Artificial Intelligence 2.2 Heuristic (Informed) Search Artificial Intelligence 2016 - Heuristic...

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Artificial Intelligence 2.2 Heuristic (Informed) Search

Prof. Dr. Jana Koehler Fall 2016

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Agenda

Using Knowledge during Search – evaluation of search states – admissible and monotone (consistent) Heuristics – finding good heuristics

(1) Greedy Best-First Search (2) A* and IDA* (3) Recursive Best-First Search

Artificial Intelligence 2016 - Heuristic Search 2

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How to Determine the next Node for Expansion?

Uninformed Search – rigid procedure, no knowledge of the cost from a node to

the goal – e.g. FIFO, LIFO queues

Informed Search

– "value" of expanding a node as knowledge that steers the search algorithm through the search space

– evaluation function f(n) assigns a real number to each node


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The Evaluation Function f(n) = g(n) + h(n)


g(n) = the costs from the initial state to the current state (a precise value)

+ h(n) = the costs from the current state to the goal state (an estimated value)

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Heuristic Function h(n)

h(n) is a heuristic – describes the estimated costs from node n to a goal state

Heuristics are problem-specific and steer the search

Important properties of h(n)

– h(n) = 0 if n is a goal state – h is admissible – h is consistent


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Admissible h(n)

h*(n) = the real costs of the optimal path from node n to the goal

h is admissible if it guarantees for all n

h is an optimistic estimate of the costs that actually occur – it underestimates the real costs


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Consistent (Monotone) h(n)


goal state

a node n

n' is a successor of n

c(n,a,n') step cost of getting from n to n' with action a

h(n)

h(n')

a

Die Dreiecksungleichung muss erfüllt sein. triangle inequality: each side of a triangle cannot be longer than the sum of the other two sides

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(1) Greedy (Best-First) Search

Uses f(n) = h(n) Epands the node first that is estimated as being closest to

the goal Does not consider the current path costs

– "counterpart" to uniform-cost search, which uses f(n) = g(n)


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Greedy Search on the Romania Travel Example


h: Aerial Distances to Bucharest

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Example of Incompleteness


Problem: From Iasi to Fagaras • Iasi to Neamt estimated shorter than Vaslui to Fagaras • Neamt is expanded, but it is a dead-end the algorithm gets stuck without finding a solution

if it expands Neamt, it puts Iasi back into the frontier (infinite loop)

Cannot use this table, why?

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Properties of Greedy Search

minimal search costs (expands a single path) not optimal incomplete (graph search is complete on finite spaces) time and space complexity is O(bm) m - maximum depth


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Greedy search only uses h(n)

Uniform-Cost search uses g(n) – finds an optimal solution if path costs grow monotonically

A* uses f(n) = g(n) + h(n)

A* combines both using preferably admissible and consistent heuristics

(2) A* (Hart, Nilsson, Raphael 1968)


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Properties of A*

Complete – if a solution exists, A* will find it provided that

1) every node has a finite number of successor nodes, and 2) each action has positive and finite costs

Optimal – first solution found has minimum path cost if h is

admissible (on trees) or if h is consistent (on graphs) • under an admissable heuristics on graphs, A* needs to expand

all nodes with f(n) <= C* (cost of optimal solution)

Complexity – space and time exponential in the path length of the

solution


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f-based Contours in a Search Space

A* fans out from the start node, adding nodes in concentric bands of increasing f-costs – with good heuristics the bands stretch towards the goal state and are

more narrowly focused around the optimal path


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Branch and Bound under admissible Heuristics


a first solution with cost C has been found

…

f(n) > C

all nodes with f > C can be pruned from the search space

A* on graphs finds a first solution s with cost C, - if f is consistent, s is the optimal solution - if f is only admissible, A* needs to expand all nodes with lower or equal

costs than C, until only nodes with f(n) > C are left to be sure that s is the optimal solution

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IDA* (Korf 1985)

A* requires exponential memory in the worst case – combine with Depth-Limited Search

Idea: use successive iterations with increasing f costs – use f bounds instead of bounding the length of the path

At each iteration, perform a depth-first search, cutting off a

branch when its total cost (g + h) exceeds a given threshold – threshold starts at the estimate of the cost of the initial

state, and increases for each iteration of the algorithm – at each iteration, the threshold used for the next iteration

is the minimum cost of all values that exceeded the current threshold


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Example of IDA*

initial f-cost limit = 366 (f-costs of the initial state) – first expansion: T=118 +329, S=140+253, Z=75+374

next f-cost limit = 393 (S)


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Heuristics

Informed search depends on good heuristics

Heuristic = method to help solve a problem, commonly informal – "rules of thumb", educated guesses, intuitive judgments

or simply common sense


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Designing Heuristic Functions

