
Outputs
Want to reuse
combinational logic
from cycle to cycle
6.884 - Spring 2005
2/18/05
L06 – Clocks 2
Digital Systems Timing Conventions
All digital systems need a conventi... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/6532a35551bb8707fa017de5e53c66c7_l06_clocks.pdf |
a Latch
Latches are a mux, clock selects
either data or output value
Q
CMOS Transmission Gate Latch
D
0
1
CLK
Optional
input buffer
D’
D
CLK
CLK
Usually have local
inverter to
generate CLK
Q
Q
Optional
output buffer
CLK
Parallel N and P transistors act as
switch, called a “transmission gate”
6.884 - ... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/6532a35551bb8707fa017de5e53c66c7_l06_clocks.pdf |
ailing Setup
CLK
CLK
Q
D
CLK
CLK
If data arrives too close to clock edge, it won’t set up
the feedback loop before clock closes the input
transmission gate.
6.884 - Spring 2005
2/18/05
L06 – Clocks 10
The Hold Time Race
CLK
CLK
Q
D
CLK
CLK
Added clock buffers to demonstrate
positive hold time on this ... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/6532a35551bb8707fa017de5e53c66c7_l06_clocks.pdf |
– only found in high performance custom devices
6.884 - Spring 2005
2/18/05
L06 – Clocks 14
Flip-Flop Timing Parameters
Clock
D
Q
TCQmin
TCQmax
Tsetup
Thold
TCQmin/TCQmax
– propagation in→out at clock edge
Tsetup/Thold
– define window around rising clock edge during which
data must be steady to be samp... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/6532a35551bb8707fa017de5e53c66c7_l06_clocks.pdf |
ps
directly, no local
buffers
6.884 - Spring 2005
2/18/05
L06 – Clocks 18
H-Trees
Recursive pattern to distribute signals uniformly with
equal delay over area
Uses much less power than grid, but has more skew
In practice, an approximate H-tree is used at the top
level (has to route around functional blocks)... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/6532a35551bb8707fa017de5e53c66c7_l06_clocks.pdf |
Oscillator
Circuit
Divide by N
6.884 - Spring 2005
2/18/05
L06 – Clocks 23
Intel Itanium Clock Distribution
Global Distribution
Regional Distribution
Local Distribution
GCLK
Regional Grid
RCD
RCD
DLCLK
OTB
Reference
clock
CLKP
CLKN
VCC
/2
PLL
Main clock
DSK
DSK
DSK
DSK = Active deskew circuits,... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/6532a35551bb8707fa017de5e53c66c7_l06_clocks.pdf |
– worst case is when CLK2 is earlier/later than CLK1
Fast path timing constraint
TCQmin + TPmin ≥ Thold + Tskew
– worst case is when CLK2 is earlier/later than CLK1
6.884 - Spring 2005
2/18/05
L06 – Clocks 27 | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/6532a35551bb8707fa017de5e53c66c7_l06_clocks.pdf |
6.S897/HST.956 Machine Learning for Healthcare
Lecture 1: What makes healthcare unique?
Instructors: David Sontag, Peter Szolovits
The Problem
Healthcare costs in the US amount to over $3 trillion and are rapidly rising. The US has some of the best
clinicians in the world, but there are still many cases of chronic... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/655ed887d3a9c977a81864aec4139ac8_MIT6_S897S19_lec1note.pdf |
not generalize well across different populations.
The RX Project is an AI designed for automated knowledge acquisition. Figure 1 displays the discovery
system that combines empirical data with a knowledge base that combines with researchers to generate and
evaluate hypotheses about causal relationships to create new ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/655ed887d3a9c977a81864aec4139ac8_MIT6_S897S19_lec1note.pdf |
This highlights a theme for the rest of the course: that new policy
can open the door to innovation.
2.2 Data
New medical datasets are now publicly available.
Mimic, the only publicly available EHR dataset, was created out of MIT, consisting of intensive care unit
patient records. From the MIMIC Physiotnet website... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/655ed887d3a9c977a81864aec4139ac8_MIT6_S897S19_lec1note.pdf |
Resources (FHIR) is a standardized API
protocol for sending EHR data. Observational Health and Data Science Informatics (OHDSI) is a common
data model for EHR data.
