Download presentation
Presentation is loading. Please wait.
Published byBlaise Mills Modified over 9 years ago
1
COMP SCI 5400 – Introduction to Artificial Intelligence Dr. Daniel Tauritz (Dr. T) Department of Computer Science tauritzd@mst.edu http://web.mst.edu/~tauritzd/ general course website: http://web.mst.edu/~tauritzd/courses/intro_AI.html http://web.mst.edu/~tauritzd/courses/intro_AI.html
2
What is AI? Systems that… –act like humans (Turing Test) –think like humans –think rationally –act rationally Play Ultimatum Game
3
Computer Agent Perceives environment Operates autonomously Persists over prolonged periods
4
Rational Agents Environment Sensors (percepts) Actuators (actions)
6
Rational Agents Environment Sensors (percepts) Actuators (actions) Agent Function Agent Program
8
Rational Behavior Depends on: Agent’s performance measure Agent’s prior knowledge Possible percepts and actions Agent’s percept sequence
9
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.”
10
PEAS description & properties: –Fully/Partially Observable –Deterministic, Stochastic, Strategic –Episodic, Sequential –Static, Dynamic, Semi-dynamic –Discrete, Continuous –Single agent, Multiagent –Competitive, Cooperative –Known, Unknown
11
Agent Types Simple Reflex Agents Model-Based Reflex Agents Goal-Based Agents Utility-Based Agents Learning Agents
12
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.
13
Environment Assumptions Fully Observable Single Agent Discrete Sequential Known & Deterministic
14
Open-loop problem-solving steps Problem-formulation (actions & states) Goal-formulation (states) Search (action sequences) Execute solution
15
Well-defined problems Initial state Action set: ACTIONS(s) Transition model: RESULT(s,a) Goal test Step cost: c(s,a,s’) Path cost Solution / optimal solution
16
Example problems Vacuum world Tic-tac-toe 8-puzzle 8-queens problem
17
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!
18
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
19
Redundant paths Loopy paths Repeated states Redundant paths
20
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
21
Search node datastructure n.STATE n.PARENT-NODE n.ACTION n.PATH-COST States are NOT search nodes!
22
function CHILD-NODE(problem,parent,action) returns a node return a node with: STATE = problem.RESULT(parent.STATE,action) PARENT = parent ACTION = action PATH-COST = parent.PATH-COST + problem.STEP-COST(parent.STATE,action)
23
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
24
Explored Set Explored Set = Set of expanded nodes Implemented typically as a hash table for constant time insertion & lookup
25
Problem-solving performance Completeness Optimality Time complexity Space complexity
26
Complexity in AI 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
27
Tree Search 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)
28
Example state space #1
29
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)
30
Uniform Cost Search (UCS) g(n) = lowest path-cost from start node to node n Frontier: priority queue ordered by g(n)
31
Depth First Tree Search (DFTS) Frontier: LIFO queue (a.k.a. stack) Complete: no (DGFS is complete for finite state spaces) Optimal: no Time complexity: O(b m ) Space complexity: O(bm) Backtracking version of DFTS: –space complexity: O(m) –modifies rather than copies state description
32
Depth-Limited Tree Search (DLTS) 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
33
Diameter example 1
34
Diameter example 2
35
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)
36
Bidirectional Search BiBFTS Complete: Yes, if b is finite Optimal: Not “out of the box” Time & Space complexity: O(b d/2 )
37
Example state space #2
38
Best First Search (BeFS) Select node to expand based on evaluation function f(n) Node with lowest f(n) selected as f(n) correlated with path-cost Represent frontier with priority queue sorted in ascending order of f-values
39
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]
40
Heuristics h(n) is a heuristic function Heuristics incorporate problem- specific knowledge Heuristics need to be relatively efficient to compute
41
Important BeFS algorithms UCS: f(n) = g(n) GBeFS: f(n) = h(n) A*S: f(n) = g(n)+h(n)
42
GBeFTS Incomplete (so also not optimal) Worst-case time and space complexity: O(b m ) Actual complexity depends on accuracy of h(n)
43
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
44
Example state space # 3
45
Admissible heuristics h(n) admissible if: Example: straight line distance A*TS optimal if h(n) admissible
46
Consistent heuristics h(n) consistent if: Consistency implies admissibility A*GS optimal if h(n) consistent
47
A* search notes 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
48
Memory-bounded heuristic search 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
49
Heuristic Functions Effective branching factor Domination Composite heuristics Generating admissible heuristics from relaxed problems
50
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
51
Generating admissible heuristics The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem.
