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CS347 – Introduction to
Artificial Intelligence
CS347 course website:
http://web.mst.edu/~tauritzd/courses/cs347/
Dr. Daniel Tauritz (Dr. T)
Department of Computer Science
[email protected]
http://web.mst.edu/~tauritzd/
What is AI?
Systems that…
–act like humans (Turing Test)
–think like humans
–think rationally
–act rationally
Play Ultimatum Game
Rational Agents
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Environment
Sensors (percepts)
Actuators (actions)
Agent Function
Agent Program
Performance Measures
Rational Behavior
Depends on:
• Agent’s performance measure
• Agent’s prior knowledge
• Possible percepts and actions
• Agent’s percept sequence
Rational Agent Definition
“For each possible percept
sequence, a rational agent selects
an action that is expected to
maximize its performance measure,
given the evidence provided by the
percept sequence and any prior
knowledge the agent has.”
Task Environments
PEAS description & properties:
–Fully/Partially Observable
–Deterministic, Stochastic, Strategic
–Episodic, Sequential
–Static, Dynamic, Semi-dynamic
–Discrete, Continuous
–Single agent, Multiagent
–Competitive, Cooperative
Problem-solving agents
A definition:
Problem-solving agents are goal
based agents that decide what
to do based on an action
sequence leading to a goal
state.
Problem-solving steps
• Problem-formulation (actions &
states)
• Goal-formulation (states)
• Search (action sequences)
• Execute solution
Well-defined problems
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Initial state
Action set
Transition model: RESULT(s,a)
Goal test
Path cost
Solution / optimal solution
Example problems
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Vacuum world
Tic-tac-toe
8-puzzle
8-queens problem
Search trees
• Root corresponds with initial state
• Vacuum state space vs. search tree
• Search algorithms iterate through
goal testing and expanding a state
until goal found
• Order of state expansion is critical!
Search node datastructure
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n.STATE
n.PARENT-NODE
n.ACTION
n.PATH-COST
States are NOT search nodes!
Frontier
• Frontier = Set of leaf nodes
• Implemented as a queue with ops:
– EMPTY?(queue)
– POP(queue)
– INSERT(element,queue)
• Queue types: FIFO, LIFO (stack), and
priority queue
Problem-solving performance
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Completeness
Optimality
Time complexity
Space complexity
Complexity in AI
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b – branching factor
d – depth of shallowest goal node
m – max path length in state space
Time complexity: # generated nodes
Space complexity: max # nodes stored
Search cost: time + space complexity
Total cost: search + path cost
Tree Search
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Breadth First Tree Search (BFTS)
Uniform Cost Tree Search (UCTS)
Depth-First Tree Search (DFTS)
Depth-Limited Tree Search (DLTS)
Iterative-Deepening Depth-First
Tree Search (ID-DFTS)
Example state space #1
Breadth First Tree Search (BFTS)
• Frontier: FIFO queue
• Complete: if b and d are finite
• Optimal: if path-cost is non-decreasing
function of depth
• Time complexity: O(b^d)
• Space complexity: O(b^d)
Uniform Cost Tree Search (UCTS)
• Frontier: priority queue ordered by g(n)
Depth First Tree Search (DFTS)
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Frontier: LIFO queue (a.k.a. stack)
Complete: no
Optimal: no
Time complexity: O(bm)
Space complexity: O(bm)
Backtracking version of DFTS has a space
complexity of: O(m)
Depth-Limited Tree Search (DLTS)
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Frontier: LIFO queue (a.k.a. stack)
Complete: not when l < d
Optimal: no
Time complexity: O(b^l)
Space complexity: O(bl)
Diameter: min # steps to get from any state
to any other state
Diameter
example 1
Diameter
example 2
Example state space #2
Graph Search
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Breadth First Graph Search (BFGS)
Uniform Cost Graph Search (UCGS)
Depth-First Graph Search (DFGS)
Depth-Limited Graph Search (DLGS)
Iterative-Deepening Depth-First
Graph Search (ID-DFGS)
Best First Search (BeFS)
• Select node to expand based on
evaluation function f(n)
• Typically node with lowest f(n)
selected because f(n) correlated with
path-cost
• Represent frontier with priority queue
sorted in ascending order of f-values
Path-cost functions
• g(n) = lowest path-cost from
start node to node n