The informedness of the heuristic is critical for the success of the search algorithm – steer the algorithm towards the most promising parts of

the search space – recognize dead ends early – find a near-optimal solution under practical conditions

Requires an understanding of the application domain

– keep heuristic and search approach separate – try out different variants in empirical tests – an art …(… and a hot topic in AI research)


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Heuristic Functions Example


(Misplaced Tiles)

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Misplaced Tiles vs. Manhattan Distance

Distance between two points measured along axes at right angles

Disadvantage: considers all tiles independently


h1 = 8 (all tiles are misplaced)

h2 = 3 + 1 + 2 + … = 18

goal position

current position

initial state goal state

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Linear Conflict Heuristic vs. Manhattan Distance

Two tiles tj and tk are in a linear conflict if tj and tk are in the same line, the goal positions of tj and tk are both in that line, tj is to the right of tk and the goal position of tj is to the left of the goal position of tk – LC will add a cost of 2 moves for each pair of conflicting

tiles to the Manhattan Distance – motivation: tiles need to surround one another


MH = 2 + 0 = 2 LC = 2 + 2 = 4

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Gaschnigs Heuristic

Relaxed Move: Any tile can be moved to the blank position (count the number of swaps) If 9 is not at the 1st position & the permutation is not a solution then swap 9 with the element whose target position 9 is taking else swap 9 with the rightmost element that is not in its proper place


9 stands for the blank transform 724596831 into 912345678

724596831 – 729546831 – 792546831 – 712546839 – 712546938 – 712549638 – 712945638 – 712345698 – 912345678 = 8 swaps

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Applying Gaschnigs Heuristic During Search

4 successor nodes Compute the number of swaps for each node Expand the node with the fewest number of swaps first Problem relaxation is a powerful idea and very successfully

used to derive good heuristics Artificial Intelligence 2016 - Heuristic Search 29

724596831

794526831 724956831 724569831 724536891

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General Technique: Problem Relaxation

Primitive Predicates in the N-Puzzle – ON(x, y) : tile x is on cell y – CLEAR(y) : cell y is clear of tiles – ADJ(y, z) : cell y is adjacent to cell z

Move(x, y, z) preconditions : ON(x, y) & CLEAR(z) & ADJ(y, z) effects : ON(x, z) & CLEAR(y) & NOT ON(x, y) & NOT CLEAR(z)

Remove CLEAR(z) & ADJ(y, z) – Misplaced Tile Distance Remove CLEAR(z) – Manhattan Distance Remove ADJ(y, z) – Gaschnigs Distance


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Pattern Database

Apply the Divide and Conquer principle – decompose the problem into subproblems (subgoals) – store solutions to the subproblems with associated costs

(patterns) – reuse these solutions

Famous example: end game libraries in chess Artificial Intelligence 2016 - Heuristic Search 31

Divide the 15 puzzle into Fringe + 8 puzzle • map the current location of the fringe tiles

into an index of the database • the data base tells us the minimal number

of moves to achieve the fringe • achieve the fringe + solve the remaining 8

puzzle

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Learning Heuristic Functions

Relaxed problem heuristic

– problem with fewer restrictions on the actions is called a relaxed problem

– cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem

Modern search algorithms analyze the domain and the given problem instance – learn a problem-instance specific heuristic before they

start searching


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Summary Questions

1. Explain the underlying ideas of greedy best-first search, A* and IDA*.

2. What are the properties of A*? 3. Why can IDA* be more efficient in practice? 4. What is the principal idea behind branch and bound? 5. What are heuristics, which role do they play in informed

search, and how can we develop good heuristics? 6. What is an admissible heuristics? 7. What can happen with A* if f is non-admissible?


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Exercise 2.2.1 A* Search of the Example

Take the path costs from the Romania map for g(n) and the aerial distance estimates for h(h) a) Perform an A* search from Arad to Bucharest! b) Do you obtain the optimal route as the solution? Why?


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a) Formalize the vacuum world with a variable number of rooms, one cleaning agent, the actions left, right, up, down, suck (each with costs 1) that are always executable in any state. A room can be clean or dirty, with some random dirt distribution that is known to the agent. The goal is to have all rooms clean.

b) Define a simple to compute heuristic and expand an A* search tree to depth 3 using your heuristic starting from the initial state shown below. Show the values for f(h), g(n), h(n)

c) Give an argument why your heuristic is or is not admissible + monotone. Hint: the goal of this exercise is to investigate how A* explores the search and how this is influenced by the heuristic. Your heuristic does not need to be admissible and monotone.

Exercise 2.2.2 A* for the Vacuum World


TESTAT bis zum 9.10. 24:00

Artificial Intelligence 2.2 Heuristic (Informed) Search Artificial Intelligence 2016 - Heuristic...

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