Download Artificial Intelligence 2.2 Heuristic (Informed) Search

Artificial Intelligence
2.2 Heuristic (Informed)
Search
Prof. Dr. Jana Koehler
Fall 2016
© HSLU - JK
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
2
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
3
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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)
4
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
5
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
6
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Consistent (Monotone) h(n)
a node n
c(n,a,n') step cost of getting from n to n' with action a
a
n' is a successor of n
h(n)
h(n')
goal state
7
Artificial Intelligence 2016 - Heuristic Search
Die Dreiecksungleichung
muss erfüllt sein.
triangle inequality: each
side of a triangle cannot
be longer than the sum of
the other two sides
©
HSLU - JK
(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)
8
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Greedy Search on the Romania Travel Example
h: Aerial Distances
to Bucharest
9
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
10
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Example of Incompleteness
Cannot use this table, why?
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)
11
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
12
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
(2) A* (Hart, Nilsson, Raphael 1968)
 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
13
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
14
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
15
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
16
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
17
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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)
18
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
22
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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)
23
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Heuristic Functions Example
(Misplaced Tiles)
24
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Misplaced Tiles vs. Manhattan Distance
h1 = 8 (all tiles are misplaced)
h2 = 3 + 1 + 2 + … = 18
initial state
goal state
 Distance between two points measured along axes at right
angles
goal position
current position
 Disadvantage: considers all tiles independently
25
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Empirical Comparison of Both Example Heuristics
26
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Linear Conflict Heuristic vs. Manhattan Distance
MH = 2 + 0 = 2
LC = 2 + 2 = 4
 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
27
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Gaschnigs Heuristic
9 stands for the blank
transform
into
724596831
912345678
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
724596831 – 729546831 – 792546831 – 712546839 – 712546938 – 712549638 –
712945638 – 712345698 – 912345678
= 8 swaps
28
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Applying Gaschnigs Heuristic During Search
724596831
794526831
724956831 724569831 724536891
 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
29
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
General Technique: Problem Relaxation
 Primitive Predicates in the N-Puzzle
– ON(x, y)
– CLEAR(y)
– ADJ(y, z)
:
:
:
tile x is on cell y
cell y is clear of tiles
cell y is adjacent to cell z
preconditions
:
ON(x, y) & CLEAR(z) & ADJ(y, z)
effects
:
ON(x, z) & CLEAR(y) &
 Move(x, y, z)
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
30
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
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
 Famous example: end game libraries in chess
31
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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
32
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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?
33
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
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?
34
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK
Exercise 2.2.2 A* for the Vacuum World
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.
TESTAT bis zum 9.10. 24:00
35
Artificial Intelligence 2016 - Heuristic Search
©
HSLU - JK