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Python logic

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Title: Python logic


1
Tell me what you do with witches? Burn And what
do you burn apart from witches? More witches!
Shh! Wood! So, why do witches burn? pause
B--... 'cause they're made of... wood? Good!
Heh heh. Oh, yeah. Oh. So, how do we tell
whether she is made of wood? . Does wood sink
in water? No. No. No, it floats! It floats!
Throw her into the pond! The pond! Throw her
into the pond! What also floats in water?
Bread! Apples! Uh, very small rocks! ARTHUR
A duck! CROWD Oooh. BEDEVERE Exactly. So,
logically... VILLAGER 1 If... she... weighs...
the same as a duck,... she's made of
wood. BEDEVERE And therefore? VILLAGER 2 A
witch! VILLAGER 1 A witch!
Python logic
2
Problematic scenarios for hill-climbing
Solution(s) ? Random restart hill-climbing
? Do the non-greedy thing with some
probability pgt0 ? Use simulated annealing
Ridges
  • When the state-space landscape has
  • local minima, any search that moves
  • only in the greedy direction cannot be
  • (asymptotically) complete
  • Random walk, on the other hand, is
  • asymptotically complete
  • Idea Put random walk into greedy hill-climbing

3
The middle ground between hill-climbing and
systematic search
  • Hill-climbing has a lot of freedom in deciding
    which node to expand next. But it is incomplete
    even for finite search spaces.
  • Good for problems which have solutions, but the
    solutions are non-uniformly clustered.
  • Systematic search is complete (because its search
    tree keeps track of the parts of the space that
    have been visited).
  • Good for problems where solutions may not exist,
  • Or the whole point is to show that there are no
    solutions (e.g. propositional entailment problem
    to be discussed later).
  • or the state-space is densely connected (making
    repeated exploration of states a big issue).

Smart idea Try the middle ground between the two?
4
Tabu Search
  • A variant of hill-climbing search that attempts
    to reduce the chance of revisiting the same
    states
  • Idea
  • Keep a Tabu list of states that have been
    visited in the past.
  • Whenever a node in the local neighborhood is
    found in the tabu list, remove it from
    consideration (even if it happens to have the
    best heuristic value among all neighbors)
  • Properties
  • As the size of the tabu list grows, hill-climbing
    will asymptotically become non-redundant (wont
    look at the same state twice)
  • In practice, a reasonable sized tabu list (say
    100 or so) improves the performance of hill
    climbing in many problems

Hill climbing ? O(1) space complexity! ? but
has no termination or completeness
guarantee (because it doesnt know
where it has been, it can loop even in
finite search spaces)
5
Making Hill-Climbing Asymptotically Complete
  • Random restart hill-climbing
  • Keep some bound B. When you made more than B
    moves, reset the search with a new random initial
    seed. Start again.
  • Getting random new seed in an implicit search
    space is non-trivial!
  • In 8-puzzle, if you generate a random state by
    making random moves from current state, you are
    still not truly random (as you will continue to
    be in one of the two components)
  • biased random walk Avoid being greedy when
    choosing the seed for next iteration
  • With probability p, choose the best child but
    with probability (1-p) choose one of the children
    randomly
  • Use simulated annealing
  • Similar to the previous ideathe probability p
    itself is increased asymptotically to one (so you
    are more likely to tolerate a non-greedy move in
    the beginning than towards the end)

With random restart or the biased random walk
strategies, we can solve very large problems
million queen problems in under minutes!
6
Ideas for improving convergence -- Random
restart hill-climbing After every N
iterations, start with a completely
random assignment --Probabilistic
greedy -with probability p do what
the greedy strategy suggests -with
probability (1-p) pick a random variable
and change its value randomly
-- p can increase as the search
progresses
A greedier version of the above (pick both the
best var and val) For each variable v, let
l(v) be the value that it can take so that
the number of conflicts are minimized. Let n(v)
be the number of conflicts with this value.
--Pick the variable v with the
lowest n(v) value. --Assign it the
value l(v)
1
2
This one basically searches the 1-neighborhood of
the current assignment (where k-neighborhood is
all assignments that differ from the current
assignment in atmost k-variable values)
7
Model-checking by Stochastic Hill-climbing
Clauses 1. (p,s,u) 2. (p, q) 3. (q, r)
4. (q,s,t) 5. (r,s) 6. (s,t) 7. (s,u)
Applying min-conflicts idea to Satisfiability
  • Start with a model (a random t/f assignment to
    propositions)
  • For I 1 to max_flips do
  • If model satisfies clauses then return model
  • Else clause a randomly selected clause from
    clauses that is false in model
  • With probability p whichever symbol in clause
    maximizes the number of satisfied clauses
    /greedy step/
  • With probability (1-p) flip the value in model of
    a randomly selected symbol from clause /random
    step/
  • Return Failure

