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CS347 – Introduction to Artificial Intelligence

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CS347 – Introduction toArtificial Intelligence

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

Dr. Daniel Tauritz (Dr. T)

Department of Computer Science

tauritzd@mst.edu

http://web.mst.edu/~tauritzd/

Systems that…

- act like humans (Turing Test)
- think like humans
- think rationally
- act rationally
Play Ultimatum Game

- Three Sisters Puzzle

A definition:

Problem-solving agents are goal based agents that decide what to do based on an action sequence leading to a goal state.

- Fully Observable
- Single Agent
- Discrete
- Sequential
- Known & Deterministic

- Problem-formulation (actions & states)
- Goal-formulation (states)
- Search (action sequences)
- Execute solution

- Initial state
- Action set: ACTIONS(s)
- Transition model: RESULT(s,a)
- Goal test
- Step cost: c(s,a,s’)
- Path cost
- Solution / optimal solution

- Vacuum world
- Tic-tac-toe
- 8-puzzle
- 8-queens problem

- 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

- 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

- n.STATE
- n.PARENT-NODE
- n.ACTION
- n.PATH-COST
States are NOT search nodes!

- 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

- Completeness
- Optimality
- Time complexity
- Space complexity

- 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

- 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)

- 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)

- g(n) = lowest path-cost from start node to node n
- Frontier: priority queue ordered by g(n)

- 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)

- 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

function ID-DFS(problem) returns solution/fail

for depth = 0 to ∞ do

result ← DLS(problem,depth)

ifresult ≠ 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)

BiBFTS

- Complete: Yes, if b is finite
- Optimal: Not “out of the box”
- Time & Space complexity: O(bd/2)

- 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

- 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]

- h(n) is a heuristic function
- Heuristics incorporate problem-specific knowledge
- Heuristics need to be relatively efficient to compute

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

- Incomplete (so also not optimal)
- Worst-case time and space complexity: O(bm)
- Actual complexity depends on accuracy of h(n)

- 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

- h(n) admissible if:

Example: straight line distance

A*TS optimal if h(n) admissible

- h(n) consistent if:

Consistency implies admissibility

A*GS optimal if h(n) consistent

- 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

- 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

- Effective branching factor
- Domination
- Composite heuristics
- Generating admissible heuristics from relaxed problems

- n-puzzle legal actions:
Move from A to B if horizontally or vertically adjacent and B is blank

Relaxed problems:

- Move from A to B if adjacent
- Move from A to B if B is blank
- Move from A to B

The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem.

Environments characterized by:

- Competitive multi-agent
- Turn-taking
Simplest type: Discrete, deterministic, two-player, zero-sum games of perfect information

- 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

- 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!

A good State Eval Heuristic should:

- order the terminal states in the same way as the utility function
- be relatively quick to compute
- strongly correlate nonterminal states with chance of winning

- 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

- 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

- Worst-case: O(bd)
- Best-case: O(bd/2) [Knuth & Moore, 1975]
- Average-case: O(b3d/4)

- 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

- 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

- 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

- Constant
- Percentage of remaining time
- State dependent
- Hybrid

- 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.

- 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

- 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

- 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)

- Balancing time versus memory
- Opening table
- Human expert knowledge
- Monte Carlo analysis

- End game database

- Beam Search (n best moves)
- ProbCut (forward pruning version of alpha-beta 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

- 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

Worst Case Time Complexity: O(bmnm) with b the average branching factor, m the deepest search depth, and n the average chance branching factor

- Interval arithmetic
- Monte Carlo simulations (for dice called a rollout)

- Complete-state formulation
- Objective function
- Global optima
- Local optima (don’t use textbook’s definition!)
- Ridges, plateaus, and shoulders
- Random search and local search

- 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)

- 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

- First-choice hill-climbing
- Random-restart hill-climbing
- Simulated Annealing

- Deterministic local beam search
- Stochastic local beam search
- Evolutionary Algorithms
- Particle Swarm Optimization
- Ant Colony 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

- Population based
- Pheromone trail and stigmergetic communication
- Shortest path searching
- Stochastic moves

- 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)

- CR – Competitive Ratio
- TAPC – Total Actual Path Cost
- C* - Optimal Path Cost
- Best case: CR = 1
- Worst case: CR = ∞

- Online-DFS-Agent
- Random Walk
- Learning Real-Time A* (LRTA*)

- René Descartes (1596-1650)
- Rationalism
- Dualism
- Materialism
- Star Trek & Souls

- Environment
- Sensors (percepts)
- Actuators (actions)
- Agent Function
- Agent Program
- Performance Measures

Depends on:

- Agent’s performance measure
- Agent’s prior knowledge
- Possible percepts and actions
- Agent’s percept sequence

“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

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