Problem Solving and Search

School of Computer Science & Engineering Chung-Ang University

Artificial Intelligence

Dae-Won Kim

Outline

• Problem-solving agents

• Problem types

• Problem formulation

• Example problems

• Basic search algorithms

Problem-Solving Agents

On holiday In Romania;

currently in Arad.

Flight leaves tomorrow for Bucharest.

Goal: be in Bucharest

Solution: sequence of cities

Solution: ???

Performance measure: ???

Problem formulation:

states – various cities

actions – drive between cities

Problem Formulation: How To

vs. Problem Modeling

A problem is defined by four items:

• Initial state

• Successor function:

set of action-state pairs

• Goal test

• Path cost

A solution is a sequence of actions leading from the initial state to a goal state.

Consider a solution in algorithm

Problem Formulation: Romania

• Initial state:

• Successor function:

• Goal test:

• Path cost:

• Initial state: x = “at Arad”

• Successor function:

S = {<AradZerind,Zerind>, …}

• Goal test: x = “at Bucharest”

• Path cost: sum of distances

Problem Formulation: Vacuum Cleaner

• States:

• Actions:

• Goal test:

• Path cost:

• States: integer dirt and robot locations

• Actions: left, right, suck, stay

• Goal test: no dirt

• Path cost: 1 per action (0 for stay)

Problem Formulation: Robot Assembly

• States: real-valued coordinates of joint angles

• Actions: continuous motions of robot joints

• Goal test: complete assembly

• Path cost: time to execute

Problem Formulation: The 8-Puzzle

• States ?

• Actions ?

• Goal test ?

• Path cost ?

How to achieve the goal state through the complex state space from the initial state?

Answer: Tree Search Algorithms

Idea: exploration of state space by generating successors of already-explored states (expanding states)

Implementation: States vs. Nodes

A state is (a representation of) a physical configuration

A node is data structure constituting part of a search tree includes parents, children, depth, path cost.

A search strategy is defined by picking the order of node expansion.

Strategies are evaluated along the following dimensions:

• Completeness

• Time complexity

• Space complexity

• Optimality

Uninformed Search Strategies

Uninformed search strategies use only the information available in the problem definition.

• Breadth-first search

• Uniform-cost search

• Depth-first search

• Depth-limited search

• Iterative deepening search

Breath-First Search

Expand shallowest unexpanded node.

Implementation: FIFO queue

• Complete?

• Time complexity?

• Space complexity?

• Optimal?

• Complete? Yes (if b is finite)

• Time complexity? O(bd+1)

• Space complexity? O(bd+1)

• Optimal? Yes (if cost = 1 per step)

Uniform-Cost Search

Expand least-cost unexpanded node

using queue ordered by path cost

Equivalent to BFS if step costs equal.

Depth-First Search

Expand deepest unexpanded node.

Implementation: LIFO queue

• Complete?

• Time complexity?

• Space complexity?

• Optimal?

• Complete? No (infinite-depth, loops)

• Time complexity? O(bm)

• Space complexity? O(bm)

• Optimal? No

Depth-Limited Search

= DFS with depth limit (L).

i.e., nodes at depth (L) have no successors

Iterative Deepening Search

• Complete?

• Time complexity?

• Space complexity?

• Optimal?

• Complete? Yes

• Time complexity? O(bd)

• Space complexity? O(bd)

• Optimal? Yes (if step cost = 1)

Informed Search Methods : The Basics

School of Computer Science & Engineering Chung-Ang University

Artificial Intelligence

Dae-Won Kim

Outline

• Best-first search

• Greedy search

• A* search

• Brach and Bound

A strategy is defined by picking the order of node expansion.

Informed search strategy can find solutions more efficiently than an uninformed search.

It uses problem-specific knowledge beyond the definition of the problem itself.

Best-First Search

Idea: use an evaluation function for each node.

• Estimate the “desirability” of each node

• Expand most desirable unexpanded node

Special cases:

• Greedy search

• A* search

• Branch and Bound

Romania Example with Step Costs

Greedy Search

We need an evaluation function : heuristic function h(n)

h(n) = estimate of cost from n to the closest goal

h(n) = straight-line distance from n to Bucharest

Greedy search expands the node that appears to be closest to goal.

Properties of greedy search

• Complete?

• Time complexity?

• Space complexity?

• Optimal?

• Complete? No (can get stuck in loops)

Yes (in finite space with repeated-state checking)

• Time complexity? O(bm), A good heuristic is needed.

• Space complexity? O(bm), Keeps all nodes in memory.

• Optimal? No

What is A* Search?

Idea: avoid expanding paths that are already expensive.

Evaluation function: f(n) = g(n) + h(n)

• g(n) = cost so far to reach n

• h(n) = estimated cost to goal from n

• f(n) = estimated total cost through n to goal

A* search uses an admissible heuristic. Thus, it is optimal.

• h(n) h*(n) where h*(n) is the true cost from n.

• h(n) 0, so h(Goal) = 0.

e.g., hstraight(n) never overestimates the actual distance.

Romania Example with A* Search

Properties of A* Search

• Complete?

• Time complexity?

• Space complexity?

• Optimal?

• Complete? Yes

• Time complexity? Exponential in [relative error in h x length of sol.]

• Space complexity? Keeps all nodes in memory.

• Optimal? Yes

Q: Explain why A* is optimal?

Admissible Heuristics for the 8-puzzle

Guess a h-function : f = g + h

h1(n) = number of misplaced tiles

h1(n) = 6

h2(n) = total Manhattan distance

h2(n) = 4 + 0 + 3 + 3 + 1 + 0 + 2 + 1 = 14

Admissible Heuristics & Dominance

If h2(n) > h1(n) for all n, then h2 dominates h1 and is better for search.

• IDS = 50,000,000,000 nodes

• A*(h1) = 39,135 nodes

• A*(h2) = 1,641 nodes

Q: How to find good heuristics?

Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem.

The optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem.