Rutgers CS440, Fall 2003 Lecture 5: On-line search Constraint Satisfaction Problems Reading: Sec. 4.5 & Ch. 5, AIMA.

Rutgers CS440, Fall 2003

Lecture 5:On-line search

Constraint Satisfaction Problems

Reading: Sec. 4.5 & Ch. 5, AIMA

Rutgers CS440, Fall 2003

Recap

• Blind search (BFS, DFS)• Informed search (Greedy, A*)• Local search (Hill-climbing, SA, GA1)• Today:

– On-line search

– Constraint satisfaction problems

Rutgers CS440, Fall 2003

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

Rutgers CS440, Fall 2003

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

Rutgers CS440, Fall 2003

DFS-Online

function action = DFS-Online(percept)s = GetState(percept);

if GoalTest(s) return stop;if NewState(s) unexplored[s] = Actions(s);if NonEmpty(sp)

Result[sp,ap] = s;Unbacktracked[s] = sp;

endIf Empty(unexpolred[s])

If Empty(unbacktracked[s]) return stop;else

a: Result[ s, a ] = Pop( unbacktracked[s] );else

a = Pop( unexplored[s] );endsp = s;return a;

endcc

Rutgers CS440, Fall 2003

Maze

s0

G

Rutgers CS440, Fall 2003

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*

Rutgers CS440, Fall 2003

LRTA*

function action = LRTA*(percept)s = GetState(percept);

if GoalTest(s) return stop;if NewState(s) H[s] = h(s);if NonEmpty(sp)

Result[sp,ap] = s;H[sp] = min b Actions(sp) { c(sp,b,s) + H(s) };

enda = arg min b Actions(s) { c(s,b, Result[s,b] ) + H(Result[s,b] ) };endsp = s;return a;

endcc

Rutgers CS440, Fall 2003

Maze – LRTA*

4

s0

4

3

3

3

4

2

2

2

3

4

1

1

2

3

4

G1

2

3

4

3 3 4

44

Rutgers CS440, Fall 2003

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, …

Rutgers CS440, Fall 2003

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) }

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Constraint graph

• Nodes: variables, links: constraints

WA

NT

SA

Q

NSW

V

T

Rutgers CS440, Fall 2003

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 < V3

• Continuous-valued variables– Functional optimization

– Linear programming

Rutgers CS440, Fall 2003

Types of constraints

• Unary: V blue• Binary: V Q• Higher order• Preferences: different values of V have different “scores”

Rutgers CS440, Fall 2003

CSP as search

• Generic: fits all CSP• Depth N, N = number of variables• DFS, but path irrelevant• Problem: # leaves is n!dN > dN

• Luckily, we only need consider one variable per depth! – Assignment is commutative.

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

Rutgers CS440, Fall 2003

Backtracking search

• DFS with single variable assignment

Rutgers CS440, Fall 2003

Map coloring example

WA

NT

SA

Q

NSW

V

T

WA

NTQ

Rutgers CS440, Fall 2003

How to improve efficiency?

• Address the following questions1. Which variable to pick next?

2. What value to assign next?

3. Can one detect failure early?

4. Take advantage of problem structure?

Rutgers CS440, Fall 2003

Most constrained variable

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

WA

NT

SA

Q

NSW

V

T

WA

NT

SA

Rutgers CS440, Fall 2003

Most constraining variable

• Tie breaker among most constrained variables

WA

NT

SA

Q

NSW

V

T

SA

NTQ

Rutgers CS440, Fall 2003

Least constraining value

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

Rutgers CS440, Fall 2003

Forward checking

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

Rutgers CS440, Fall 2003

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

Rutgers CS440, Fall 2003

Arc consistency

• Simplest form of propagation makes each arcs consistent:• X -> 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

Rutgers CS440, Fall 2003

Problem structure

• Knowing something about the problem can significantly reduce search time

WA

NT

SA

Q

NSW

V

T

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

Rutgers CS440, Fall 2003

Tree-structured CSP

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

WA

NT

SA

Q

NSW

V

T

WA NT SAQ NSW V

1. Flatten tree into a “chain” where each node is preceded by its parent

2. Propagate constraints from last to second node

3. Make assignment from first to last

Rutgers CS440, Fall 2003

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.

WA

NT

SA

Q

NSW

V

T

WA

NT

SA

Q

NSW

V

T

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

Rutgers CS440, Fall 2003

Iterative CSP

• Local search for CSP• Start from (invalid) initial assignment, modify it to reduce the

number of violated constraints1. Randomly pick a variable

2. Reassign a value such that constraints are least violated (min-conflict heuristic)

– Hill-climbing with h(s) = min-conflict