Consider the assignment all false -- clauses
1 (p,s,u) 5 (r,s) are violated --Pick
onesay 5 (r,s) if we flip r, 1 (remains)
violated if we flip s, 4,6,7 are violated
So, greedy thing is to flip r we get all
false, except r otherwise, pick either
randomly
Remarkably good in practice!! --So good that
people startedwondering if there actually are any
hard problems out there
8
If most sat problems are easy, then exactly
where are the hard ones?
?
9
Hardness of 3-sat as a function of
clauses/variables
Probability that there is a satisfying
assignment
Cost of solving (either by finding a solution
or showing there aint one)
4.3
clauses/variables
10
Phase Transition in SAT
Theoretically we only know that phase transition
ratio occurs between 3.26 and 4.596.
Experimentally, it seems to be close to 4.3 (We
also have a proof that 3-SAT has sharp threshold)
11
Progress in nailing the bound.. (just FYI)
http//www.ipam.ucla.edu/publications/ptac2002/pta
c2002_dachlioptas_formulas.pdf
12
Beam search for Hill-climbing
  • Hill climbing, as described, uses one seed
    solution that is continually updated
  • Why not use multiple seeds?
  • Stochastic hill-climbing uses multiple seeds (k
    seeds kgt1). In each iteration, the neighborhoods
    of all k seeds are evaluated. From the
    neighborhood, k new seeds are selected
    probabilistically
  • The probability that a seed is selected is
    proportional to how good it is.
  • Not the same as running k hill-climbing searches
    in parallel
  • Stochastic hill-climbing is sort of almost
    close to the way evolution seems to work with one
    difference
  • Define the neighborhood in terms of the
    combination of pairs of current seeds (Sexual
    reproduction Crossover)
  • The probability that a seed from current
    generation gets to mate to produce offspring in
    the next generation is proportional to the seeds
    goodness
  • To introduce randomness do mutation over the
    offspring
  • This type of stochastic beam-search hillclimbing
    algorithms are called Genetic algorithms.
  • Genetic algorithms limit number of matings to
    keep the num seeds the same

13
Illustration of Genetic Algorithms in Action
Very careful modeling needed so the things
emerging from crossover and mutation are
still potential seeds (and not monkeys
typing Hamlet) Is the genetic metaphor
really buying anything?
14
Hill-climbing in continuous search spaces
Example cube root Finding using newton- Raphson
approximation
  • Gradient descent (that you study in calculus of
    variations) is a special case of hill-climbing
    search applied to continuous search spaces
  • The local neighborhood is defined in terms of the
    gradient or derivative of the error function.
  • Since the error function gradient will be zero
    near the minimum, and higher farther from it, you
    tend to take smaller steps near the minimum and
    larger steps farther away from it. just as you
    would want
  • Gradient descent is guranteed to converge to the
    global minimum if alpha (see on the right) is
    small, and the error function is uni-modal
    (I.e., has only one minimum).
  • Versions of gradient-descent algorithms will be
    used in neuralnetwork learning.
  • Unfortunately, the error function is NOT unimodal
    for multi-layer neural networks. So, you will
    have to change the gradient descent with ideas
    such as simulated annealing to increase the
    chance of reaching global minimum.

