A4B36ZUI - Introduction to ARTIFICIAL INTELLIGENCE · Introduction to AI. Search-based AI, ......
Transcript of A4B36ZUI - Introduction to ARTIFICIAL INTELLIGENCE · Introduction to AI. Search-based AI, ......
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A4B36ZUI - Introduction to ARTIFICIAL INTELLIGENCEhttps://cw.fel.cvut.cz/wiki/courses/Michal Pechoucek & Jiri KlemaDepartment of Computer Science Czech Technical University in Prague
In parts based on cs121.stanford.edu & S. Russell and P. Norvig, Artificial Intelligence: A Modern Approach. 3rd edition, Prentice Hall, 2010
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What is Artificial Intelligence?
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What is Artificial Intelligence?
Artificial Intelligence is family of technologies and scientific field that allows/studies:• automation, acceleration and scalability of• human (i) perception, (ii) reasoning and (iii)
decision making
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Why Artificial Intelligence Matters?
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Why Artificial Intelligence Matters?
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Why Artificial Intelligence Matters?
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Why Artificial Intelligence Matters?
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Types of Artificial Intelligence?
WEAK AI 𝕏 STRONG AI
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Types of Artificial Intelligence?
• Strong AI: – Searle's strong AI hypothesis: The appropriately programmed
computer with the right inputs and outputs would thereby have a mind in exactly the same sense human beings have minds.
– Artificial general intelligence is a hypothetical artificial intelligence that demonstrates the intelligence of a machine that could successfully perform any intellectual task that a human being can.
• Weak AI: – Turing’s hypothesis: If a machine behaves as intelligently as a human
being, then it is as intelligent as a human being = passes the Turing Test
• AI according to Smith:– The machine shall exhibit indistinguishable behaviour (according to
the weak AI definition) by means of the knowledge structures and reasoning process identical to those used by human
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Types of Artificial Intelligence?
WEAK AI 𝕏 STRONG AI
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Types of Artificial Intelligence?
WEAK AI 𝕏 STRONG AI
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Turing Test
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• Test proposed by Alan Turing in 1950• The computer is asked questions by a human interrogator. It passes
the test if the interrogator cannot tell whether the responses come from a person
• Required capabilities: natural language processing, knowledge representation, automated reasoning, learning,…
• CAPTCHA: Completely Automatic Public Turing tests to tell Computers and Humans Apart
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Types of Artificial Intelligence?
WEAK AI 𝕏 STRONG AI
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Types of Artificial Intelligence?
WEAK AI 𝕏 STRONG AI
Narrow AI→Extended AI→General AI→Super AI
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Artificial Intelligence Approaches
1. Statistical AI: Machine Learning (Computational Statistics, Mathematical Optimisation)perception, understanding, prediction, classification
2. Symbolic AI: Automated Reasoning (Symbolic AI, Search based AI)problem solving, decision making, planning
3. Sub-symbolic AI: (Control theory, Computational intelligence, Softcomputing) robotics, alternative problem solving, alternative understanding
4. Collective AI: Multiagent systems (Agent Architectures, Game Theory, Mechanism Design, Combinatorial Auctions)robotics, distributed systems, market mechanisms, AI simulations
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Artificial Intelligence Approaches
1. Statistical AI: Machine Learning (Statistics, Optimisation, Neural Networks, Deep Learning)perception, understanding, prediction, classification
2. Symbolic AI: Automated Reasoning (Symbolic AI, Search based AI)problem solving, decision making, planning
3. Sub-symbolic AI: (Control theory, Computational intelligence, Softcomputing) robotics, alternative problem solving, alternative understanding
4. Collective AI: Multiagent systems (Agent Architectures, Game Theory, Mechanism Design, Combinatorial Auctions)robotics, distributed systems, market mechanisms, AI simulations
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Artificial Intelligence Approaches
1. Statistical AI: Machine Learning (Computational Statistics, Mathematical Optimisation)perception, understanding, prediction, classification
2. Symbolic AI: Automated Reasoning (Symbolic AI, Search based AI)problem solving, decision making, planning
3. Sub-symbolic AI: (Control theory, Computational intelligence, Softcomputing, Neural Networks, Deep Learning) robotics, alternative problem solving, alternative understanding
4. Collective AI: Multiagent systems (Agent Architectures, Game Theory, Mechanism Design, Combinatorial Auctions)robotics, distributed systems, market mechanisms, AI simulations
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Artificial Intelligence in OI
1. Statistical AI: Machine Learning. (Computational Statistics, Mathematical Optimisation)perception, understanding, prediction, classification
