Problem-solving as search

History Problem-solving as search – early insight of AI. Newell and Simon’s theory of human intelligence and problemsolving. 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

Classic AI Toy Problem: 8-puzzle initial state

goal state

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8

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1

1

6

4

8

5

7

7

2

4 6

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 tree 28 3 16 4 7 5 28 3 1 4 76 5

28 3 16 4 7 5 28 3 6 4 1 7 5

28 3 1 4 76 5

2 3 18 4 76 5

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

28 3 16 4 7 5 28 3 1 64 7 5

What is size of state space?

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 1. 

Define search space •  Initial, goal, and intermediate states

2. 

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

3. 

Apply search algorithm (tree search) to find path from initial to goal state, while avoiding (if possible) repeating a state during the search.

4. 

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: 1.  completeness – does it always find a solution if one exists? 2.  optimality – does it always find a optimal (least-cost or highest value) solution? 3.  time complexity – number of nodes generated/expanded 4.  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

•  Uninformed search: 1.  Breadth-first 2.  Depth-first 3.  Depth-limited 4.  Iterative deepening depth-first 5.  Bidirectional

•  Informed (or heuristic) search (deterministic or stochastic): 1.  Greedy best-first 2.  A* (and many variations) 3.  Hill climbing

4.  Simulated annealing 5.  Genetic algorithm 6.  Tabu search 7.  Ant colony optimization

•  Adversarial search: 1.  Minimax with alpha-beta pruning

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 1. current state = initial state 2.  Expand current state 3.  Evaluate offspring states s with heuristic h(s), which estimates cost of path from s to goal state 4. current state = argmins h(s) for s ∈ offspring (current state) 5. 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. 1.  g is a cost function •  Total cost incurred so far from initial state at node n 2.  h is an heuristic 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.

n  

Let n be an unexpanded node on a shortest path to an optimal goal G1. f(G2)  =  g(G2)  since  h(G2)  =  0   start   >  g(G1)  since  G2  is  subop/mal  

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

G1   Since  f(G2)  >  f(n),  A*  will  never  select   G2  for  expansion  

Variations of A* •  IDA* (iterative deepening A*) •  ARA* (anytime repairing A*) •  D* (dynamic A*)

(From http://aima.eecs.berkeley.edu/slides-pdf/)

(From http://aima.eecs.berkeley.edu/slides-pdf/)

(From http://aima.eecs.berkeley.edu/slides-pdf/)

Example of Simulated Annealing Netlogo simulation

Simulated Annealing is complete (if you run it for a long enough time!)

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