1. Toy Problem: Missionaries and Cannibals
On one bank of a river are three
missionaries (black triangles) and
three cannibals (red circles). There is
one boat available that can hold up
to two people and that they would like to use to cross the river. If the cannibals ever outnumber the missionaries on either of the river’s banks, the missionaries will get eaten.
How can the boat be used to safely carry all the missionaries and cannibals across the river?
Solving Problems by Searching

2. Solving Problems by Searching
Search Problems
initial state
set of possible actions/applicability conditions
successor function: state  set of <action, state>
successor function + initial state = state space
path (solution)
goal
goal test function or goal state
path cost function
assumption: path cost = sum of step costs
for optimality
Initial state: current state the world is in (state = situation)
States: symbol structures representing real world objects and relations  physical symbols systems;
Possible actions aka. operators or production rules (problem formulation)
Successor function: action is applicable in given state; result of applying action in given state is paired state
State space: directed graph with states as nodes and actions as arcs
Path: in the graph
Goal (goal formulation)
Function for multiple goal states (chess) or state for unique goal state (Bucharest)
solution path from initial state to goal state
Path cost function
Step cost: cost of applying a given action in a given state
Optimality: solution path with minimal path cost
Solving Problems by Searching

3. Missionaries and Cannibals: Initial State and Actions
all missionaries, all cannibals, and the boat are on the left bank
5 possible actions:
one missionary crossing
one cannibal crossing
two missionaries crossing
two cannibals crossing
one missionary and one cannibal crossing
Actions: not every action applicable in every state
Example: first action not applicable in initial state
Solving Problems by Searching

4. Missionaries and Cannibals: State Space
1m 1c 1m 1m
1m 1c 2c 2c 1c 1c
16 possible world states
Actions reversible and reversing action is same action; hence bidirectional arcs 1c 1c 2m 2m
1m 1c
Solving Problems by Searching

5. Solving Problems by Searching
Breadth-First Search
strategy:
expand root node
expand successors of root node
expand successors of successors of root node
etc.
implementation:
use FIFO queue to store fringe nodes in general tree search algorithm
Strategy: all nodes at one level are expanded before any nodes at the next level are expanded
FIFO queue: new nodes queued at the end of the queue meaning shallower nodes will get expanded first
Solving Problems by Searching

6. Breadth-First Search: Missionaries and Cannibals
depth = 0
depth = 1
depth = 2
depth = 3
Solving Problems by Searching

7. Solving Problems by Searching
Depth-First Search
strategy:
always expand the deepest node in the current fringe first
when a sub-tree has been completely explored, delete it from memory and “back up”
implementation:
use LIFO queue (stack) to store fringe nodes in general tree search algorithm
alternative: recursive function that calls itself on each of its children in turn
Strategy:
Search proceeds immediately to the deepest level of the tree, where the nodes have no successors;
Deletion of (unsuccessfully) explored sub-trees saves memory;
“backing up” means going to a shallower node in the search tree that will be expanded next;
LIFO queue:
New nodes queued at the beginning of the queue meaning deeper nodes will get expanded first;
Effectively, this is a stack;
Solving Problems by Searching

8. Depth-First Search: Missionaries and Cannibals
Always expand one of the newly generated children until progress is blocked
depth = 2
depth = 3
Solving Problems by Searching

9. Depth-First Search: Finding the Optimal Path
first solution discovered may not be optimal  must keep searching
continued search:
assumption: non-decreasing path costs
remember path cost of cheapest path found so far
do not expand nodes for which path cost exceeds cheapest found so far
General problem: need to search complete tree with DF search to find optimal path  use BF search instead
Solving Problems by Searching