2. Symbolic AI: Automated Reasoning (Symbolic AI, Search based AI)problem solving, decision making, planning
3. Sub-symbolic AI: (Control theory, Computational intelligence, Softcomputing, Neural Networks, Deep Learning) robotics, alternative problem solving, alternative understanding
4. Collective AI: Multiagent systems (Agent Architectures, Game Theory, Mechanism Design, Combinatorial Auctions)robotics, distributed systems, market mechanisms, AI simulations
B4B33RPZ
B4B36ZUI
B4B36FUP
B4M33SSU
B4M33MAS
B4M33PUIB4M33LUP
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B4B36ZUI Content
1. Introduction to AI. Search-based AI, Uninformed search 2. Informed A* search 3. Advanced A*: RBFS, SMA* 4. Local search, Online search 5. Constrain Satisfaction Problem 6. Two players games 7. Monte Carlo Tree Search 8. Knowledge representation - Introduction 9. Knowledge representation in FOL 10. Rational Decisions under Uncertainty 11. Sequential Decisions under Uncertainty 12. Knowledge in Multi-agent Systems 13. AI Applications
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Introduction to AI Uninformed Search
R&N: Chap. 3, Sect. 3.1–3.6
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Example: 8-Puzzle
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Initial state Goal state
State: any arrangement of 8 numbered tiles and an empty tile
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8-Puzzle: Successor Function
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search is about the exploration of alternatives
SUCC(state) → subsetofstates The successor function is knowledgeabout the 8-puzzle game, but it does not tell us which outcome to use, nor towhich state of the board to apply it.
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(n2-1)-puzzle
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4321 ....
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15-Puzzle Sam Loyd offered $1,000 of his own money to the
first person who would solve the following problem:
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Stating a Problem as a Search Problem
▪ State space S ▪ Successor function: x ∈ S → SUCCESSORS(x) ∈ 2S
▪ Initial state s0 ▪ Goal test: x∈S → GOAL?(x) =T or F ▪ Arc cost
S1
3 2
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State Graph
▪ Each state is represented by a distinct node
▪ An arc (or edge) connects a node s to a node s’ if s’ ∈ SUCC(s)
▪ The state graph may contain more than one connected component
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Solution to the Search Problem▪ A solution is a path connecting the initial node to
a goal node (any one)
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Solution to the Search Problem▪ A solution is a path connecting the initial node to
a goal node (any one) ▪ The cost of a path is the sum of the arc costs
along this path ▪ An optimal solution is a solution path of minimum
cost ▪ There might be
no solution ! I
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How big is the state space of the (n2-1)-puzzle?
▪ 8-puzzle → 9! = 362,880 states ▪ 15-puzzle → 16! ~ 2.09 x 1013 states ▪ 24-puzzle → 25! ~ 1025 states
But only half of these states are reachable from any given state(but you may not know that in advance)
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8-puzzle ! 362,880 states
15-puzzle ! 2.09 x 1013 states
24-puzzle ! 1025 states
100 millions states/sec
0.036 sec
~ 55 hours
> 109 years
8-, 15-, 24-Puzzles
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Searching the State Space
▪ Often it is not feasible (or too expensive) to build a complete representation of the state graph
▪ A problem solver must construct a solution by exploring a small portion of the graph
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Other examples
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8-Queens ProblemPlace 8 queens in a chessboard so that no two queens are in the same row, column, or diagonal.
A solution Not a solution
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Formulation #1
▪ States: all arrangements of 0, 1, 2, ..., 8 queens on the board
▪ Initial state: 0 queens on the board
▪ Successor function: each of the successors is obtained by adding one queen in an empty square
▪ Arc cost: irrelevant ▪ Goal test: 8 queens are on the
board, with no queens attacking each other
! ~ 64x63x...x57 ~ 3x1014 states
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Formulation #2
▪ States: all arrangements of k = 0, 1, 2, ..., 8 queens in the k leftmost columns with no two queens attacking each other
▪ Initial state: 0 queens on the board ▪ Successor function: each
successor is obtained by adding one queen in any square that is not attacked by any queen already in the board, in the leftmost empty column
▪ Arc cost: irrelevant ▪ Goal test: 8 queens are on the
board! 2,057 states
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n-Queens Problem▪ A solution is a goal node, not a path to this node
(typical of design problem) ▪ Number of states in state space:
8-queens → 2,057 100-queens → 1052
▪ But techniques exist to solve n-queens problems efficiently for large values of n
They exploit the fact that there are many solutions well distributed in the state space
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Path Planning
What is the state space?
