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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
How difficult is it to achieve AI?
• Three Sisters Puzzle
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.
Environment Assumptions
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Fully Observable
Single Agent
Discrete
Sequential
Known & Deterministic
Open-loop 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: ACTIONS(s)
Transition model: RESULT(s,a)
Goal test
Step cost: c(s,a,s’)
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!
function TREE-SEARCH(problem) returns solution/fail
initialize frontier using initial problem state
loop do
if empty(frontier) then return fail
choose leaf node and remove it from frontier
if chosen node contains goal state then return corresponding solution
expand chosen node and add resulting nodes to frontier
Redundant paths
• Loopy paths
• Repeated states
• Redundant paths
function GRAPH-SEARCH(problem) returns solution/fail
initialize frontier using initial problem state
initialize explored set to be empty
loop do
if empty(frontier) then return fail
choose leaf node and remove it from frontier
if chosen node contains goal state then return corresponding solution
add chosen node to explored set
expand chosen node and add resulting nodes to frontier only if not yet in
frontier or explored set
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 Search (UCS)
• g(n) = lowest path-cost from start node to
node n
• 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
Iterative-Deepening Depth-First Tree
Search (ID-DFTS)
function ID-DFS(problem) returns solution/fail
for depth = 0 to ∞ do
result ← DLS(problem,depth)
if result ≠ cutoff then return result
• Complete: Yes, if b is finite
• Optimal: Yes, if path-cost is nondecreasing
function of depth
• Time complexity: O(b^d)
• Space complexity: O(bd)
Bidirectional Search
BiBFTS
• Complete: Yes, if b is finite
• Optimal: Not “out of the box”
• Time & Space complexity: O(bd/2)
Example state space #2
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 non-negative
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): which player has the move
Actions(s): set of legal moves
Result(s,a): defines transitional model
Terminal test: game over!
Utility function: associates playerdependent values with terminal states
Minimax
Example
game tree 1
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!
Heuristic Depth-Limited Minimax
State Eval Heuristic Qualities
A good State Eval Heuristic should:
(1)order the terminal states in the same
way as the utility function
(2)be relatively quick to compute
(3)strongly correlate nonterminal states
with chance of winning
Weighted Linear State Eval
Heuristic
EVAL(s)  i 1 wifi(s)
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Heuristic Iterative-Deepening Minimax
• IDM(s,d) calls DLM(s,1), DLM(s,2),
…, DLM(s,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)
Example
game tree 2
Move Ordering Heuristics
• Knowledge based (e.g., try captures first
in chess)
• Principal Variant (PV) based
• Killer Move: the last move at a given
depth that caused αβ-pruning or had best
minimax value
• History Table: track how often a
particular move at any depth caused αβpruning or had best minimax value
History Table (HT)
• Option 1: generate set of legal moves
and use HT value as f-value
• Option 2: keep moves with HT values in
a sorted array and for a given state
traverse the array to find the legal move
with the highest HT value
Example
game tree 3
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
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 approach that XOR’s the “from”
location and the “to” location to move a piece)
Search versus lookup
• Balancing time versus memory
• Opening table
– Human expert knowledge
– Monte Carlo analysis
• End game database
Forward pruning
• Beam Search (n best moves)
• ProbCut (forward pruning version of
alpha-beta pruning)
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
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)
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
Dynamic, nondeterministic, unknown domains
Exploration problems, safely explorable
Agents have access to:
– ACTIONS(s)
– c(s,a,s’)
 cannot be used until RESULT(s,a)
– 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
Descartes Mind-Body Connection
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René Descartes (1596-1650)
Rationalism
Dualism
Materialism
Star Trek & Souls
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.”
PEAS description & properties:
–Fully/Partially Observable
–Deterministic, Stochastic, Strategic
–Episodic, Sequential
–Static, Dynamic, Semi-dynamic
–Discrete, Continuous
–Single agent, Multiagent
–Competitive, Cooperative
–Known, Unknown
AI courses at S&T
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CS301 Intro to Data Mining (FS2014)
CS345 Intro to Robotics (FS2014)
CS346 Intro to Computer Vision (FS2014)
CS347 Introduction to Artificial Intelligence
CS348 Evolutionary Computing (FS2015)
CS444 Data Mining & Knowledge Discovery (SP2015)
CS447 Advanced Topics in AI (SP2015)
CS448 Advanced Evolutionary Computing (SP2016)
CpE358 Computational Intelligence (FS2014)
SysEng378 Intro to Neural Networks & Applications