State Space Representations and Search Algorithms in Artificial Intelligence, Slides of Artificial Intelligence

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State Space Representa,ons

and Search Algorithms

CS171, Fall 2016

Introduc,on to Ar,ficial Intelligence

Prof. Alexander Ihler

Reading: R&N 3.1-3.

Architectures for Intelligence

• Search?
  • Determine how to achieve some goal; “what to do”
• Logic & inference?
  • Reason about what to do
  • Encode knowledge / “expert” systems
  • Know what to do
• Learning?
  • Learn what to do
• Modern view: complex & mul,-faceted

Why search?

• Engaged in bigger, important problem
  • Hit a search subproblem we need to solve
  • Search, solve it, get back to original problem
• Predict the results of our ac,ons in the future
• Many sequences of ac,ons, each with some u,lity
  • Maximize performance measure
• Want to achieve some goal by some (any?) means
• Or, find the best (op,mal) way to achieve it

Example: Romania

• On holiday in Romania
  • Currently in Arad
  • Flight leaves tomorrow from Bucharest
• Formulate goal:
  • Be in Bucharest
• Formulate problem:
  • States: various ci,es
  • Ac,ons: drive between ci,es / choose next city
• Find a solu,on:
  • Sequence of ci,es, e.g.: Arad, Sibiu, Fagaras, Bucharest

Problem types

Classifying the environment:

• Sta,c / Dynamic Previous problem was sta,c: no aaen,on to changes in environment
• Determinis,c / Stochas,c Previous problem was determinis,c: no new percepts were necessary, we can predict the future perfectly
• Observable / Par,ally Observable / Unobservable Previous problem was observable: it knew the ini,al state, &c
• Discrete / Con,nuous Previous problem was discrete: we can enumerate all possibili,es

Why not Dijkstra’s Algorithm?

• D’s algorithm inputs the en,re graph
  • Want to search in unknown spaces
  • Combine search with “explora,on”
  • Ex: autonomous rover on Mars must search an unknown space
• D’s algorithm takes connec,ons as given
  • Want to search based on agent’s ac,ons, w/ unknown connec,ons
  • Ex: web crawler may not know what connec,ons are available on a URL before visi,ng it
  • Ex: agent may not know the result of an ac,on before trying it
• D’s algorithm won’t work on infinite spaces
  • Many ac,ons spaces are infinite or effec,vely infinite
  • Ex: logical reasoning space is infinite
  • Ex: real world is essen,ally infinite to a human-size agent

Vacuum world state space graph R R L L R R L L R L R L R L L R S S S S S S S S

Ex: Vacuum World

• Unobservable, start in {1,2,3,4,5,6,7,8}
• Solu,on? [Right, Suck, Le[, Suck]

State-Space Problem Formula,on A problem is defined by four items: ini5al state e.g., "at Arad“ ac5ons Ac#ons(s) = set of ac,ons avail. in state s transi5on model Results(s,a) = state that results from ac,on a in states Alt: successor func,on S(x) = set of ac,on–state pairs

• e.g., S(Arad) = { , … } goal test , (or goal state) e.g., x = "at Bucharest”, Checkmate(x) path cost (addi,ve)
• e.g., sum of distances, number of ac,ons executed, etc.
• c(x,a,y) is the step cost, assumed to be ≥ 0 A solu5on is a sequence of ac,ons leading from the ini,al state to a goal state 86 98 142 92 87 90 85 101 211 138 146 97 (^75120) 70 111 118 140 151 71 75 Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Sibiu (^) Fagaras Rimnicu Vilcea Pitesti Cralova Bucharest Giurgiu Urziceni Neamt Iasi Vaslui Hirsova Eforie 99 80

Selec,ng a state space

• Real world is absurdly complex à state space must be abstracted for problem solving
• (Abstract) state ß set of real states
• (Abstract) ac,on ß complex combina,on of real ac,ons
  • e.g., "Arad à Zerind" represents a complex set of possible routes, detours, rest stops, etc.
• For guaranteed realizability, any real state "in Arad” must get to some real state "in Zerind”
• (Abstract) solu,on ß set of real paths that are solu,ons in the real world
• Each abstract ac,on should be "easier" than the original problem

Ex: 8-Queens

Place as many queens as possible on the chess board without capture

• states? - any arrangement of n<=8 queens
  • or arrangements of n<=8 queens in le[most n columns, 1 per column, such that no queen aaacks any other.
• ini,al state? no queens on the board
• ac,ons? - add queen to any empty square
  • or add queen to le[most empty square such that it is not aaacked by other queens.
• goal test? 8 queens on the board, none aaacked.
• path cost? 1 per move

Ex: 8-Queens

Place as many queens as possible on the chess board without capture

• states? - any arrangement of n<=8 queens
  • or arrangements of n<=8 queens in le[most n columns, 1 per column, such that no queen aaacks any other.
• ini,al state? no queens on the board
• ac,ons? - add queen to any empty square
  • or add queen to le[most empty square such that it is not aaacked by other queens.
• goal test? 8 queens on the board, none aaacked.
• path cost? 1 per move (not relevant…)

Ex: Sliding ,le puzzle

• states?
• ini,al state?
• ac,ons?
• goal test?
• path cost? 2 8 3 1 5 4 7 6 1 2 3 4 8 6 7 5 Start State Goal State Try it yourselves…

Ex: Sliding ,le puzzle

• states? Loca,ons of ,les
• ini,al state? Given
• ac,ons? Move blank square up / down / le[ / right
• goal test? Goal state (given)
• path cost? 1 per move 2 8 3 1 5 4 7 6 1 2 3 4 8 6 7 5 Start State Goal State

of states: n+1! / 2

8-puzzle: 181,440 states 15-puzzle: 1.3 trillion 24-puzzle: 10^ Optimal solution of n-Puzzle family is NP-hard