An Introduction to Artificial Intelligence Lecture 4a: Informed Search and Exploration Ramin Halavati (halavati@ce.sharif.edu) In which we see how information about the state space can prevent algorithms from blundering about the dark.

Outline Best-first search Greedy best-first search A* search Heuristics Local search algorithms Hill-climbing search Simulated annealing search Local beam search Genetic algorithms

UNINFORMED? Uninformed: To search the states graph/tree using Path Cost and Goal Test.

INFORMED? Informed: More data about states such as distance to goal. Best First Search Almost Best First Search Heuristic h(n): estimated cost of the cheapest path from n to goal. h(goal) = 0. Not necessarily guaranteed, but seems fine.

Greedy Best First Search Compute estimated distances to goal. Expand the node which gains the least estimate.

Greedy Best First Search Example Heuristic: Straight Line Distance (HSLD)

Greedy Best First Search Example

Properties of Greedy Best First Search Complete? No, can get stuck in loop. Time? O(bm), but a good heuristic can give dramatic improvement Space? O(bm), keeps all nodes in memory Optimal? No, it depends

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 from n to goal f(n) = estimated total cost of path through n to goal

A* search example

A* vs Greedy

Admissible Heuristics h(n) is admissible: if for every node n, h(n) ≤ h*(n), h*(n): the true cost from n to goal. Never Overestimates. It’s Optimistic. Example: hSLD(n) (never overestimates the actual road distance)

A* is Optimal Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal TREE-SEARCH: To re-compute the cost of each node, each time you reach it. GRAPH-SEARCH: To store the costs of all nodes, the first time you reach em.

Optimality of A* ( proof ) Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. f(G2) = g(G2) since h(G2) = 0 g(G2) > g(G) since G2 is suboptimal f(G) = g(G) since h(G) = 0 f(G2) > f(G) from above h(n) ≤ h* (n) since h is admissible g(n) + h(n) ≤ g(n) + h*(n) f(n) ≤ f(G) Hence f(G2) > f(n), and A* will never select G2 for expansion

Consistent Heuristics h(n) is consistent if: for every node n, every successor n' of n generated by any action a, h(n) ≤ c(n,a,n') + h(n') Consistency: Monotonicity Triangular Inequality. Usually at no extra cost! Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal

Optimality of A* A* expands nodes in order of increasing f value Gradually adds "f-contours" of nodes Contour i has all nodes with f=fi, where fi < fi+1

Properties of A* Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) ) Time? Exponential Space? Keeps all nodes in memory (bd) Optimal? Yes A* prunes all nodes with f(n)>f(Goal). A* is Optimally Efficient.

How to Design Heuristics? E.g., for the 8-puzzle: h1(n) = number of misplaced tiles h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile)

Admissible heuristics h1(n) = Number of misplaced tiles h2(n) = Total Manhattan distance

Effective Branching Factor If A* finds the answer by expanding N nodes, using heuristic h(n), At depth d, b* is effective branching factor if: 1+b*+(b*)2+…+(b*)d = N+1

Dominance If h2(n) ≥ h1(n) for all n (both admissible) then h2 dominates h1. h2 is better for search. h2 is more realistic. h (n)=max(h1(n), h2(n),… ,hm(n)) Heuristic must be efficient.

How to Generate Heuristics? Formal Methods Relaxed Problems Pattern Data Bases Disjoint Pattern Sets Learning ABSOLVER, 1993 A new, better heuristic for 8 puzzle. First heuristic for Rubik’s cube.

“Relaxed Problem” Heuristic A problem with fewer restrictions on the actions. The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem. 8-puzzle: Main Rule: A tile can be moved from square A to B if A is horizontally or vertically adjacent to B and B is empty. Relaxed Rules: A tile can move from square A to square B if A is adjacent to B. (h2) A tile can move from square A to square B if B is blank. A tile can move from square A to square B. (h1)

“Sub Problem” Heuristic The cost to solve a subproblem. It IS admissible.

“Pattern Database” Heuristics To store the exact solution cost to some sub-problems.

“Disjoint Pattern” Databases To add the result of several Pattern-Database heuristics. Speed Up: 103 times for 15-Puzzle and 106 times for 24-Puzzle. Separablity: Rubik’s cube vs. 8-Puzzle.

Learning Heuristics from Experience Machine Learning Techniques. Feature Selection Linear Combinations

BACK TO MAIN SEARCH METHOD What’s wrong with A*? It’s both Optimal and Optimally Efficient. MEMORY

Memory Bounded Heuristic Search Iterative Deepening A* (IDA*) Similar to Iterative Deepening Depth First Search Bounded by f-cost. Memory: b*d

Recursive Best First Search Main Idea: To search a level with limited f-cost, based on other open nodes with continuous update.

Recursive Best First Search

Recursive Best First Search, Sample

Recursive Best First Search, Sample Complete? Yes, given enough space. Space? b * d Optimal? Yes, if admissible. Time? Hard to analyze. It depends…

Memory, more memory… A*: bd IDA*, RBFS: b*d What about exactly 10 MB?

Memory-Bounded A* MA* Simplified Memory Bounded A* (SMA*) To store as many nodes as possible (the A* trend). When memory is full, remove the worst current node and update its parent.

SMA* Example

SMA* Code

To be continued…