52
Adversarial Search Environments characterized by: Competitive multi-agent Turn-taking Simplest type: Discrete, deterministic, two-player, zero-sum games of perfect information
53
Search problem formulation S 0 : 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 player- dependent values with terminal states
54
Minimax
55
Example game tree 1
56
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!
57
Heuristic Depth-Limited Minimax
58
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
59
Weighted Linear State Eval Heuristic
60
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
61
Redundant info example
62
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
63
DLM w/ Alpha-Beta Pruning Time Complexity Worst-case: O(b d ) Best-case: O(b d/2 ) [Knuth & Moore, 1975] Average-case: O(b 3d/4 )
64
Example game tree 2
65
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 αβ- pruning or had best minimax value
66
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
67
Example game tree 3
68
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
69
Time Per Move Constant Percentage of remaining time State dependent Hybrid
70
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.
71
QS game tree Ex. 1
72
QS game tree Ex. 2
74
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
75
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
76
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)
77
Search versus lookup Balancing time versus memory Opening table –Human expert knowledge –Monte Carlo analysis End game database
78
Forward pruning Beam Search (n best moves) ProbCut (forward pruning version of alpha-beta pruning)
79
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
80
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
81
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
82
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
84
Adversarial Search in Stochastic Environments Worst Case Time Complexity: O(b m n m ) with b the average branching factor, m the deepest search depth, and n the average chance branching factor
85
Example “chance” game tree
86
Expectiminimax & Pruning Interval arithmetic Monte Carlo simulations (for dice called a rollout)
87
State-Space Search Complete-state formulation Objective function Global optima Local optima (don’t use textbook’s definition!) Ridges, plateaus, and shoulders Random search and local search
88
Steepest-Ascent Hill-Climbing Greedy Algorithm - makes locally optimal choices Example 8 queens problem has 8 8 ≈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)
89
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
90
More Local Search Algorithms First-choice hill-climbing Random-restart hill-climbing Simulated Annealing
91
Population Based Local Search Deterministic local beam search Stochastic local beam search Evolutionary Algorithms Particle Swarm Optimization Ant Colony Optimization
92
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.htmlhttp://www.borgelt.net//psopt.html
93
Ant Colony Optimization Population based Pheromone trail and stigmergetic communication Shortest path searching Stochastic moves
94
Online Search 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)
95
Online Search Optimality CR – Competitive Ratio TAPC – Total Actual Path Cost C* - Optimal Path Cost Best case: CR = 1 Worst case: CR = ∞
96
Online Search Algorithms Online-DFS-Agent Random Walk Learning Real-Time A* (LRTA*)
97
Online Search Example Graph 1
98
Online Search Example Graph 2
99
Online Search Example Graph 3
100
Descartes Mind-Body Connection René Descartes (1596-1650) Rationalism Dualism Materialism Star Trek & Souls
101
Key historical events for AI 4 th century BC Aristotle propositional logic 1600’s Descartes mind-body connection 1805 First programmable machine Mid 1800’s Charles Babbage’s “difference engine” & “analytical engine” Lady Lovelace’s Objection 1847 George Boole propositional logic 1879 Gottlob Frege predicate logic
102
Key historical events for AI 1931 Kurt Godel: Incompleteness Theorem In any language expressive enough to describe natural number properties, there are undecidable (incomputable) true statements 1943 McCulloch & Pitts: Neural Computation 1956 Term “AI” coined 1976 Newell & Simon’s “Physical Symbol System Hypothesis” A physical symbol system has the necessary and sufficient means for general intelligent action.
103
AI courses at S&T 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
104
How difficult is it to achieve AI? Three Sisters Puzzle
Similar presentations
© 2025 SlidePlayer.com Inc.
All rights reserved.