1. Constraint Satisfaction Problems
2/10
Constraint Satisfaction Problems

2. 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

4. Constraint Satisfaction Problems (a brief animated overview)Z X Y
X: red
Y: blue
Z: green
Red
green
blue X
Search backtracking, variable/value heuristics
Inference Consistency enforcement, forward checking Y Z
Coloring Problem
Constraint Graph
CSP Representation
Variables
Problem Statement
CSP Algorithm
Values
Solution
Constraints
December 2, 1998
Sqalli, Tutorial on Constraint Satisfaction Problems

6. Example: N-queen problem
Variables: Queen per column
Values: N rows that queen
can be in
Constraints: no pair in same
row, column or diagonal

7. Constraint Graphs
will be hyper-graphs
for non-binary CSPs

8. “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

9. 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

11. How about:
Breadth-first search?
IDDFS?
All solutions
are at depth d!

13. Review of CSP/SAT concepts
x,y,u,v: {A,B,C,D,E}
w: {D,E} l : {A,B}
x=A wE
y=B uD
u=C lA
v=D lB
Constraint Satisfaction Problem (CSP)
Given
A set of variables
(Normally, discrete—but 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:
x=B, y=C, u=D, v=E, w=D, l=B x A N
: {x=A} 1 y B
MAY BE WORTH PUTTING THIS IN TH EMIDDLE OF THE TUTORIAL__ASK FOR A SHOW OF HANDS AS TO HOW MANY KNOW CSPs. IF they don’t;, kEEP the CSP in a different window and show it to them…
MAY BE WORTH PUTTING SOME NUMBERS FROM BACCHUS’s SLIDES. 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 N
: {x=
A & y =
B & v = D & u = C & w= E } w D 5 N
: {x=
A & y =
B & v = D & u = C & w= D } 6

15. “Most Constrained Variable First”
“Least-constraining Value First”

16. 2/12 I do not think much of a man who
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
2/12
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.

17. “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.”

18. [They brought..] the change from soul to mind as the engine of existence, and then from angels to ages as the overseers of life

19. 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.

20. y n y y n n n n n
Dynamic variable ordering: Pick the variable with the
smallest “live” (remaining) domain next.

21. 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..

22. 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”
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 k=2 (“arc consistency”) or 3 (“path consistency”)
One measure of hardness of a problem is the minimum level of consistency that needs to be enforced on the problem so it becomes strongly n-consistent

23. (measured in k-strong-consistency)
How much consistency
should we enforce?
Total cost
incurred in search
Cost of enforcing the
consistency
Cost of searching
with the heuristic 1 2 3 h0 n
Degree of consistency
(measured in k-strong-consistency)
Overloading new semantics
on an old graphic 

24. 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 Z=1, then Y can’t have any valid values. So, we remove 1 from Z’s domain. If Z=2, or 3 then Y can have a valid value (since Y can be 1 or 2).
Now we go to Y. If Y=1, then X can’t have any value. So, we remove 1 from X’s domain. If Y=3, then Z can’t have any value. So, we remove 3 from Y’s domain. So Y has just 2 in its domain!
Now notice that Y’s domain has changed. So, we re-consider Z (since anytime Y’s domain changes, it is possible that Z’s domain gets affected). Sure enough, Z=2 no longer works since Y can only be 2 and so it can’t take any value if Z=2. So, we remove 2 also from Z’s domain. So, Z’s domain is now just 3!
Now, we go to X. X can’t 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).
X:{1,2,3}
X<Y
Y:{1,2,3}
Y<Z
Z:{1,2,3}
X:{1}
X<Y
Y:{2}
Y<Z
Z:{3}
You can do arc-consistency
Either as pre-processing or
In lieu of forward checking

25. 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? Can’t 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 (n>2) 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

26. Not enough to show the correct configuration
of the 18-puzzle problem or rubik’s cube..
(although by including the list of actions as part of
the state, you can support hill-climbing)

27. 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

29. Applying min-conflicts based hill-climbing to 8-puzzle
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

30. Problematic scenarios for hill-climbing
Solution(s):
 Random restart hill-climbing
 Do the non-greedy thing with
some probability p>0
 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

31. A greedier version of the above (pick both the best var and val):
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
The neighborhood
1 is subsumed by
Neighborhood 2
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)

32. 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 idea—the 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!

33. “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 k>1). 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 seed’s 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

34. 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?

35. 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

36. --didn’t discuss the remaining slides--

40. 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.

41. 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

42. 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 node’s actual distance from its neighbors
Running value iteration for a few iterations has exactly this effect..

43. 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 {xi=v1 & xj=v2} there is no possible consistent value of xk, then we write this as an additional constraint
{xi=v1}=> {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 & Inference—The 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 enumeration”which 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).

44. More on arc-consistency
Arc-consistency doesn’t 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 domain—thus 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.

45. 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..

46. Arc-Consistency > directed arc-consistency > Forward Checking
> : “stronger than”
DAC:For 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
X<Y<Z
X:{1,2}
X:{1,2,3}
X<Y
X<Y
Y:{2}
Y:{1,2,3}
Y<Z
Y<Z
AC is the strongest
It propagates Changes in
all directions Until we
reach a fixed point
(no further changes)
Z:{1,2,3}
Z:{1,2,3}
After forward checking assuming
X<Y<Z, and X has been set to value 1
After arc-consistency
X:{1}
X<Y
X:{1}
Y:{2}
X<Y
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 don’t try
To see if that changes domain of Z)
Y<Z
Y:{2,3}
Y<Z
Z:{3}
Z:{1,2,3}
ADDED AFTER CLASS
IMPORTANT