2.4 Advances in Machine Learning
Breakthroughs in machine learning for tasks such as image recognition and language translation have inspired
new confi... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/655ed887d3a9c977a81864aec4139ac8_MIT6_S897S19_lec1note.pdf |
.
3.1
Imagining the emergency department of the future
Professor Sontag has been working with Beth Israel Deaconess Medical Center to use technology in their
Emergency Department. The ER is an interesting setting because it faces extreme constraints on a daily
basis, these include: limited resources, time-sensitiv... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/655ed887d3a9c977a81864aec4139ac8_MIT6_S897S19_lec1note.pdf |
practices.
Professor Sontag believes this could be particularly useful in academic medical centers to aid in the training
of new doctors and in less populated areas where doctors might need to cover a broader set of conditions.
3.2 Efficient healthcare workflows
Machine learning can make many healthcare workflows more ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/655ed887d3a9c977a81864aec4139ac8_MIT6_S897S19_lec1note.pdf |
, patient data is particularly sensitive, so de-identification is necessary and negotiating
data sharing agreements is time-consuming. Due to the complexity of the healthcare system, there is often
missing data and different training data vs testing data distributions. Production EHR systems might be
difficult to work w... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/655ed887d3a9c977a81864aec4139ac8_MIT6_S897S19_lec1note.pdf |
.05225, 2017.
[SA75] Edward H Shortliffe and Stanton G Axline. Computer-based consultations in clinical therapeutics:
Explanation and rule acquisition capabilities of the mycin. 1975.
6.S897/HST.956 Machine Learning for Healthcare — Lec1 — 6
... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/655ed887d3a9c977a81864aec4139ac8_MIT6_S897S19_lec1note.pdf |
Lecture 2
L t
2
Semiconductor Physics (I)
Outline
Intrinsic bond model : electrons and holes
• Intrinsic bond model : electrons and holes
•
• Generation and recombination
• Intrinsic semiconductor
• Doping: Extrinsic semiconductor
• Charge Neutrality
ead g ss g
Reading Assignment:
e t
Howe and Sodini; Chapter 2. ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6583f0018f5bb90464ea68b550f73c95_MIT6_012S09_lec02.pdf |
-
3
Simple “flattened” model of Si
crystal
4 valence electrons (– 4 q),
contributed by each ion
silicon ion (+ 4 q)
border of bulk silicon region
two electrons in bond
At 0K
At 0K:
• All bonds are satisfied
– ⇒ all valence electrons engaged in bonding
N “f
• No “free” electrons
”
t
l
6.012 Lecture 2
Elec... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6583f0018f5bb90464ea68b550f73c95_MIT6_012S09_lec02.pdf |
bringing together electron and hole
• Releases energy in thermal or optical form
• Recombination rate:
• 1 recombination event requires 1 electron + 1
R = [cm−3 • s−1 ]
hole ⇒
R ∝ n • p
Generation and recombination most likely at surfaces
where periodic crystalline structure is broken
6.012 Lecture 2
Electroni... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6583f0018f5bb90464ea68b550f73c95_MIT6_012S09_lec02.pdf |
34
Zn
Ga
Ge
As
Se
48
49
50
51
52
CdCd
InIn SnSn SbSb TeTe
6.012 Lecture 2
Electronic Devices and Circuits -
9
Doping: Donors Cont’d...
• 4 electrons participate in bonding
• 5th electron easy to release ⇒
– at room temperature, each donor releases
1 electron that is available for conduction
1 electr... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6583f0018f5bb90464ea68b550f73c95_MIT6_012S09_lec02.pdf |
15
15
16
16
Al
Si
P
S
IIB
30
31
32
33
34
ZnZn GaGa Ge AsAs SeSe
Ge
48
49
50
51
52
Cd
In Sn Sb Te
6.012 Lecture 2
Electronic Devices and Circuits -
12
Doping: Acceptors Cont’d...