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Formulation #1
Cost of one horizontal/vertical step = 1 Cost of one diagonal step = √2
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Optimal Solution
This path is the shortest in the discretized state space, but not in the original continuous space
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Formulation #2sweep-line
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Formulation #2
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States
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Successor Function
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Solution Path
A path-smoothing post-processing step is usually needed to shorten the path further
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Formulation #3
Cost of one step: length of segment
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Formulation #3
Cost of one step: length of segment
Visibility graph
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Solution Path
The shortest path in this state space is also the shortest in the original continuous space
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Simple Problem-Solving-Agent
1. s0 ← sense/read initial state 2. GOAL? ← select/read goal test 3. Succ ← read successor function 4. solution ← search(s0, GOAL?, Succ)
5. perform(solution)
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Search Nodes and States
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If states are allowed to be revisited,the search tree may be infinite even
when the state space is finite
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Data Structure of a Node
PARENT-NODESTATE
Depth of a node N = length of path from root to N
(depth of the root = 0)
BOOKKEEPING
5Path-Cost
5DepthRightAction
Expanded yes...
CHILDREN
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Node expansion The expansion of a node N consists of:
1) Evaluating the successor function on STATE(N)
2) Generating a child of N for each state returned by the function
node generation ≠ node expansion
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▪ The Open-List is the set of all search nodes that haven’t been expanded yet
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Open-List of Search Tree
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Search Strategy
▪ The Open-List is the set of all search nodes that haven’t been expanded yet
▪ The Open-List is implemented as a priority queue INSERT(node,Open-List) REMOVE(Open-List) ▪ The ordering of the nodes in Open-List defines
the search strategy
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Search Algorithm #1SEARCH#1 1. If GOAL?(initial-state) then return initial-state 2. INSERT(initial-node,Open-List) 3. Repeat:
a. If empty(Open-List) then return failure b. N ← REMOVE(Open-List) c. s ← STATE(N) d. For every state s’ in SUCCESSORS(s)
i. Create a new node N’ as a child of N ii. If GOAL?(s’) then return path or goal state iii.INSERT(N’,Open-List)
Expansion of N
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Performance Measures
▪ CompletenessA search algorithm is complete if it finds a solution whenever one exists[What about the case when no solution exists?]
▪ OptimalityA search algorithm is optimal if it returns a minimum-cost path whenever a solution exists
▪ ComplexityIt measures the time and amount of memory required by the algorithm
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Blind vs. Heuristic Strategies
▪ Blind (or un-informed) strategies do not exploit state descriptions to order Open-List. They only exploit the positions of the nodes in the search tree
▪ Heuristic (or informed) strategies exploit state descriptions to order Open-List (the most “promising” nodes are placed at the beginning of Open-List)
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ExampleFor a blind strategy, N1 and N2 are just two nodes (at some position in the search tree)
Goal state
N1
N2STATE
STATE1
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ExampleFor a heuristic strategy counting the number of misplaced tiles, N2 is more promising than N1
Goal state
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Remark
▪ Some search problems, such as the (n2-1)-puzzle, are NP-hard
▪ One can’t expect to solve all instances of such problems in less than exponential time (in n)
▪ One may still strive to solve each instance as efficiently as possible ! This is the purpose of the search strategy
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Blind Strategies
▪ Breadth-first - Bidirectional
▪ Depth-first - Depth-limited - Iterative deepening
▪ Uniform-Cost(variant of breadth-first)
Arc cost = 1
Arc cost = c(action) ≥ ε > 0
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Breadth-First Strategy
New nodes are inserted at the end of Open-List
2 3
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Open-List = (1)
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Breadth-First Strategy
New nodes are inserted at the end of Open-List
Open-List = (2, 3)2 3
4 5
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Breadth-First Strategy
New nodes are inserted at the end of Open-List
Open-List = (3, 4, 5)2 3
4 5
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Breadth-First Strategy
New nodes are inserted at the end of Open-List
Open-List = (4, 5, 6, 7)2 3
4 5
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6 7
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Important Parameters
1) Maximum number of successors of any state! branching factor b of the search tree
2) Minimal length (≠ cost) of a path between the initial and a goal state! depth d of the shallowest goal node in the search tree
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Evaluation
▪ b: branching factor ▪ d: depth of shallowest goal node ▪ Breadth-first search is:
- Complete? Not complete? - Optimal? Not optimal?
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Evaluation
▪ b: branching factor ▪ d: depth of shallowest goal node ▪ Breadth-first search is:
- Complete - Optimal if step cost is 1
▪ Number of nodes generated: ???
