CS347Introduction to

Artificial Intelligence

Dr. Daniel Tauritz (Dr. T)tauritzd@umr.eduhttp://web.umr.edu/~tauritzd

What is AI?

Systems that…think like humansact like humansthink rationallyact rationally

Play Ultimatum Game

Key historical events for AI

1600’s Descartes mind-body connection 1805 First programmable machine Mid 1800’s Charles Babbage’s “difference

engine” & “analytical engine” Lady Lovelace’s Objection 1943 McCulloch & Pitts: Neural Computation 1976 Newell & Simon’s “Physical Symbol

System Hypothesis”

Rational Agents

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 PEAS properties:

Fully/Partially ObservableDeterministic, Stochastic, StrategicEpisodic, SequentialStatic, Dynamic, SemidynamicDiscrete, ContinuousSingle agent, MultiagentCompetitive, 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 Goal-formulation Search Execute solution

Play Three Sisters Puzzle

Well-defined problems

Initial state Successor function Goal test Path cost Solution Optimal solution

Example problems

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! Water jug example

Search node datastructure

STATE PARENT-NODE ACTION PATH-COST DEPTH

States are NOT search nodes!

Fringe

Fringe = Set of leaf nodes Implemented as a queue with ops:

MAKE-QUEUE(element,…) EMPTY?(queue) FIRST(queue) REMOVE-FIRST(queue) INSERT(element,queue) INSERT-ALL(elements,queue)

Problem-solving performance

Completeness Optimality Time complexity Space complexity

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

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)

Graph Search

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)

Example state space

Diameter example 1

Diameter example 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 fringe 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]

Important BeFS algorithms

UCS: f(n) = g(n)

GBeFS: f(n) = h(n)

A*S: f(n) = g(n)+h(n)

Heuristics

h(n) is a heuristic function Heuristics incorporate problem-

specific knowledge Heuristics need to be relatively

efficient to compute

GBeFS

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

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

Example graph

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

Initial state: board position & starting player

Successor function: returns list of (legal move, state) pairs

Terminal test: game over! Utility function: associates player-

dependent 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

criticalGuiding information for deeper

searches

Time Per Move

Constant Percentage of remaining time State dependent Hybrid

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

Example game tree

Example game tree

Example game tree

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)

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.

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

Search Depth Heuristics 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 based State based

Move Ordering Heuristics

Knowledge based Killer Move: the last move at a given

depth that caused an AB-pruning or had best minimax value

History Table

Adversarial Search inStochastic 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!

Local Search

Steepest-ascent hill-climbing Stochastic hill-climbing First-choice hill-climbing Random-restart hill-climbing Simulated Annealing Deterministic local beam search Stochastic local beam search Evolutionary Algorithms

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

Particle Swarm Optimization

PSO is a stochastic population-based optimization technique which assigns velocities to population members encoding trial solutions

PSO update rules:

Ant Colony Optimization

Population based Pheromone trail and stigmergetic

communication Shortest path searching Stochastic moves