3 electrons participate in bonding
• 3 electrons participate in bonding
• 1 bonding site “unsatisfied” making it easy ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6583f0018f5bb90464ea68b550f73c95_MIT6_012S09_lec02.pdf |
charge neutrality must be satisfied
In general:
•
ρ q po no + Nd Na
ρ = q po − no + Nd − Na
(
(
)
)
Let us examine this for Nd = 1017 cm-3, Na = 0
We solved this in an earlier example:
n o = Nd = 1017cm−3, po =
2
2
n i
Nd
= 103cm−3
Hence:
ρ ≠ 0 !!
ρ
What is wrong??
6.012 Lecture 2
Electronic Devices a... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6583f0018f5bb90464ea68b550f73c95_MIT6_012S09_lec02.pdf |
Verilog
L3: Introduction to Verilog
L3: Introduction to
(Combinational Logic)
(Combinational Logic)
Acknowledgements : Rex Min
Verilog References:
• Samir Palnitkar, Verilog HDL, Pearson Education (2nd edition).
• Donald Thomas, Philip Moorby, The Verilog Hardware Description Language, Fifth
Edition, Kluwer Academic ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
...
always @ (posedge clk)
begin
count <= count + 1;
end
...
HDL Code
“This section of code looks
like a counter. My FPGA has
some of those...”
counter
Inferred Macro
(cid:132) Place-and-route: with area and/or speed in mind, choose
the needed macros by location and route the interconnect
M
M
M
M
M
M
M
M
M
M
M
M
M
M... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
outbar, sel);
input a, b, sel;
output out, outbar;
assign out = sel ? a : b;
assign outbar = ~out;
endmodule
a
b
1
0
sel
out
outbar
(cid:132) Continuous assignments use the assign keyword
(cid:132) A simple and natural way to represent combinational logic
(cid:132) Conceptually, the right-hand expression is continuous... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
3in (out, in1, in2,in3);
(cid:134) bufif1 and bufif0 are tri-state buffers
(cid:132) Net represents connections between hardware elements. Nets are
declared with the keyword wire.
L3: 6.111 Spring 2006
Introductory Digital Systems Laboratory
7
always
Procedural Assignment with always
Procedural Assignment with
(cid... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
Match Assignments
MixMix--andand--Match Assignments
(cid:132) Procedural and continuous assignments can (and often do) co-exist
within a module
(cid:132) Procedural assignments update the value of reg. The value will remain
unchanged till another procedural assignment updates the variable.
This is the main differenc... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
111 Spring 2006
Introductory Digital Systems Laboratory
11
The Power of Verilog
The Power of
bit Signals
Verilog: : nn--bit Signals
(cid:132) Multi-bit signals and buses are easy in Verilog.
(cid:132) 2-to-1 multiplexer with 8-bit operands:
module mux_2_to_1(a, b, out,
outbar, sel);
input[7:0] a, b;
input sel;
outpu... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
module maybe_mux_3to1(a, b, c,
sel, out);
input [1:0] sel;
input a,b,c;
output out;
reg out;
always @(a or b or c or sel)
begin
case (sel)
2'b00: out = a;
2'b01: out = b;
2'b10: out = c;
endcase
end
endmodule
Is this a 3-to-1 multiplexer?
L3: 6.111 Spring 2006
Introductory Digital Systems Laboratory
14
Incomplete Spe... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
end
endmodule
(cid:132) …or, fully specify all
branches of conditionals and
assign all signals from all
branches
(cid:134) For each if, include else
(cid:134) For each case, include default
L3: 6.111 Spring 2006
Introductory Digital Systems Laboratory
16
Dangers of Verilog
Dangers of
: Priority Logic
Verilog: Prior... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
if (i[1]) e = 2’b01;
else if (i[2]) e = 2’b10;
else if (i[3]) e = 2’b11;
else e = 2’bxx;
end
Inferred
Result:
2’b11
2’bxx
1
0
2’b10
1
0
2’b01
1
0
2’b00
1
0
e[1:0]
i[3]
i[2]
i[1]
i[0]
(cid:132) if-else and case statements are interpreted very literally!