Err x3-a
a1/3
xo
X?
Tons of variations based on how alpha is set
15
Origins of gradient descentNewton-Raphson
applied to function minimization
  • Newton-Raphson method is used for finding roots
    of a polynomial
  • To find roots of g(x), we start with some value
    of x and repeatedly do
  • x ? x g(x)/g(x)
  • To minimize a function f(x), we need to find the
    roots of the equation f(x)0
  • X ? x f(x)/f(x)
  • If x is a vector then
  • X ? x f(x)/f(x)

Because hessian is costly to Compute (will have
n2 double Derivative entries for an
n-dimensional vector), we try approximations
f(x)
D
Hf(x)
16
Between Hill-climbing and systematic search
  • You can reduce the freedom of hill-climbing
    search to make it more complete
  • Tabu search
  • You can increase the freedom of systematic search
    to make it more flexible in following local
    gradients
  • Random restart search

17
Tabu Search
  • A variant of hill-climbing search that attempts
    to reduce the chance of revisiting the same
    states
  • Idea
  • Keep a Tabu list of states that have been
    visited in the past.
  • Whenever a node in the local neighborhood is
    found in the tabu list, remove it from
    consideration (even if it happens to have the
    best heuristic value among all neighbors)
  • Properties
  • As the size of the tabu list grows, hill-climbing
    will asymptotically become non-redundant (wont
    look at the same state twice)
  • In practice, a reasonable sized tabu list (say
    100 or so) improves the performance of hill
    climbing in many problems

18
Random restart search
  • Because of the random permutation, every time
    the search is restarted, you are likely to follow
    different paths through the search tree. This
    allows you to recover from the bad initial moves.
  • The higher the cutoff value the lower the amount
    of restarts (and thus the lower the freedom to
    explore different paths).
  • When cutoff is infinity, random restart search is
    just normal depth-first searchit will be
    systematic and complete
  • For smaller values of cutoffs, the search has
    higher freedom, but no guarantee of completeness
  • A strategy to guarantee asymptotic completeness
  • Start with a low cutoff value, but keep
    increasing it as time goes on.
  • Random restart search has been shown to be very
    good for problems that have a reasonable
    percentage of easy to find solutions (such
    problems are said to exhibit heavy-tail
    phenomenon). Many real-world problems have this
    property.
  • Variant of depth-first search where
  • When a node is expanded, its children are first
    randomly permuted before being introduced into
    the open list
  • The permutation may well be a biased random
    permutation
  • Search is restarted from scratch anytime a
    cutoff parameter is exceeded
  • There is a Cutoff (which may be in terms of
    of backtracks, of nodes expanded or amount of
    time elapsed)

19
Tell me what you do with witches? Burn And what
do you burn apart from witches? More witches!
Shh! Wood! So, why do witches burn? pause
B--... 'cause they're made of... wood? Good!
Heh heh. Oh, yeah. Oh. So, how do we tell
whether she is made of wood? . Does wood sink
in water? No. No. No, it floats! It floats!
Throw her into the pond! The pond! Throw her
into the pond! What also floats in water?
Bread! Apples! Uh, very small rocks! ARTHUR
A duck! CROWD Oooh. BEDEVERE Exactly. So,
logically... VILLAGER 1 If... she... weighs...
the same as a duck,... she's made of
wood. BEDEVERE And therefore? VILLAGER 2 A
witch! VILLAGER 1 A witch!
Python logic
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Representation
Reasoning
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Facts Objects relations
FOPC
Prob FOPC
Ontological commitment
Prob prop logic
Prop logic
facts
t/f/u
Deg belief
Epistemological commitment
Assertions t/f
24
Think of a sentence as the stand-in for a set of
worlds (where it is true)
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Proof by model checking
KBa
False False False False False False False False
So, to check if KB entails a, negate a, add it
to the KB, try to show that the resultant
(propositional) theory has no solutions (must
have to use systematic methods)
33
Connection between Entailment and Satisfiability
  • The Boolean Satisfiability problem is closely
    connected to Propositional entailment
  • Specifically, propositional entailment is the
    conjugate problem of boolean satisfiability
    (since we have to show that KB f has no
    satisfying model to show that KB f)
  • Of late, our ability to solve very large scale
    satisfiability problems has increased quite
    significantly