• h(n) = estimated path-cost of
cheapest path from node n to a
goal node [with h(goal)=0]
Heuristics
• h(n) is a heuristic function
• Heuristics incorporate problemspecific knowledge
• Heuristics need to be relatively
efficient to compute
Important BeFS algorithms
• UCS: f(n) = g(n)
• GBeFS: f(n) = h(n)
• A*S: f(n) = g(n)+h(n)
GBeFTS
• Incomplete (so also not optimal)
• Worst-case time and space
complexity: O(bm)
• Actual complexity depends on
accuracy of h(n)
A*S
• f(n) = g(n) + h(n)
• f(n): estimated cost of optimal
solution through node n
• if h(n) satisfies certain
conditions, A*S is complete &
optimal
Example state space # 3
Admissible heuristics
• h(n) admissible if:
Example: straight line distance
A*TS optimal if h(n) admissible
Consistent heuristics
• h(n) consistent if:
Consistency implies admissibility
A*GS optimal if h(n) consistent
A* search notes
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Optimally efficient for consistent heuristics
Run time is a function of the heuristic error
Suboptimal variants
Not strictly admissible heuristics
A* Graph Search not scalable due to
memory requirements
Memory-bounded heuristic search
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Iterative Deepening A* (IDA*)
Recursive Best-First Search (RBFS)
IDA* and RBFS don’t use all avail. memory
Memory-bounded A* (MA*)
Simplified MA* (SMA*)
Meta-level learning aims to minimize total
problem solving cost
Heuristic Functions
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Effective branching factor
Domination
Composite heuristics
Generating admissible heuristics from
relaxed problems
Sample relaxed problem
• n-puzzle legal actions:
Move from A to B if horizontally or
vertically adjacent and B is blank
Relaxed problems:
(a)Move from A to B if adjacent
(b)Move from A to B if B is blank
(c) Move from A to B
Generating admissible heuristics
The cost of an optimal solution to a
relaxed problem is an admissible
heuristic for the original problem.
Adversarial Search
Environments characterized by:
• Competitive multi-agent
• Turn-taking
Simplest type: Discrete, deterministic,
two-player, zero-sum games of
perfect information
Search problem formulation
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S0: Initial state (initial board setup)
Player(s):
Actions(s):
Result(s,a):
Terminal test: game over!
Utility function: associates playerdependent values with terminal states
Minimax
Depth-Limited Minimax
• State Evaluation Heuristic
estimates Minimax value of a node
• Note that the Minimax value of a
node is always calculated for the
Max player, even when the Min
player is at move in that node!
Iterative-Deepening Minimax
• IDM(n,d) calls DLM(n,1),
DLM(n,2), …, DLM(n,d)
• Advantages:
–Solution availability when time is
critical
–Guiding information for deeper
searches
Redundant info example
Alpha-Beta Pruning
• α: worst value that Max will accept at this
point of the search tree
• β: worst value that Min will accept at this
point of the search tree
• Fail-low: encountered value <= α
• Fail-high: encountered value >= β
• Prune if fail-low for Min-player
• Prune if fail-high for Max-player
DLM w/ Alpha-Beta Pruning
Time Complexity
• Worst-case: O(bd)
• Best-case: O(bd/2) [Knuth & Moore, 1975]
• Average-case: O(b3d/4)
Move Ordering Heuristics
• Knowledge based
• Killer Move: the last move at a given
depth that caused AB-pruning or had
best minimax value
• History Table
Example game tree
Example game tree
Search Depth Heuristics
• Time based / State based
• Horizon Effect: the phenomenon of
deciding on a non-optimal principal variant
because an ultimately unavoidable
damaging move seems to be avoided by
blocking it till passed the search depth
• Singular Extensions / Quiescence Search
Time Per Move
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Constant
Percentage of remaining time
State dependent
Hybrid
Quiescence Search
• When search depth reached, compute
quiescence state evaluation heuristic
• If state quiescent, then proceed as usual;
otherwise increase search depth if
quiescence search depth not yet reached
• Call format: QSDLM(root,depth,QSdepth),
QSABDLM(root,depth,QSdepth,α,β), etc.
QS game tree Ex. 1
QS game tree Ex. 2
Forward pruning
• Beam Search (n best moves)
• ProbCut (forward pruning version of
alpha-beta pruning)
Transposition Tables (1)
• Hash table of previously calculated
state evaluation heuristic values
• Speedup is particularly huge for
iterative deepening search
algorithms!