Beware of unintended priority logic.
L3: 6.111 Spring 2006
Introd... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
Example: A 32-bit ALU
Function Table
A[31:0]
B[31:0]
32’d1
32’d1
0 1
0 1
F[0]
+
-
*
F[2:0]
F2 F1 F0
Function
0 0 0
0 0 1
0 1 0
0 1 1
1 0 X
A + B
A + 1
A - B
A - 1
A * B
00 01 10
F[2:1]
R[31:0]
L3: 6.111 Spring 2006
Introductory Digital Systems Laboratory
20
Module Definitions
Module Definitions
2-to-1 MUX
3-to-1 MUX
m... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
// signed arithmetic later
assign prod = i0 * i1;
endmodule
L3: 6.111 Spring 2006
Introductory Digital Systems Laboratory
21
Level ALU Declaration
TopTop--Level ALU Declaration
(cid:132) Given submodules:
A[31:0]
B[31:0]
module mux32two(i0,i1,sel,out);
module mux32three(i0,i1,i2,sel,out);
module add32(i0,i1,sum);
modu... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
SE which allows
simulation at different levels including Behavioral and Post-Place-and-
Route
L3: 6.111 Spring 2006
Introductory Digital Systems Laboratory
23
More on Module Interconnection
More on Module Interconnection
(cid:132) Explicit port naming allows port mappings in arbitrary
order: better scaling for large... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
XNOR
Note distinction between ~a and !a
&a
~&
|
~|
^
AND
NAND
OR
NOR
XOR
a < b
a > b
a <= b
a >= b
a == b
a != b
a === b
a !== b
Relational
[in]equality
returns x when x
or z in bits. Else
returns 0 or 1
case
[in]equality
returns 0 or 1
based on bit by bit
comparison
L3: 6.111 Spring 2006
Introductory Digital Syst... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
4'b0000;
b = 4'b0000;
cin = 1'b0;
#50;
a = 4'b0101;
b = 4'b1010;
// sum = 1111, cout = 0
#50;
a = 4'b1111;
b = 4'b0001;
// sum = 0000, cout = 1
a = 4'b0000;
b = 4'b1111;
cin = 1'b1;
// sum = 0000, cout = 1
#50;
a = 4'b0110;
b = 4'b0001;
// sum = 1000, cout = 0
#50;
end // initial begin
endmodule // test_adder
L3: 6.111... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/65a101becabd787ab2a6514e967c300b_lec3.pdf |
Lectures 8 & 9
M/G/1 Queues
Eytan Modiano
MIT
Eytan Modiano
Slide 1
M/G/1 QUEUE
Poisson
M/G/1
General independent
Service times
• Poisson arrivals at rate λ
• Service time has arbitrary distribution with given E[X] and E[X2]
– Service times are independent and identically distributed (IID)
–
– E[service tim... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/65d9ab519ec4851af812f4b89ffaeedc_Lectures8_9.pdf |
1
j=i- N i
• E[Wi] = E[Ri] + E[X]E[Ni] = R + NQ/µ
– Here we have used PASTA property plus independent service time property
• W = R + λW/µ => W = R/(1-ρ)
– Using little’s formula
Eytan Modiano
Slide 5
What is R?
(Time Average Residual Service Time)
Residual Service
Time R(t)
X
1
X
3
X
2
X 2
X1
X
4
X... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/65d9ab519ec4851af812f4b89ffaeedc_Lectures8_9.pdf |
i
Eytan Modiano
Slide 7
E[Wi] = E[Ri] + E[X]E[Ni] = R + NQ/µ = R/(1-ρ)
Average Residual Service Time
(with vacations)
Residual Service
Time R(t)
X
1
V
1
X
2
X
4
X
3
X1
X 2
V1
X4
time ->
X 3
M(t)
R = [R(t)]=
t
0
1
t
L(t)
2
R( τ )d τ = 1 (� X
i + �
t
i=1
2
j=1
V
2
j )
2
R =
lim
t →∞
E[... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/65d9ab519ec4851af812f4b89ffaeedc_Lectures8_9.pdf |
1/µ
Each slot = one packet transmission time = 1/µ
• Transmission can begin only at start of a slot
•
If system is empty at the start of a slot, server not available for the
duration of the slot (vacation)
• E[X] = E[v] = 1/µ
• E[X2] = E[v2] = 1/µ2
W =
λ / µ2
2(1 − λ/µ)
+
1/ µ2
2 /µ
=
λ / µ
2(µ− λ)
+
1... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/65d9ab519ec4851af812f4b89ffaeedc_Lectures8_9.pdf |
time units when there is nothing to transmit
E[V] = m; E[V2] = m2.
WSFDM = WFDM + E[V2]/2E[V]
= WFDM + m/2
Eytan Modiano
Slide 12
TDM EXAMPLE
TDM
Frame
slot m
slot 1
slot 2
. . .
slot m
• TDM with one packet slots is the same (a session has to wait for
its own slot boundary), so
W = R/(1-ρ)
R = λ=
E[X2] ... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/65d9ab519ec4851af812f4b89ffaeedc_Lectures8_9.pdf |
6.852: Distributed Algorithms
Fall, 2009
Class 7
Today’s plan
z Asynchronous systems
z Formal model
I/O automata
Executions and traces
Operations: composition, hiding
Properties and proof methods:
Invariants
Simulation relations
z Reading: Chapter 8
• Next:
– Asynchronous network algorithms: Leader elec... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/666ea5c2d621cd26c6badf37a750d290_MIT6_852JF09_lec07.pdf |
model.
• Based on earlier algorithm [Dwork, Lynch, Stockmeyer].
• Algorithm idea:
– Processes use unreliable leader election subalgorithm to choose
coordinator, who tries to achieve consensus.
– Coordinator decides based on active support from majority of processes.
– Does not assume anything based on not receivin... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/666ea5c2d621cd26c6badf37a750d290_MIT6_852JF09_lec07.pdf |
Automata
z General mathematical modeling framework for reactive
components.
Little structure---must add structure to specialize it for networks,
shared-memory systems,…
z Designed for describing systems in a modular way:
Supports description of individual system components, and how they
compose to yield a larger... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/666ea5c2d621cd26c6badf37a750d290_MIT6_852JF09_lec07.pdf |
Used to define fairness (give turns to all tasks).
Needed to guarantee liveness properties (e.g., the system keeps
making progress, or eventually terminates).
Channel automaton
send(m)
C
receive(m)
z Reliable unidirectional FIFO channel between
two processes.
Fix message alphabet M.
z signature
input a... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/666ea5c2d621cd26c6badf37a750d290_MIT6_852JF09_lec07.pdf |
S s2 S s3 S s4 S s5 ... (if finite, ends in state)
s0 is a start state
(si, Si, si+1) is a step (i.e., in trans)
Ȝ, send(a), a, send(b), ab, receive(a), b, receive(b), Ȝ
Execution fragments
z An I/O automaton executes as follows:
Start at some start state.
Repeatedly take step from current state t... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/666ea5c2d621cd26c6badf37a750d290_MIT6_852JF09_lec07.pdf |
ynchronize on actions”).
• Composing finitely many or countably infinitely many
automata Ai, i I:
• Need compatibility conditions:
– Internal actions aren’t shared:
• int(Ai) acts(Aj) =
– Only one automaton controls each output:
• out(Ai) out(Aj) =
– But output of one automaton can be an input of one or ... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/666ea5c2d621cd26c6badf37a750d290_MIT6_852JF09_lec07.pdf |
“looks good” to each component, it
is good overall.
z Substitutivity
Can replace a component with one that
implements it.
Composition: Basic results
Theorem 1: Projection
If D execs(3 Ai) then D|Ai execs(Ai) for every i.
If E traces(3 Ai) then E|Ai traces(Ai) for every i.
Composition: Basic resul... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/666ea5c2d621cd26c6badf37a750d290_MIT6_852JF09_lec07.pdf |
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