Constraint Satisfaction Problems

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Constraint Satisfaction Problems
2/10
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Search when states are factored

• Until now, we assumed states are black-boxes.
• We will now assume that states are made up of
  state-variables and their values
• Two interesting problem classes
• CSP SAT (Constraint Satisfaction Problems)
• Planning

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Constraint Satisfaction Problems (a brief
animated overview)
X
Z
Y
Coloring Problem
Constraint Graph
Problem Statement
Values
Constraints
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Example N-queen problem
N4
Variables Queen per column Values N rows that
queen can be in Constraints
no pair in same row, column or diagonal
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Constraint Graphs will be hyper-graphs for
non-binary CSPs
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Real world CSP problems..

• Most assignment problems including
• Time-tabling
• Variables Courses Values Rooms, times
• Jobshop Scheduling
• Variables jobs values machines
• Sudoku KenKen
• Cross-word puzzle
• Boolean satisfiability

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Complexity of CSP..

• Boolean Satisfiability is a special case of
  discrete variable CSP problem
• So, CSP is NP-hard
• Specific types of CSP may be tractable.
• E.g. if all the variables are boolean and all the
  constraints are binary, you have 2-SAT which is
  tractable.
• The topology of the constraint graph also
  affects the complexity of the CSP problem
• E.g. If the constraint graph is a chain graph or
  a multi-tree, we can solve it polynomially

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How about Breadth-first search? IDDFS?
All solutions are at depth d!
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Review of CSP/SAT concepts
x,y,u,v A,B,C,D,E w D,E l A,B xA
??w?E yB ??u?D uC ??l?A vD ??l?B

• Constraint Satisfaction Problem (CSP)
• Given
• A set of variables
• (Normally, discretebut can be continuous)
• Legal domains for each of the variables
• A set of constraints on values groups of
  variables can take
• Constraints can be Unary, binary or
  multi-ary based on how many variables they
  connect
• Find an assignment of values to all the variables
  so that none of the constraints are violated
• SAT Problem CSP with boolean variables

A solution xB, yC, uD, vE, wD, lB

x
A
N
xA
1

y
B


N
x
A y
B
2

v
D


N
x
A y
B v D
3

u
C


N
x
A y
B v D u C
4

w
E

w
D


N
x
A y
B v D u C w E
5


N
x
A y
B v D u C w D
6
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Most Constrained Variable First Least-constrain
ing Value First
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2/12
I will say then that I am not, nor ever have been
in favor of bringing about in anyway the social
and political equality of the white and black
races - that I am not nor ever have been in favor
of making voters or jurors of negroes, nor of
qualifying them to hold office, nor to intermarry
with white people and I will say in addition to
this that there is a physical difference between
the white and black races which I believe will
forever forbid the two races living together on
terms of social and political equality. And
inasmuch as they cannot so live, while they do
remain together there must be the position of
superior and inferior, and I as much as any other
man am in favor of having the superior position
assigned to the white race. I say upon this
occasion I do not perceive that because the white
man is to have the superior position the negro
should be denied everything." 
-- September 18, 1858
I do not think much of a man who is not wiser
today than he was yesterday.
My paramount object in this struggle is to save
the Union, and is not either to save or to
destroy slavery. If I could save the Union
without freeing any slave I would do it, and if I
could save it by freeing all the slaves I would
do it and if I could save it by freeing some and
leaving others alone I would also do that. What I
do about slavery, and the colored race, I do
because I believe it helps to save the Union and
what I forbear, I forbear because I do not
believe it would help to save the Union. I shall
do less whenever I shall believe what I am doing
hurts the cause, and I shall do more whenever I
shall believe doing more will help the cause.
---August 22, 1862
(born 2/12/1809)
In giving freedom to the slave, we assure freedom
to the free - honorable alike in what we give,
and what we preserve. We shall nobly save, or
meanly lose, the last best hope of earth. Other
means may succeed this could not fail. The way
is plain, peaceful, generous, just - a way which,
if followed, the world will forever applaud, and
God must forever bless. ---December,
1, 1862.
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It is not the strongest of the species that
survives, nor the most intelligent that survives.
It is the one that is the most adaptable to
change.
The universe we observe has precisely the
properties we should expect if there is, at
bottom, no design, no purpose, no evil, no good,
nothing but blind, pitiless indifference
(b 2/12/1809)
We can allow satellites, planets, suns,
universe, nay whole systems of universes, to be
governed by laws, but the smallest insect, we
wish to be created at once by special act.
We must, however, acknowledge, as it seems to
me, that man with all his noble qualities...
still bears in his bodily frame the indelible
stamp of his lowly origin.
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They brought.. the change from soul to mind as
the engine of existence, and then from angels to
ages as the overseers of life
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General Search vs. CSP