• Good for chess because often
repeated states in same search
Transposition Tables (2)
• Datastructure: Hash table indexed by
position
• Element:
–State evaluation heuristic value
–Search depth of stored value
–Hash key of position (to eliminate
collisions)
–(optional) Best move from position
Transposition Tables (3)
• Zobrist hash key
– Generate 3d-array of random 64-bit numbers
(piece type, location and color)
– Start with a 64-bit hash key initialized to 0
– Loop through current position, XOR’ing hash
key with Zobrist value of each piece found
(note: once a key has been found, use an
incremental apporach that XOR’s the “from”
location and the “to” location to move a piece)
MTD(f)
MTDf(root,guess,depth) {
lower = -∞;
upper = ∞;
do {
beta=guess+(guess==lower);
guess = ABMaxV(root,depth,beta-1,beta);
if (guess<beta) upper=guess; else lower=guess;
} while (lower < upper);
return guess; } // also needs to return best move
IDMTD(f)
IDMTDf(root,first_guess,depth_limit) {
guess = first_guess;
for (depth=1; depth ≤ depth_limit; depth++)
guess = MTDf(root,guess,depth);
return guess; }
// actually needs to return best move
Adversarial Search in
Stochastic Environments
Worst Case Time Complexity: O(bmnm)
with b the average branching factor,
m the deepest search depth, and n the
average chance branching factor
Example “chance” game tree
Expectiminimax & Pruning
• Interval arithmetic
• Monte Carlo simulations (for dice called
a rollout)
Null Move Forward Pruning
• Before regular search, perform
shallower depth search (typically
two ply less) with the opponent at
move; if beta exceeded, then prune
without performing regular search
• Sacrifices optimality for great speed
increase
Futility Pruning
• If the current side to move is not in check,
the current move about to be searched is
not a capture and not a checking move,
and the current positional score plus a
certain margin (generally the score of a
minor piece) would not improve alpha, then
the current node is poor, and the last ply of
searching can be aborted.
• Extended Futility Pruning
• Razoring
State-Space Search
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Complete-state formulation
Objective function
Global optima
Local optima (don’t use textbook’s
definition!)
• Ridges, plateaus, and shoulders
• Random search and local search
Steepest-Ascent Hill-Climbing
• Greedy Algorithm - makes locally optimal choices
Example
8 queens problem has 88≈17M states
SAHC finds global optimum for 14% of instances
in on average 4 steps (3 steps when stuck)
SAHC w/ up to 100 consecutive sideways moves,
finds global optimum for 94% of instances in on
average 21 steps (64 steps when stuck)
Stochastic Hill-Climbing
• Chooses at random from among uphill
moves
• Probability of selection can vary with the
steepness of the uphill move
• On average slower convergence, but
also less chance of premature
convergence
More Local Search Algorithms
• First-choice hill-climbing
• Random-restart hill-climbing
• Simulated Annealing
Population Based Local Search
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Deterministic local beam search
Stochastic local beam search
Evolutionary Algorithms
Particle Swarm Optimization
Ant Colony Optimization
Particle Swarm Optimization
• PSO is a stochastic population-based
optimization technique which assigns
velocities to population members
encoding trial solutions
• PSO update rules:
PSO demo: http://www.borgelt.net//psopt.html
Ant Colony Optimization
• Population based
• Pheromone trail and stigmergetic
communication
• Shortest path searching
• Stochastic moves
Online Search
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Offline search vs. online search
Interleaving computation & action
Exploration problems, safely explorable
Agents have access to:
– ACTIONS(s)
– c(s,a,s’)
– GOAL-TEST(s)
Online Search Optimality
• CR – Competitive Ratio
• TAPC – Total Actual Path Cost
• C* - Optimal Path Cost
TAPC
CR 
C*
• Best case: CR = 1
• Worst case: CR = ∞
Online Search Algorithms
• Online-DFS-Agent
• Random Walk
• Learning Real-Time A* (LRTA*)
Online Search Example Graph 1
Online Search Example Graph 2
Online Search Example Graph 3
AI courses at S&T
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CS345 Computational Robotic Manipulation (SP2012)
CS347 Introduction to Artificial Intelligence (SP2012)
CS348 Evolutionary Computing (FS2011)
CS434 Data Mining & Knowledge Discovery (FS2011)
CS447 Advanced Topics in AI (SP2013)
CS448 Advanced Evolutionary Computing (SP2012)
CpE358 Computational Intelligence (FS2011)
SysEng378 Intro to Neural Networks & Applications