CS 347 Introduction to Artificial Intelligence CS 347

CS 347 – Introduction to Artificial Intelligence CS 347 course website: http: //web. mst. edu/~tauritzd/courses/cs 347/ Dr. Daniel Tauritz (Dr. T) Department of Computer Science tauritzd@mst. edu 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

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

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 Mc. Culloch & 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.

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. ”

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

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 (Be. FS) • 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 Be. FS algorithms • UCS: f(n) = g(n) • GBe. FS: f(n) = h(n) • A*S: f(n) = g(n)+h(n)

Heuristics • h(n) is a heuristic function • Heuristics incorporate problemspecific knowledge • Heuristics need to be relatively efficient to compute

GBe. FS • 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

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

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 playerdependent 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 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(b 3 d/4)

Move Ordering Heuristics • Knowledge based • Killer Move: the last move at a given depth that caused an ABpruning or had best minimax value • History Table

Example game tree

Example game tree

Example game tree

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

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.

Time Per Move • • Constant Percentage of remaining time State dependent Hybrid

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 3 d-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)

MTD(f) MTDf(root, guess, depth) { lower = -∞; upper = ∞; do { beta=guess+(guess==lower); guess = ABMax. V(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

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!

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

Online Search • • Offline search vs. online search Interleaving computation & action Exploration problems, safely explorable Agents have access to: – ACTIONS(s) – c(s, a, s’) – 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 = ∞

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

Online Search Example Graph

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