34
Entailment Satisfiability
  • SAT (boolean satisfiability) problem
  • Given a set of propositions
  • And a set of (CNF) clauses
  • Find a model (an assignment of t/f values to
    propositions) that satisfies all clauses
  • k-SAT is a SAT problem where all clauses are
    length less than or equal to k
  • SAT is NP-complete
  • 1-SAT and 2-SAT are polynomial
  • k-SAT for kgt 2 is NP-complete (so 3-SAT is the
    smallest k-SAT that is NP-Complete)
  • If we have a procedure for solving SAT problems,
    we can use it to compute entailment
  • If the sentence S is entailed, if negation of S,
    when added to the KB, gives a SAT theory that is
    unsatisfiable (NO MODEL)
  • CO-NP-Complete
  • SAT is useful for modeling many other
    assignment problems
  • We will see use of SAT for planning it can also
    be used for Graph coloring, n-queens, Scheduling
    and Circuit verification etc (the last thing
    makes SAT VERY interesting for Electrical
    Engineering folks)
  • Our ability to solve very large scale SAT
    problems has increased quite phenomenally in the
    recent years
  • We can solve SAT instances with millions of
    variables and clauses very easily
  • To use this technology for inference, we will
    have to consider systematic SAT solvers.

35
Davis-Putnam-Logeman-Loveland Procedure
?detect failure
36
DPLL Example
Pick p set ptrue unit propagation
(p,s,u) satisfied (remove) p(p,q) ? q
derived set qT (p,q) satisfied
(remove) (q,s,t) satisfied (remove)
q(q,r)?r derived set rT (q,r)
satisfied (remove) (r,s) satisfied
(remove) pure literal elimination in
all the remaining clauses, s occurs negative
set sTrue (i.e. sFalse) At this point
all clauses satisfied. Return
pT,qTrTsFalse
Clauses (p,s,u) (p, q) (q, r) (q,s,t)
(r,s) (s,t) (s,u)
37
Lots of work in SAT solvers
  • DPLL was the first (late 60s)
  • Circa 1994 came GSAT (hill climbing search for
    SAT)
  • Circa 1997 came SATZ
  • Circa 1998-99 came RelSAT
  • 2000 came CHAFF
  • Current best can be found at
  • http//www.satlive.org/SATCompetition/2003/results
    .html

38
Inference rules
Kb true but theorem not true ?
  • Sound (but incomplete)
  • Modus Ponens
  • AgtB, A B
  • Modus tollens
  • AgtB,B A
  • Abduction (??)
  • A gt B,A B
  • Chaining
  • AgtB,BgtC AgtC
  • Complete (but unsound)
  • Python logic

A B AgtB KB A
T T T F F
T F F F F
F T T F T
F F T T T
How about SOUND COMPLETE? --Resolution
(needs normal forms)
39
Need something that does case analysis
If WMDs are found, the war is justified
WgtJ If WMDs are not found, the war is still
justified WgtJ Is the war justified anyway?
J? Can Modus Ponens derive it?
40
Need something that does case analysis
If WMDs are found, the war is justified
WgtJ If WMDs are not found, the war is still
justified WgtJ Is the war justified anyway?
J? Can Modus Ponens derive it?
41
Modus ponens, Modus Tollens etc are special
cases of resolution!
Forward apply resolution steps until the
fact f you want to prove appears as a resolvent
Backward (Resolution Refutation) Add negation
of the fact f you want to derive to KB
apply resolution steps until you derive an
empty clause
42
If WMDs are found, the war is justified W V
J If WMDs are not found, the war is still
justified W V J Is the war justified anyway?
J?
43
Resolution does case analysis
If WMDs are found, the war is justified W V
J If WMDs are not found, the war is still
justified W V J Either WMDs are found or they
are not found W V W Is the war justified
anyway? J?
44
Aka the product of sums form From CSE/EEE 120
Aka the sum of products form
Prolog without variables and without the cut
operator Is doing horn-clause theorem proving
For any KB in horn form, modus ponens is a
sound and complete inference
45
Conversion to CNF form
ANY propositional logic sentence can be converted
into CNF form Try (PQ)gt(R V W)
  • CNF clause Disjunction of literals
  • Literal a proposition or a negated proposition
  • Conversion
  • Remove implication
  • Pull negation in
  • Use demorgans laws to distribute disjunction over
    conjunction
  • Separate conjunctions
  • into clauses

