CS347 – Introduction toArtificial Intelligence

Dr. Daniel Tauritz (Dr. T)Department of Computer Science

tauritzd@mst.eduhttp://web.mst.edu/~tauritzd/

CS347 course website: http://web.mst.edu/~tauritzd/courses/cs347/

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

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.”

PEAS description & properties:–Fully/Partially Observable–Deterministic, Stochastic, Strategic–Episodic, Sequential–Static, Dynamic, Semi-dynamic–Discrete, Continuous–Single agent, Multiagent–Competitive, Cooperative–Known, Unknown

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

• 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• 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

• 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/failinitialize frontier using initial problem state

loop doif 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/failinitialize frontier using initial problem state

initialize explored set to be empty

loop doif 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

• 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

• 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)

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)

• 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)

• 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/failfor 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 problem-specific 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

• 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

• 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

• 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 SearchEnvironments characterized by:• Competitive multi-agent• Turn-taking

Simplest type: Discrete, deterministic, two-player, zero-sum games of perfect information

Search problem formulation• 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 player-

dependent 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

1( ) ( )

ni i

iEVAL s w f s

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 αβ-pruning 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

• 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

• 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

• 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

• 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

• Best case: CR = 1

• Worst case: CR = ∞

*

TAPCCR

C

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• 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