# Administrivia

Exam 1 is next Thursday (25 Sep) at 1900 in HAS 134.

HW3 is due next *Wednesday* (extended from Friday)

# Today 

- defining and representing constraint satisfaction problems (CSPs)
- CSPs as search
- being smart about search in CSPs
- variable and value ordering
- propagating information through constraints
- intelligent backtracking
- CSPs as local search

a potpourri of approaches, best approach will vary by problem: you gotta think about it!

# What is constraint satisfaction?

A family of methods for solving problems where the internal structure of solutions must satisfy a set of specified constraints

Big ideas:

- Variable and value ordering: Choose next states wisely

- Propagate information: Provide good information to your heuristics

- Intelligent backtracking: Backtrack in a smart way

# You've already done constraint satisfaction 

- your class schedule, both by semester and overall
- logic puzzles
- sudoku

# AI Jeopardy

These entities are one part of the definition of a CSP, and they define candidate solutions : Variables

These entities define the possible set of values that variables can take on: Domains

These entities are the other part of the definition of a CSP, and they define limitations to possible assignments: Constraints

# CSPs are a subset of general search

In search (BFS, A*, etc.), states are essentially opaque, and serve only as input to transition fn and evaluation fn.

In CSP, states are defined by assignments of values to a set of variables X1...Xn. Each variable Xi has a domain Di of possible values.

States are evaluated based on their consistency with a set of constraints C1...Cm over the values of the variables.

A goal state is a complete assignment to all variables that satisfies all the constraints.

# Example: Map coloring

aka the god damn map of australia, also in ppt

       NT Q
    WA SA NSW
          V
          T

      /NT--Q
     / |  /|
    WA-SA -NSW
         \ |
          V
          
           T

Variables: WA, NT, Q, SA, NWS, V, T

Domains: Di = {R, G, B}

Constraints: adjacent regions must be different colors

- e.g. WA != NT
- or (WA, NT) in {(R,G), (R,B), ... }
- or (WA, NT) not in {(R,R), (G,G), (B, B) }

# Example: solution

WA = red, NT = green, Q = red, NWS = green, V = red, SA = blue, T = free

# Constraint graph

nodes are variables; links between any two nodes that participate in a constraint

Q1: Draw the constraint graph for 

Australia:

      /NT--Q
     / |  /|
    WA-SA -NSW
         \ |
          V
          
           T


# Cryptoarithmetic

      T W O
    + T W O
    -------
    F O U R

Q2a. What are the variables?
Q2b. What are the domains?

Note the hidden variables (carry digits).

Q2c. What are the constraints? Unlike map, some involve >2 variables. 

- AI Jeopardy?
 unary, binary, n-ary constraints

# n-ary constraints

For n-ary constraints, we use a hyper-constraint graph (otherwise you end up with a clique in the worst case)

n-ary constraints are more concise than the binary decomposition, and enable special purpose inference

# Two in-class exercises

clapping for time constraints

numbers for search under constraint

# we're bound for South Australia

is coloring the map of Australia a search problem?

- what is a state?
- what is the successor fn (i.e. action and transition)
- goal test?
- branching factor?
- what search method would you apply?
- heuristics?

# CSP solutions as search

formulation is identical for all CSPs

solution found at depth n (for n variables).

path is irrelevant

depth first search is a good fit (with backtracking!)

branching factor b at the top level is nd. 

b=(n-l)d at depth l, hence how many leaves? (n!)(d^n)

only d^n complete assignments

# example DFS on australia

ppt

# simple backtracking search

Depth-first search

Choose values for one variable at a time 

Backtrack when a variable has no legal values left to assign.

If search is uninformed, then general performance is relatively poor

How can we do better?

# improving backtracking efficiency

Basic question: What next step should our search procedure take?

Approaches:

- Minimum remaining values heuristic
- Degree heuristic
- Least-constraining value heuristic

see ppt for illustration

# what about special algorithms for constraint satisfaction?

we can propagate information through constraints, and substantially prune the search tree as we go (in essence, improving the heuristics)

# providing better information to heuristics

Forward checking:

 - Precomputing information needed by MRV anyway
 - enables early stopping

Constraint propagation:

 - enforces constraints locally on constraint graph
 - Arc consistency (2-consistency)
 - n-consistency

# more ppt for examples

# n-consistency

Arc consistency does not detect all inconsistencies:
Partial assignment {WA=red, NSW=red} is inconsistent.

Stronger forms of propagation can be defined using the notion of n-consistency.

A CSP is k-consistent if for any set of k-1 variables and for any consistent assignment to those variables, a consistent value can always be assigned to any kth variable.

e.g.:

- 1-consistency or node-consistency
- 2-consistency or arc-consistency
- 3-consistency or path-consistency (in binary constraint graphs)

# Other techniques

Check global constraints:

- AllDiff can be checked many ways. One simple way is pigeonhole principle; m variables, n values, if m > n, no satisfying assignment is possible. 

- AtMost() and other integer domain constraints are handled by bounds propagation (propagate intervals, rather than assignments)

Intelligent backtracking:

Standard  is chronological backtracking i.e. try different value for preceding / most recently assigned variable. Another option:

- determine conflict set (set of assignments that cause a conflict at dead-end)
- back jump to the most recent element of the conflict set
- forward checking computes the conflict set (and in fact obviates it if run after every assignment)

# is this amenable to local search?

Do we need the path to the solution or only the solution itself?
Is memory a constraint?
Can we apply local search methods?

- Hillclimbing
- Simulated annealing
- Genetic algorithms

What are good heuristics for choosing moves in local search?

# min-conflicts heuristic

To enable local search:

 - allow states with unsatisfied constraints
 - operators reassign variable values

Variable selection: randomly select any conflicted variable

Value selection by min-conflicts heuristic:

 - choose value that violates the fewest constraints
 - then gradient descent with h(n) = total number of violated constraints

# Example: n-queens

States: 4 queens in 4 columns (4^4 = 256 states)
Actions: move queen in column
Goal test: no attacks; h(n) = 0
Evaluation: h(n) = number of attacks

    . . . .    . Q . .    . Q . .
    . Q . Q -> . . . Q -> . . . Q
    Q . Q .    Q . Q .    Q . . .
    . . . .    . . . .    . . Q .
    
can solve random  n-queens in almost constant time for arbitrary n with high probability 

(works because state space is dense with possible solutions; sparse state space requires more advanced techniques)