46
Need for resolution
Resolution does case analysis
Yankees win, it is Destiny YVD Dbacks win,
it is Destiny Db V D Yankees or Dbacks win
Y V Db Is it Destiny either way? D? Can
Modus Ponens derive it? Not until Sunday, when
Db won
47
Solving problems using propositional logic
  • Need to write what you know as propositional
    formulas
  • Theorem proving will then tell you whether a
    given new sentence will hold given what you know
  • Three kinds of queries
  • Is my knowledge base consistent? (i.e. is there
    at least one world where everything I know is
    true?) Satisfiability
  • Is the sentence S entailed by my knowledge base?
    (i.e., is it true in every world where my
    knowledge base is true?)
  • Is the sentence S consistent/possibly true with
    my knowledge base? (i.e., is S true in at least
    one of the worlds where my knowledge base holds?)
  • S is consistent if S is not entailed
  • But cannot differentiate between degrees of
    likelihood among possible sentences

48
Steps in Resolution Refutation
Is there search in inference? Yes!! Many
possible inferences can be done Only few are
actually relevant --Idea Set of Support
At least one of the resolved
clauses is a goal clause, or
a descendant of a clause
derived from a goal clause -- Used in the
example here!!
  • Consider the following problem
  • If the grass is wet, then it is either raining or
    the sprinkler is on
  • GW gt R V SP GW V R V SP
  • If it is raining, then Timmy is happy
  • R gt TH R V TH
  • If the sprinklers are on, Timmy is happy
  • SP gt TH SP V TH
  • If timmy is happy, then he sings
  • TH gt SG TH V SG
  • Timmy is not singing
  • SG SG
  • Prove that the grass is not wet
  • GW? GW

49
Search in Resolution
  • Convert the database into clausal form Dc
  • Negate the goal first, and then convert it into
    clausal form DG
  • Let D Dc DG
  • Loop
  • Select a pair of Clauses C1 and C2 from D
  • Different control strategies can be used to
    select C1 and C2 to reduce number of resolutions
    tries
  • Idea 1 Set of Support At least one of C1 or C2
    must be either the goal clause or a clause
    derived by doing resolutions on the goal clause
    (COMPLETE)
  • Idea 2 Linear input form Atleast one of C1 or
    C2 must be one of the clauses in the input KB
    (INCOMPLETE)
  • Resolve C1 and C2 to get C12
  • If C12 is empty clause, QED!! Return Success (We
    proved the theorem )
  • D D C12
  • End loop
  • If we come here, we couldnt get empty clause.
    Return Failure
  • Finiteness is guaranteed if we make sure that
  • we never resolve the same pair of clauses more
    than once AND
  • we use factoring, which removes multiple copies
    of literals from a clause (e.g. QVPVP gt QVP)

50
Mad chase for empty clause
  • You must have everything in CNF clauses before
    you can resolve
  • Goal must be negated first before it is converted
    into CNF form
  • Goal (the fact to be proved) may become converted
    to multiple clauses (e.g. if we want to prove P V
    Q, then we get two clauses P Q to add to the
    database
  • Resolution works by resolving away a single
    literal and its negation
  • PVQ resolved with P V Q is not empty!
  • In fact, these clauses are not inconsistent (P
    true and Q false will make sure that both clauses
    are satisfied)
  • PVQ is negation of P Q. The latter will
    become two separate clauses--P , Q. So, by
    doing two separate resolutions with these two
    clauses we can derive empty clause

51
Complexity of Propositional Inference
  • Any sound and complete inference procedure has to
    be Co-NP-Complete (since model-theoretic
    entailment computation is Co-NP-Complete (since
    model-theoretic satisfiability is NP-complete))
  • Given a propositional database of size d
  • Any sentence S that follows from the database by
    modus ponens can be derived in linear time
  • If the database has only HORN sentences
    (sentences whose CNF form has at most one ve
    clause e.g. A B gt C), then MP is complete for
    that database.
  • PROLOG uses (first order) horn sentences
  • Deriving all sentences that follow by resolution
    is Co-NP-Complete (exponential)
  • Anything that follows by unit-resolution can be
    derived in linear time.
  • Unit resolution At least one of the clauses
    should be a clause of length 1

52
Example
  • Pearl lives in Los Angeles. It is a high-crime
    area. Pearl installed a burglar alarm. He asked
    his neighbors John Mary to call him if they
    hear the alarm. This way he can come home if
    there is a burglary. Los Angeles is also
    earth-quake prone. Alarm goes off when there is
    an earth-quake.
  • Burglary gt Alarm
  • Earth-Quake gt Alarm
  • Alarm gt John-calls
  • Alarm gt Mary-calls
  • If there is a burglary, will Mary call?
  • Check KB E M
  • If Mary didnt call, is it possible that Burglary
    occurred?
  • Check KB M doesnt entail B

