Lecture 5: Online search Constraint Satisfaction Problems

1
Lecture 5On-line searchConstraint Satisfaction
Problems

• Reading Sec. 4.5 Ch. 5, AIMA

2
Recap

• Blind search (BFS, DFS)
• Informed search (Greedy, A)
• Local search (Hill-climbing, SA, GA1)
• Today
• On-line search
• Constraint satisfaction problems

3
On-line search

• So far, off-line search
• Observe environment, determine best set of
  actions (shortest path) that leads from start to
  goal
• Good for static environments
• On-line search
• Perform action, observe environment, perform,
  observe,
• Dynamic, stochastic domains exploration
• Similar to
• local search, but to get to all successor
  states, agent has to explore all actions

4
Depth-first online

• Online algorithms cannot simultaneously explore
  distant states (i.e., jump from current state to
  a very distant state, like, e.g., A)
• Similar to DFS
• Try unexplored actions
• Once all tried, and the goal is not reached,
  backtrack to unbacktracked previous state

5
DFS-Online
function action DFS-Online(percept) s
GetState(percept) if GoalTest(s) return
stop if NewState(s) unexploreds
Actions(s) if NonEmpty(sp) Resultsp,ap
s Unbacktrackeds sp end If
Empty(unexpolreds) If Empty(unbacktrackeds)
return stop else a Result s, a Pop(
unbacktrackeds ) else a Pop( unexploreds
) end sp s return a endcc
6
Maze
G
s0
7
Online local search

• Hill-climbing
• Is an online search!
• However, it does not have any memory.
• Can one use random restarts?
• Instead
• Random wandering by choosing random actions
  (random walk) --- eventually, visits all points
• Augment HC with memory of sorts keep track of
  estimates of h(s) and updated them as the search
  proceeds
• LRTA - learning real-time A

8
LRTA
function action LRTA(percept) s
GetState(percept) if GoalTest(s) return
stop if NewState(s) Hs h(s) if
NonEmpty(sp) Resultsp,ap s Hsp min b?
Actions(sp) c(sp,b,s) H(s) end a arg min
b? Actions(s) c(s,b, Results,b )
H(Results,b ) end sp s return a endcc
9
Maze LRTA
2
1
G
3
2
1
1
4
3
2
2
2
3
4
3
s0
3
3
3
3
4
4
4
4
4
4
4
10
Constraint satisfaction problems - CSP

• Slightly different from standard search
  formulation
• Standard search abstract notion of state
  successor function goal test
• CSP
• State a set of variables V V1, , VN and
  values that can be assigned to them specified by
  their domains D D1, , DN, Vi ? Di
• Goal a set of constraints on the values that
  combinations of V can take
• Examples
• Map coloring, scheduling, transportation,
  configuration, crossword puzzle, N-queens,
  minesweeper,

11
Map coloring

• Vi WA, NT, SA, Q, NSW, V, T
• Di R, G, B
• Goal / constraint adjacent regions must have
  different color

• Solution (WA,R), (NT,G), (SA,B), (Q,R),
  (NSW,G), (V,R), (T,G)

12
Constraint graph

• Nodes variables, links constraints

NT
Q
WA
NSW
SA
V
T
13
Types of CSPs

• Discrete-valued variables
• Finite O(dN) assignments, d D
• Map coloring, Boolean satisfiability
• Infinite D is infinite, e.g., strings or natural
  numbers
• Scheduling
• Linear constraints aV1 bV2 lt V3
• Continuous-valued variables
• Functional optimization
• Linear programming

14
Types of constraints

• Unary V ? blue
• Binary V ? Q
• Higher order
• Preferences different values of V have different
  scores

15
CSP as search
function assignment NaiveCSP(V,D,C) 1) Initial
state 2) Successor function assign value
to unassigned variable that does not conflict
with current assignments. 3) Goal Assignment
complete? end

• Generic fits all CSP
• Depth N, N number of variables
• DFS, but path irrelevant
• Problem leaves is n!dN gt dN
• Luckily, we only need consider one variable per
  depth! Assignment is commutative.

16
Backtracking search

• DFS with single variable assignment

17
Map coloring example
NT
Q
WA
18
How to improve efficiency?

• Address the following questions
• Which variable to pick next?
• What value to assign next?
• Can one detect failure early?
• Take advantage of problem structure?

19
Most constrained variable

• Choose next the variable that is most constrained
  based on current assignment

NT
WA
SA
20
Most constraining variable

• Tie breaker among most constrained variables

NT
Q
SA
21
Least constraining value

• Select the value that least constrains the other
  variables (rules our fewest other variables)

22
Forward checking

• Terminate search early, if necessary
• keep track of values of unassigned variables

23
Constraint propagation

• FC propagates assignment from assigned to
  unassigned variables, but does not provide early
  detection of all failures

• NT SA cannot both be blue!
• CP repeatedly enforces constraints

24
Arc consistency

• Simplest form of propagation makes each arcs
  consistent
• X -gt Y is consistent iff for all x ? X, there is
  y ? Y that satisfies constraints

• Detects failure earlier than FC.
• Either preprocessor or after each assignment
• AC-3 algorithm

25
Problem structure

• Knowing something about the problem can
  significantly reduce search time

• T is independent of the rest! Connected
  components.
• c-components gt O(dcn/c)

26
Tree-structured CSP

• If graph is a tree, CSP can be accomplished in
  O(nd2)

• Flatten tree into a chain where each node is
  preceded by its parent
• Propagate constraints from last to second node
• Make assignment from first to last

27
Nearly tree-structured problems

• What if a graph is not a tree?
• Maybe can be made into a tree by constraining it
  on a variable.

• If c is cutset size, then O(dc (N-c)d2 )

28
Iterative CSP

• Local search for CSP
• Start from (invalid) initial assignment, modify
  it to reduce the number of violated constraints
• Randomly pick a variable
• Reassign a value such that constraints are least
  violated (min-conflict heuristic)
• Hill-climbing with h(s) min-conflict