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Evaluation
▪ b: branching factor ▪ d: depth of shallowest goal node ▪ Breadth-first search is:
- Complete - Optimal if step cost is 1
▪ Number of nodes generated:1 + b + b2 + … + bd = ???
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Evaluation
▪ b: branching factor ▪ d: depth of shallowest goal node ▪ Breadth-first search is:
- Complete - Optimal if step cost is 1
▪ Number of nodes generated: 1 + b + b2 + … + bd. = (bd+1-1)/(b-1) = O(bd)
▪ Time and space complexity is O(bd)
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Big O Notation
g(n) = O(f(n)) if there exist two positive constants a and N such that:
for all n > N: g(n) ≤ a × f(n)
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Time and Memory Requirements
d # Nodes Time Memory2 111 .01 msec 11 Kbytes4 11,111 1 msec 1 Mbyte6 ~106 1 sec 100 Mb8 ~108 100 sec 10 Gbytes10 ~1010 2.8 hours 1 Tbyte12 ~1012 11.6 days 100 Tbytes14 ~1014 3.2 years 10,000 Tbytes
Assumptions: b = 10; 1,000,000 nodes/sec; 100bytes/node
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RemarkIf a problem has no solution, breadth-first may run for ever (if the state space is infinite or states can be revisited arbitrary many times)
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Bidirectional Strategytwo Open-List queues: Open-List1 and Open-List2
s
Time and space complexity is O(bd/2) << O(bd) if both trees have the same branching factor b
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Depth-First Strategy
New nodes are inserted at the front of Open-List
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Open-List = (1)
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Depth-First Strategy
New nodes are inserted at the front of Open-List
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Open-List = (2, 3)
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Depth-First Strategy
New nodes are inserted at the front of Open-List
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Open-List = (4, 5, 3)
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Depth-First Strategy
New nodes are inserted at the front of Open-List
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Depth-First Strategy
New nodes are inserted at the front of Open-List
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Depth-First Strategy
New nodes are inserted at the front of Open-List
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Depth-First Strategy
New nodes are inserted at the front of Open-List
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Depth-First Strategy
New nodes are inserted at the front of Open-List
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2 3
4 5
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Depth-First Strategy
New nodes are inserted at the front of Open-List
1
2 3
4 5
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Depth-First Strategy
New nodes are inserted at the front of Open-List
1
2 3
4 5
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Depth-First Strategy
New nodes are inserted at the front of Open-List
1
2 3
4 5
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Evaluation▪ b: branching factor ▪ d: depth of shallowest goal node ▪ m: maximal depth of a leaf node ▪ Depth-first search is:
- Complete?- Optimal?
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Evaluation▪ b: branching factor ▪ d: depth of shallowest goal node ▪ m: maximal depth of a leaf node ▪ Depth-first search is:
- Complete only for finite search tree- Not optimal
▪ Number of nodes generated (worst case): 1 + b + b2 + … + bm = O(bm)
▪ Time complexity is O(bm) ▪ Space complexity is O(bm) [or O(m)] [Reminder: Breadth-first requires O(bd) time and space]
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Depth-Limited Search
▪ Depth-first with depth cutoff k (depth at which nodes are not expanded)
▪ Three possible outcomes: - Solution - Failure (no solution) - Cutoff (no solution within cutoff)
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Iterative Deepening Search
the best of both breadth-first and depth-first search
IDS: for k = 0,1,2, … d do: Depth-first search with depth cutoff k
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Iterative Deepening
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Iterative Deepening
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Iterative Deepening
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Performance
▪ Iterative deepening search is: - Complete - Optimal if step cost =1
▪ Time complexity is:(d+1)(1) + db + (d-1)b2 + … + (1) bd = O(bd)
▪ Space complexity is: O(bd) or O(d)
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d = 5 and b = 2BF ID1 1 x 6 = 62 2 x 5 = 104 4 x 4 = 168 8 x 3 = 2416 16 x 2 = 3232 32 x 1 = 3263 120
120/63 ~ 2
Number of Generated Nodes (Breadth-First & Iterative Deepening)
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d = 5 and b = 10BF ID1 610 50100 4001,000 3,00010,000 20,000100,000 100,000111,111 123,456
123,456/111,111 ~ 1.111
Number of Generated Nodes (Breadth-First & Iterative Deepening)
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Comparison of Strategies
▪ Breadth-first is complete and optimal, but has high space complexity
▪ Depth-first is space efficient, but is neither complete, nor optimal
▪ Iterative deepening is complete and optimal, with the same space complexity as depth-first and almost the same time complexity as breadth-first
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Revisited States
8-queens
No
assembly planning
Few
1 2 34 5
67 8
8-puzzle and robot navigation
Many
search tree is finitesearch tree is infinite
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Avoiding Revisited States
▪ Requires comparing state descriptions ▪ Breadth-first search:
• Store all states associated with generated nodes in Closed-List
• If the state of a new node is in Closed-List, then discard the node
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Avoiding Revisited States
▪ Depth-first search: Solution 1:
– Store all states associated with nodes in current path in Closed-List
– If the state of a new node is in Closed-List, then discard the node
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Avoiding Revisited States
▪ Depth-first search: Solution 1:
– Store all states associated with nodes in current path in Closed-List
– If the state of a new node is in Closed-List, then discard the node
Only avoids loops
Solution 2: – Store all generated states in Closed-List – If the state of a new node is in Closed-List, then
discard the node Same space complexity as breadth-first !