• Blackbox State
• External Child-generator
• State-space can be infinite
• External goal test
• Goals can occur at any depth
• Goals can have different costs
• All the search algorithms we discussed until now
  are appropriate.
• Heuristics are aimed at estimating the cost to
  goal node..

• State is made-up of state variables
• Children generation involves assigning values to
  more variables
• State space is finite
• A state is a goal state if all variables are
  assigned and no constraints are violated
• All goals occur at the same depth
• In the basic formulation, all goals have the same
  cost
• This can be generalized
• Only the Depth-first search makes sense!
• Heuristics are aimed at picking the right
  variable to assign next, and deciding the right
  value to assign to it.

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y
n
y
y
n
n
n
n
n
Dynamic variable ordering Pick the variable with
the smallest live (remaining) domain next.
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Why are CSP problems hard?

• Because what seems like a locally good decision
  (value assignment to a variable), may wind up
  leading to global inconsistency
• But what if we pre-process the CSP problem such
  that the local choices are more likely to be
  globally consistent?
• Two CSP problems CSP1 and CSP2 are considered
  equivalent if both of them have the same
  solutions.

Related to the way artificial potential fields
can be set up for improving hill-climbing..
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Pre-processing to enforce consistency
Special terminology for binary CSPs
2-consistency is called Arc
consistency (since you need only considers
pairs of variables connected by an edge
in the constraint graph) 3-consistency is
called path consistency

• An n-variable CSP problem is said to be
  k-consistent iff every consistent assignment for
  (k-1) of the n variables can be extended to
  include any k-th variable
• Strongly k-consistent if it is j-consistent for
  all j from 1 to k
• Higher the level of (strong) consistency of
  problem, the lesser the amount of backtracking
  required to solve the problem
• A CSP with strong n-consistency can be solved
  without any backtracking
• We can improve the level of consistency of a
  problem by explicating implicit constraints
• Enforcing k-consistency is of O(nk) complexity
• Break-even seems to be around k2 (arc
  consistency) or 3 (path consistency)

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How much consistency should we enforce?
Total cost incurred in search
Cost of enforcing the consistency
Cost of searching with the heuristic
2
3
1
n
h0
Degree of consistency (measured in
k-strong-consistency)
Overloading new semantics on an old graphic ?
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Enforcing Arc Consistency An example
When we enforce arc-consistency on the top left
CSP (shown as a constraint graph), we get the CSP
shown in the bottom left. Notice how the domains
have reduced. Here is an explanation of what
happens. Suppose we start from Z. If Z1, then Y
cant have any valid values. So, we remove 1 from
Zs domain. If Z2, or 3 then Y can have a valid
value (since Y can be 1 or 2). Now we go to Y. If
Y1, then X cant have any value. So, we remove 1
from Xs domain. If Y3, then Z cant have any
value. So, we remove 3 from Ys domain. So Y has
just 2 in its domain! Now notice that Ys domain
has changed. So, we re-consider Z (since anytime
Ys domain changes, it is possible that Zs
domain gets affected). Sure enough, Z2 no longer
works since Y can only be 2 and so it cant take
any value if Z2. So, we remove 2 also from Zs
domain. So, Zs domain is now just 3! Now, we go
to X. X cant be 2 or 3 (since for either of
those values, Y will not have a value. So, X has
domain 1! Notice that in this case,
arc-consistency SOLVES the problem, since X,Y and
Z have exactly 1 value each and that is the only
possible solution. This is not always the case
(see next example).
X1,2,3
XltY
YltZ
X1
XltY
YltZ
You can do arc-consistency Either as
pre-processing or In lieu of forward checking
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Things discussed on board

