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Problem-solving as searchPowerPoint Presentation

Problem-solving as search

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History

- Problem-solving as search – early insight of AI.
- Newell and Simon’s theory of human intelligence and problem-solving.
- Early examples:
- 1956: Logic Theorist (Allen Newell & Herbert Simon)
- 1958: Geometry problem solver (Herbert Gelernter)
- 1959: General Problem Solver (Herbert Simon & Alan Newell)
- 1971: STRIPS (Stanford Research Institute Problem Solver, Richard Fikes & Nils Nilsson)

Real-World Problem-Solving as Search

- Examples:
- Route/Path finding: Robots, cars, cell-phone routing, airline routing, characters in video games, …
- Layout of circuits
- Job-shop scheduling
- Game playing (e.g., chess, go)
- Theorem proving
- Drug design

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Classic AI Toy Problem: 8-puzzleinitial

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Notion of “searching a state space”

Pictures from http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp

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8-puzzle search treePictures from http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp

What is size of state space for 8-puzzle?

Size of state space 9! = 181,440

- Size of 15-puzzle state space? 16! = 2 x 1013
- Size of 24-puzzle state space? 25! = 1.5 x 1025

What is size of state space for 8-puzzle?

Size of state space 9! = 181,440

- Size of 15-puzzle state space? 16! = 2 x 1013
- Size of 24-puzzle state space? 25! = 1.5 x 1025
- Can’t do exhaustive search!

Approximate number of states

- Tic-Tac-Toe: 39
- Checkers: 1040
- Rubik’s cube: 1019
- Chess: 10120

- In general, a search problem is formalized as :
- state space
- special start and goal state(s)
- operators that perform allowable transitions between states
- cost of transitions
- All these can be either deterministic or probabilistic.

- State space as a tree/graph
- Search as tree search
- Solutions: “winning” state, or path to winning state

How to solve a problem by searching

- Define search space
- Initial, goal, and intermediate states

- Define operators for expanding a given state into its possible successor states
- Defines search tree

- Apply search algorithm (tree search) to find path from initial to goal state, while avoiding (if possible) repeating a state during the search.
- Solution is
- path from initial to goal state (e.g., traveling salesman problem)
- or, simply a goal state, which might not be initially known (e.g., drug design)

Missionaries and cannibals

- Three missionaries and three cannibals are on the left bank of a river.
- There is one canoe which can hold one or two people.
- Find a way to get everyone to the right bank, without ever leaving a group of missionaries in one place outnumbered by cannibals in that place.

From http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp

Missionaries and cannibals

- Three missionaries and three cannibals are on the left bank of a river.
- There is one canoe which can hold one or two people.
- Find a way to get everyone to the right bank, without ever leaving a group of missionaries in one place outnumbered by cannibals in that place.

How to set this up as a search problem?

From http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp

Missionaries and cannibals

- State space:
- Size?
- Initial state:
- Goal state:
- Operators:
- Cost of transitions:
- Search tree:

Drug design

- Example: Search for sequence of up to N amino acids that forms protein shape that matches a particular receptor on a pathogen.
- (Note: There are 20 amino acids to choose from at each locus in the string.)

Drug design

- State space:
- Size?
- Initial state:
- Goal state:
- Operators:
- Cost of transitions:
- Search tree:

Search Strategies

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

Strategies are evaluated along the following dimensions:

- completeness – does it always find a solution if one exists?
- optimality– does it always find a optimal (least-cost or highest value) solution?
- time complexity– number of nodes generated/expanded
- space complexity– maximum number of nodes in memory
Time and space complexity are often measured in terms of:

b – maximum branching factor of the search tree

d – depth of the least-cost solution

m – maximum depth of the state space (may be infinite)

Adapted from http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp

Search methods

Simulated annealing

Genetic algorithm

Tabu search

Ant colony optimization

Adversarial search:

Minimax with alpha-beta pruning

- Uninformed search:
- Breadth-first
- Depth-first
- Depth-limited
- Iterative deepening depth-first
- Bidirectional

- Informed (or heuristic) search (deterministic or stochastic):
- Greedy best-first
- A* (and many variations)
- Hill climbing

Uninformed strategies

- Breadth-first: Expand all nodes at depth d before proceeding to depth d+1
- Depth-first: Expand deepest unexpanded node
- Depth-limited: Depth-first search with a cutoff at a specified depth limit
- Iterative deepening: Repeated depth-limited searches, starting with a limit of zero and incrementing once each time

http://www.cse.unl.edu/~choueiry/S03-476-876/searchapplet/index.html

Uninformed Search Properties

- Breadth-first: Complete? Optimal? Time? Space?
- Depth-first: Complete? Optimal?Time?Space?
- Depth-limited: Complete?Optimal?Time?Space?
- Iterative deepening: Complete? Optimal?Time?Space?

Informed (heuristic) Search

- What is a “heuristic”?
- Examples:
- 8 puzzle
- Missionaries and Cannibals
- Tic Tac Toe
- Traveling Salesman Problem
- Drug design

Best-first greedy search

- current state = initial state
- Expand current state
- Evaluate offspring states s with heuristic h(s), which estimates cost of path from s to goal state
- current state = argminsh(s) for s offspring(current state)
- If current state ≠ goal state, go to step 2.
- http://alumni.cs.ucr.edu/~tmatinde/projects/cs455/TSP/heuristic/Travellinganimation.htm

Search Terminology

- Completeness
- solution will be found, if it exists

- Optimality
- least cost solution will be found

- Admissable heuristic h
- s, h never overestimates true cost from state s to goal state
- Best first greedy search: Complete? Optimal?
- 8-puzzle heuristics: Hamming distance, Manhattan distance: Admissible?
- Example of non-admissable heuristic for 8-puzzle?

A* Search

- Uses evaluation function f (n)= g(n)+ h(n)
- where n is a node.
- g is a cost function
- Total cost incurred so far from initial state at node n

- h is an heuristic

- g is a cost function
- Best first search is A* with g = 0.

h1(start state) =

h2(start state) =

A* Pseudocode

- give code and show example on 8-puzzle

A* Pseudocode

- create the open list of nodes, initially containing only our starting node
- create the closed list of nodes, initially empty
- while (we have not reached our goal) {
- consider the best node in the open list (the node with the lowest f value)
- if (this node is the goal) { then we're done }
- else {
- move the current node to the closed list and consider all of its successors
- for (each successor) {
- if (this suceessor is in the closed list and our current g value is lower) {
- update the successor with the new, lower, g value
- change the sucessor’s parent to our current node }
- else if (this successor is in the open list and our current g value is lower) {
- update the suceessor with the new, lower, g value
- change the sucessor’s parent to our current node }
- else this sucessor is not in either the open or closed list {
- add the successor to the open list and set its g value } } } }

Adapted from: http://en.wikibooks.org/wiki/Artificial_Intelligence/Search/Heuristic_search/Astar_Search#Pseudo-code_A.2A

- A* search is complete, and is optimal if h is admissible

Proof of Optimality of A*

- Suppose a suboptimal goal G2 has been generated and is in the OPEN list.
- Let n be an unexpanded node on a shortest path to an optimal goal G1.

f(G2) = g(G2) since h(G2) = 0

g(G1) since G2 is suboptimal

f(G2) f(n) since h is admissible

Since f(G2) f(n), A* will never select

G2 for expansion

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Variations of A*

- IDA* (iterative deepening A*)
- ARA* (anytime repairing A*)
- D* (dynamic A*)

Example of Simulated Annealing

- Netlogo simulation

Genetic Algorithms enough time!)

- Similar to hill-climbing, but with a population of “initial states”, and stochastic mutation and crossover operations for search.

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