53
Example (Real)
  • Pearl lives in Los Angeles. It is a high-crime
    area. Pearl installed a burglar alarm. He asked
    his neighbors John Mary to call him if they
    hear the alarm. This way he can come home if
    there is a burglary. Los Angeles is also
    earth-quake prone. Alarm goes off when there is
    an earth-quake.
  • Pearl lives in real world where (1) burglars can
    sometimes disable alarms (2) some earthquakes may
    be too slight to cause alarm (3) Even in Los
    Angeles, Burglaries are more likely than Earth
    Quakes (4) John and Mary both have their own
    lives and may not always call when the alarm goes
    off (5) Between John and Mary, John is more of a
    slacker than Mary.(6) John and Mary may call even
    without alarm going off
  • Burglary gt Alarm
  • Earth-Quake gt Alarm
  • Alarm gt John-calls
  • Alarm gt Mary-calls
  • If there is a burglary, will Mary call?
  • Check KB E M
  • If Mary didnt call, is it possible that Burglary
    occurred?
  • Check KB M doesnt entail B
  • John already called. If Mary also calls, is it
    more likely that Burglary occurred?
  • You now also hear on the TV that there was an
    earthquake. Is Burglary more or less likely now?

54
Example (Real)
  • Pearl lives in Los Angeles. It is a high-crime
    area. Pearl installed a burglar alarm. He asked
    his neighbors John Mary to call him if they
    hear the alarm. This way he can come home if
    there is a burglary. Los Angeles is also
    earth-quake prone. Alarm goes off when there is
    an earth-quake.
  • Pearl lives in real world where (1) burglars can
    sometimes disable alarms (2) some earthquakes may
    be too slight to cause alarm (3) Even in Los
    Angeles, Burglaries are more likely than Earth
    Quakes (4) John and Mary both have their own
    lives and may not always call when the alarm goes
    off (5) Between John and Mary, John is more of a
    slacker than Mary.(6) John and Mary may call even
    without alarm going off
  • Burglary gt Alarm
  • Earth-Quake gt Alarm
  • Alarm gt John-calls
  • Alarm gt Mary-calls
  • If there is a burglary, will Mary call?
  • Check KB E M
  • If Mary didnt call, is it possible that Burglary
    occurred?
  • Check KB M doesnt entail B
  • John already called. If Mary also calls, is it
    more likely that Burglary occurred?
  • You now also hear on the TV that there was an
    earthquake. Is Burglary more or less likely now?

55
How do we handle Real Pearl?
  • Eager way
  • Model everything!
  • E.g. Model exactly the conditions under which
    John will call
  • He shouldnt be listening to loud music, he
    hasnt gone on an errand, he didnt recently have
    a tiff with Pearl etc etc.
  • A c1 c2 c3 ..cn gt J
  • (also the exceptions may have interactions
  • c1c5 gt c9 )
  • Ignorant (non-omniscient) and Lazy
    (non-omnipotent) way
  • Model the likelihood
  • In 85 of the worlds where there was an alarm,
    John will actually call
  • How do we do this?
  • Non-monotonic logics
  • certainty factors
  • probability theory?

Qualification and Ramification problems make
this an infeasible enterprise
56
Probabilistic Calculus to the Rescue
  • Suppose we know the likelihood
  • of each of the (propositional) worlds (aka Joint
    Probability distribution )
  • Then we can use standard rules of probability to
    compute the likelihood of all queries (as I will
    remind you)
  • So, Joint Probability Distribution is all that
    you ever need!
  • In the case of Pearl example, we just need the
    joint probability distribution over B,E,A,J,M (32
    numbers)
  • --In general 2n separate numbers (which should
    add up to 1)

Only 10 (instead of 32) numbers to specify!
  • If Joint Distribution is sufficient for
    reasoning, what is domain knowledge supposed to
    help us with?
  • --Answer Indirectly by helping us specify
    the joint probability distribution with fewer
    than 2n numbers
  • ---The local relations between propositions
    can be seen as constraining the form the joint
    probability distribution can take!

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If BgtA then P(AB) ? P(BA) ?
P(BA) ?
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