• If the constraint graph is disconnected, then you
  essentially have independent subproblems.
• For example, suppose you mixed up a coloring
  problem CSP with a queens problm CSP
• You are better off solving them separately and
  concatenating the results
• You may ask Why should I solve them separately?
  Cant my search algorithm find the independence
  itself?
• The answer is that normal search algorithms that
  do chronological backtracking are unable to
  recognize and exploit problem independence
  dynamically.
• You need dependency directed backtracking
• Another question is how to do constraint graphs
  when you have non-binary (ternary etc.)
  constraints
• When you have n-ary (ngt2) constraints, your
  constraint graph is a hyper graph (with edges
  connecting a set rather than a pair of vertices)
• It is possible to convert every non-binary CSP
  into a binary CSP (by introducing new variables.
  If there is a constraint between X, Y, and Z, I
  can introduce a super variable called x-y and
  make a binary constraint between it and Z)
• Of course, when you do this, the resulting
  constraints may not be natural for someone who
  knows the domain
• Just as an assembly language program may not make
  as much sense to a domain expert as does a
  high-level language program
• Binary CSPs and Boolean CSPs are canonical
  classes of CSP in that any arbitrary CSP can be
  compiled down to an equivalent binary or
  boolean CSP

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Not enough to show the correct configuration of
the 18-puzzle problem or rubiks cube..
?(although by including the list of actions as
part of the state, you can support
hill-climbing)
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What is needed --A neighborhood function
The larger the neighborhood you consider,
the less myopic the search (but the
more costly each iteration) --A goodness
function needs to give a value to
non-solution configurations too
for 8 queens (-ve) of number of pair-wise
conflicts
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Understand the tradeoffs in defining smaller vs.
larger neighborhood
Applying min-conflicts based hill-climbing to
8-puzzle
Local Minima
No single queen move can improve h
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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

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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)
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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!
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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

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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?
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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
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N-queens vs. Boolean Satisfiability

• Given nxn board, bind assignment of positions to
  n queens so no queen constraints are violated
• Assign Each queen can take values 1..8
  corresponding to its position in its column
• Find a complete assignment for all queens
• The approach we discussed is called
  min-conflict search which does hill climbing in
  terms of number of conflicts

• Given n boolean variables and m clauses that
  constrain the values that those variables can
  take
• Each clause is of the form
• v1, v2, v7
• Meaning that one of those must hold (either v1 is
  true or v7 is true or v2 is false)
• Find an assignment of T/F values to the n
  variables that ensures that all clauses are
  satisified
• So boolean variable is like a queen, T/F values
  are like queens positions clauses are like queen
  constraints number of violated clauses are like
  number of queen conflicts.
• You can do min-conflict search!
• Extremely useful in large-scale circuit
  verification etc.

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Consistency and Hardness

• In the worst case, a CSP can be solved
  efficiently (i.e., without backtracking) only if
  it is strongly n-consistent
• However, in many problems, enforcing
  k-consistency automatically renders the problem
  n-consistent as a side-effect
• In such a case, we can clearly see that the
  problem is solvable in O(nk) time (basically the
  time taken for pre-processing)
• The hardness of a CSP problem can be thought of
  in terms of the degree of consistency that
  needs to be enforced on that CSP before it can be
  solved efficiently (backtrack-free)

added
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Graph rectification as an analog for local
consistency in normal search

• Local consistency involves pre-processing the
  search space so later search is faster.
• One way we could do it for normal graph search is
  to do a k-lookahead from each state and revise a
  nodes actual distance from its neighbors
• Running value iteration for a few iterations has
  exactly this effect..

