Homework 3 returned

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10/28
Where hard problems are
Phase Transition

• Homework 3 returned
• Homework 4 socket opened
• (No office hours today)

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Davis-Putnam-Logeman-Loveland Procedure
?detect failure
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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)
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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)

• 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!!
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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)
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Progress in nailing the bound.. (just FYI)
Not discussed in class
http//www.ipam.ucla.edu/publications/ptac2002/pta
c2002_dachlioptas_formulas.pdf
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A longer proof for 5.19 bound (FYI)
Probability that a 3-sat clause is satisfied is
7/8 3 vars 23 possible assignments only one
is all false
P(X gt tm) lt 1/t
Not discussed in class
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An easy upper bound for 3-sat transition
(optional)

• Suppose there are n variables and c clauses
• Probability that a random assignment satisfied a
  clause if 7/8
• Each clause contains 3 variables so 23 possible
  assignments for the variables. Of these, just
  one, the all false one, makes the clause false
• Probability that all c clauses are satisfied is
  (7/8)c (assuming clause independence holds when
  ngtgt3)
• There are 2n possible random assignments. So, the
  number of assignments which will satisfy the
  entire 3SAT instance is 2n (7/8)c This is the
  expected number of satisfying assignments
• We want to know when the expected num of
  satisfying assignments becomes less than 1 (i.e.,
  unsatisfiable)
• 2n (7/8)c lt 1
• n c log2 7/8 lt 0
• Taking log to the base 2 on both sides
• n lt - c log2 7/8
• n lt c log2 8/7
• c/n gt 1/log2 8/7 1/0.1926 5.1921

Not discussed in class
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CSPSAT with multi-valued variables

• It is easy to generalize the SAT problem to
  handle non-boolean variables
• CSP problem
• Given a set of discrete variables and their
  domains
• And a set of constraints (expressed as legal
  value combinations that can be taken by various
  subsets of the variables)
• Clausal constraints (such as pgtq ) can be seen
  in this way too
• Find a model (an assignment of domain values to
  the variables) that satisfies all constraints
• SAT is a boolean CSP

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We can model any CSP problem also as a SAT
problem Consider the variables
WA-is-red WA-is-green. VD
boolean variables --but this leads to some
loss of structure
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Planning as CSP

• Variables propositions at each level
• Domains the actions supporting them at that
  level
• NULL value
• Goal constraints Certain
• Constraints
• Activity constraints

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We will mostly talk about discrete domain
variables
All CSPs can be compiled to binary CSPs (by
introducing additional variables) ?Most
early work on CSP concentrated on
binary CSPs
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10/30

• Counseling today during office hours

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Backtracking Search
As is the case for SAT, basic search For CSP is
in the space of partial Assignments --extend the
assignment by selecting a variable and
considering all its values (in different
branches) --prune any (even partial) assignment
if it violates a constraint
Red Violates
Most of the SAT improvements work for CSP too.
Main differences are --Variable selection
heuristics CSP solvers typically consider
most constrained variable first heuristic
(i.e., heuristic with the smallest
domain) --Value selection heuristics CSP
solvers consider least constraining value first
heuristic -- Lookahead Instead of unit
propagation, CSP solvers use variety of
constraint propagation algorithms (forward
checking, arc-consistency, path consistency)
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Variable Ordering Strategies
Notice that for boolean CSPs (SAT), most
constrained variable heuristics are less
effective (since all variables have domains of
size 2 (normal), 1 (have a specific value) or 0
(backtrack)
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Value Ordering Heuristics
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Lookahead
Variable ordering can be improved in the presence
of FC DVO (Dynamic variable ordering)
--Consider the variable with the smallest
remaining domain next
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K-consistency

• Forward checking is a weak form of look-ahead
• --stronger look-ahead are based on the notion
  of
• enforcing k-consistency
• A CSP is k-consistent, if any legal assignment of
  values for
• any k-1 sized subset of variables can be
  extended to cover
• any kth variable.
• If a CSP is n-consistent, then it can be
  solved without
• backtracking! Make sure you get this
• Enforcing higher than 2 or 3 consistency is
  rarely worth it

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2-consistency
Forward checking will stop thinking everyone
has non-empty domains. However, NT is
blue if you propagate that we know that SA
cannot be blue. This means SA is empty domain
Graphplan mutex propagation can be Seen as a form
of 3-consistency Enforcement..
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Consistency enforcement as inference
2-consistency
A1,2 B1,2 AltB A1 V A2 B1 V B2 (A1) V
(B1) (A2) V (B1) (A2) V (B2)
One of the resolvers is Derived from As
domain Constraint. The other is a Inter-variable
constraint of Size 2
A1,2 B1,2 AltB Currently, B2 A1 V A2 B1
V B2 (A1) V (B1) (A2) V (B1) (A2) V
(B2) B2
Forward Checking
(A2)
1-level unit resolution
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Inference/Satisfaction (Conditioning) Duality
Satisfaction Conditioning
Inference
Try to split cases (disjunction) into search
tree (by committing)
satisfaction
Inference/ Theorem Proving
Try to explicate hidden structure
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Summary of Propositional Logic

• Syntax
• Semantics (entailment)
• Entailment computation
• Model-theoretic
• Using CSP techniques
• Proof-theoretic
• Resolution refutation
• Heuristics to limit type of resolutions
• Set of support
• Connection to CSP
• K-consistency can be seen as a form of limited
  inference

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Why FOPC

• If your thesis is utter vacuous
• Use first-order predicate calculus.
• With sufficient formality
• The sheerest banality
• Will be hailed by the critics "Miraculous!"

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Tarskian Interpretations
Left-leg-of
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Inference in first order logic

• For ground sentences (i.e., sentences without
  any quantification), all the old rules work
  directly
• P(a,b)gt Q(a) P(a,b) Q(a)
• P(a,b) V Q(a) resolved with P(a,b) gives Q(a)
• What about quantified sentences?
• Universal Instantiation (a universally quantified
  statement entails every instantiation of it)
• Can we combine these (so we can avoid unnecessary
  instantiations?) Yes. Generalized modus ponens
• Needs UNIFICATION