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Uniform-Cost Search▪ Each arc has some cost c ≥ ε > 0, The cost of the path to each node N is g(N) = Σ costs of arcs ▪ The goal is to generate a solution path of minimal cost ▪ The nodes in the Open-List are sorted in increasing g(N)
▪ Need to modify search algorithm
S0
1A
5B
15C
S G
A
B
C
51
15
10
5
5
G11
G10
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Search Algorithm #1SEARCH#1 1. If GOAL?(initial-state) then return initial-state 2. INSERT(initial-node,Open-List) 3. Repeat:
a. If empty(Open-List) then return failure b. N ← REMOVE(Open-List) c. s ← STATE(N) d. For every state s’ in SUCCESSORS(s)
i. Create a new node N’ as a child of N ii. If GOAL?(s’) then return path or goal state iii.INSERT(N’,Open-List)
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Search Algorithm #2SEARCH#2 1. INSERT(initial-node,Open-List) 2. Repeat:
a. If empty(Open-List) then return failure b. N ← REMOVE(Open-List) c. s ← STATE(N) d. If GOAL?(s) then return path or goal state e. For every state s’ in SUCCESSORS(s)
i. Create a node N’ as a successor of N ii.INSERT(N’,Open-List)
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Avoiding Revisited States in Uniform-Cost Search
▪ For any state S, when the first node N such that STATE(N) = S is expanded, the path to N is the best path from the initial state to S
▪ So:
• When a node is expanded, store its state into CLOSED
• When a new node N is generated:
– If STATE(N) is in CLOSED, discard N
– If there exits a node N’ in the Open-List such that STATE(N’) = STATE(N), discard the node (N or N’) with the highest-cost path
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Search Algorithm #3SEARCH#3 1. INSERT(initial-node,Open-List) 2. Repeat:
a. If empty(Open-List) then return failure b. N ← REMOVE(Open-List) c. s ← STATE(N) d. INSERT(N,Closed-List) e. If GOAL?(s) then return path or goal state f. For every state s’ in SUCCESSORS(s)
i. Create a node N’ as a successor of N ii.If N is not in Closed-List and
If N is not on Open-List with lower cost then INSERT(N’,Open-List)
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113
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114Homework
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17
" A tile j appears after a tile i if either j appears on the same row as i to the right of i, or on another row below the row of i.
" For every i = 1, 2, ..., 15, let ni be the number of tiles j < i that appear after tile i (permutation inversions)
" N = n2 + n3 + … + n15 + row number of empty tile
Permutation Inversions
121511
1410
1395 6 7 8
4321
12
15
11
14
6
13
9
5 10 7 8
4321n2 = 0 n3 = 0 n4 = 0 n5 = 0 n6 = 0 n7 = 1 n8 = 1 n9 = 1 n10 = 4 n11 = 0 n12 = 0 n13 = 0 n14 = 0 n15 = 0
! N = 7 + 4
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" Proposition: (N mod 2) is invariant under any legal move of the empty tile
" Proof:• Any horizontal move of the empty tile leaves N
unchanged • A vertical move of the empty tile changes N by an
even increment (± 1 ± 1 ± 1 ± 1)
1215
11
14
10
13
9
5 6 7
8
4321
s =
1215
11
14
10
13
9
5 6 7
8
4321
s’ = N(s’) = N(s) + 3 + 1
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" Proposition: (N mod 2) is invariant under any legal move of the empty tile
" ! For a goal state g to be reachable from a state s, a necessary condition is that N(g) and N(s) have the same parity
" It can be shown that this is also a sufficient condition
" ! The state graph consists of two connected components of equal size