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Consistency enforcement as inferring implicit
constraints

• In general, enforcing consistency involves
  explicitly adding constraints that are implicit
  given the existing constraints
• E.g. In enforcing 3-consistency if we find that
  for a particular 2-label xiv1 xjv2 there is
  no possible consistent value of xk, then we write
  this as an additional constraint
• xiv1gt xj ! v2
• Domain reduction is just a special case When
  enforcing 2-consistency (or arc-consistency), the
  new constraints will be of the form xi!v1 , and
  so these can be represented by just contracting
  the domain of xi by pruning v1 from it
• Unearthing implicit constraints can also be
  interpreted as inferring new constraints that
  hold (are entailed) given the existing
  constraints
• In the context of boolean CSPs (I.e.,
  propositional satisfiability problems), the
  analogy is even more striking since unearthing
  new constraints means writing down new clauses
  (or facts) that are entailed given the existing
  clauses
• This interpretation shows that consistency
  enforcement is just a form of inference/
  entailment computation process.
• Conditioning InferenceThe Yin and Yang There
  is a general idea that in solving a search
  problem, you can interleave two different
  processes
• inference trying to either infer the solution
  itself or saying no solution exists
• conditioning or enumerationwhich attempts to
  systematically go through potential solutions
  looking for a real solution.
• Good search algorithms interleave both inference
  and conditioning
• E.g. the CSP algorithm we discussed in the class
  uses backtracking search (enumeration), and
  forward checking (inference).

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More on arc-consistency
Arc-consistency doesnt always imply that the CSP
has a solution or that there is no search
required. In the previous example, if each
variable had domains 1,2,3,4, then at the end of
enforcing arc-consistency, each variable will
still have 2 values in its domainthus
necessitating search. Here is another example
which shows that the search may find that there
is no solution for the CSP, even though it is
arc-consistent.
Here is a binary CSP that Is arc-consistent but
has no Solution.
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Approximating K-Consistency

• K-consistency enforcement takes O(nk) effort.
  Since we are doing this only to reduce the
  overall search effort (rather than to get a
  separate badge for consistency), we can cut
  corners
• Directional K-consistency Given a fixed
  permutation (order) over the n variables,
  assignment to first k-1 variables can be extended
  to the k-th variable
• Clearly cheaper than K-consistency
• If we know that the search will consider
  variables in a fixed order, then enforcing
  directional consistency w.r.t. that order is
  better.
• Partial K-consistency enforcement Run the
  K-consistency enforcement algorithm partially
  (i.e., stop before reaching fixed-point)
• Put a time-limit on the consistency computation
• Recall how we could cut corners in computing
  Pattern Database heuristics by spending only a
  limited time on the PDB and substituting other
  cheaper heuristics in their place
• Only do one pass of the consistency enforcement
• This is what forward checking does..

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Arc-Consistency gt directed arc-consistency gt
Forward Checking
gt stronger than
DACFor each variable u, we only consider The
effects on the variables that Come after u in the
ordering
After directional arc-consistency Assuming the
variable order XltYltZ
X1,2
X1,2,3
XltY
XltY
YltZ
YltZ
AC is the strongest It propagates Changes in all
directions Until we reach a fixed point (no
further changes)
After forward checking assuming XltYltZ, and X
has been set to value 1
After arc-consistency
X1
XltY
X1
XltY
FC We start with the current Assignment for some
of the Variables, and only consider
their Effects on the future variables. (only a
single level Propagation is done. After we find
that a value of Y is pruned, we dont try To
see if that changes domain of Z)
YltZ
YltZ
ADDED AFTER CLASS IMPORTANT