CSCI 360 Introduc/on to Ar/ficial Intelligence Week 2: Problem Solving and Op/miza/on. Instructor: Wei-Min Shen

Transcription

1 CSCI 360 Introduc/on to Ar/ficial Intelligence Week 2: Problem Solving and Op/miza/on Instructor: Wei-Min Shen

2 Status Check and Review Status check Have you registered in Piazza? Have you run the Project-1? Have you submimed your pre-reading reports? Here are some good ones for your reference Review of last lecture Agent? Types of agents? Environment? Types of environment? Today s lecture =?

3 Class Outline Wk2 Ch3: Solving Problems How to represent a problem? states, ac/ons, ini/als, goals, (e.g., /c-tac-toe) How to solve a problem? Search: from here to there, ini/als to goals Depth-first, breadth-first How good is your solu/on? (fig 6.1, ALFE) How good is your state? How costly is an ac/on? Best-first, Dynamic Programming, A*, etc. Can you guarantee anything? (op/mal vs heuris/c) How much do you want to pay for your solu/on? How deep/wide can you go? Predetermined vs dynamic (e.g., itera/ve deepening) One way or many ways (bi-direc/onal)? How big is a problem? Can you put the whole world in your head? Tower of Hanoi, chess, robot-and-world, HM: state space for TOH, assign values for state and ac/ons

4 Class Outline Wk2 Ch4: Op/miza/on How to find the extremes of f(x)? Why is it so important? Why is it so hard? (no one has offered a general solu/on!) Which way to go? How much can you see? (local vs global) How many points can you remember? How big is your step? (skip THE point?) How well can you guess? How much do you know about the func/on? How do you know you are done? Will the func/on change by itself? HW: your own answers for the above ques/ons

5 This Week s Lecture Problem Solving and Search Techniques Op/miza/on Techniques Home Work 1 assignments

6 Solving Problems by Search How to represent a real-world problem? Examples, state space, states, ac/ons, ini/als, goals. How to solve a problem? Search: from here to there, ini/als to goals Depth-first, breadth-first How good is your solu/on? (fig 6.1, ALFE) How good is your state? How costly is an ac/on? Best-first, Dynamic Programming, A*, etc. Can you guarantee anything? (op/mal vs heuris/c) How much do you want to pay for your solu/on? How deep/wide can you go? Predetermined vs dynamic (e.g., itera/ve deepening) One way or many ways (bi-direc/onal)? How big is a problem? Can you put the whole world in your head? Tower of Hanoi, chess, robot-and-world,

7 What is a Problem? Any ideas? Hint: Think it as related to Agent

8 What is a Problem? Any ideas? Hint: Think it as related to Agent Percepts, ac/ons, environment, goals, u/li/es Problem: States, ini/al state, goal states, Solu/on: a sequence of ac/ons leads to the goal

9 Problem Examples: 8Puzzle, PAC-Man Ini/al State Goal State(s)

10 Tower of Hanoi, A Cleaning Robot Can dirt be a goal? 10

11 Travel Example and Project-1

12 State Space How to represent a real-world problem Graph Search Graph Ver/ces Edges Start ver/ces Goal ver/ces Solu/on is a (minimum cost) path from the start to the goal Search Problem State space States Ac/ons=operators Ini/al state Goal states or goal test Solu/on is a (minimum cost) ac/on sequence from the start to the goal state (or a state that sa/sfies the goal test) A BIG Ques/on: How do the sensors and percepts enter the picture?

13 Vacuum Robot and Its World

14 Traveling from Arad To Bucharest What is a state here?

15 State Graphs vs. Search Trees s b d a c e f G Each NODE in the search tree is an en/re PATH in the state graph (note how many nodes in the graph appear mul/ple /mes in the search tree) s p h r d e p q b c e h r q We almost always construct both on demand and we construct as lible as possible a a p q h q c r f G p q q c a f G a

16 State Space and Search Tree A Search Tree State space with ac/ons, costs/rewards (Fig6.1, ALFE)

17 Search Tree Nodes A node in the search tree is a data structure containing the state plus other data 17

18 Solving Problems by Search How to represent a problem? Examples (e.g., 8puzzle, TOH, Roomba, Travel),, states, ac/ons, ini/als, goals. How to solve a problem? Search: from here to there, from the ini/als to the goals Depth-first, breadth-first How good is your solu/on? (fig 6.1, ALFE) How good is your state? How costly is an ac/on? Best-first, Dynamic Programming, A*, etc. Can you guarantee anything? (op/mal vs heuris/c) How much do you want to pay for your solu/on? How deep/wide can you go? Predetermined vs dynamic (e.g., itera/ve deepening) One way or many ways (bi-direc/onal)? How big is a problem? Can you put the whole world in your head? Tower of Hanoi, chess, robot-and-world, HM: state space for TOH, assign values for state and ac/ons.

19 Problem Solving Methods (Overview) Breadth-first search Depth-first search Best-first search Goodness = (past + future) Uniform cost (es/mate past and future) A* search algorithm (es/mate the future) Dynamic Programming (back from the future)

20 Tree search algorithms Basic idea: offline, systematic exploration of simulated state-space by generating successors of explored states func(on TREE-SEARCH(problem, strategy) returns a solu/on, or failure ini/alize the search tree using the ini/al state problem loop do if there are no candidates for expansion, then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solu/on else expand the node and add resul/ng nodes to the search tree end Breath-first strategy: the least depth node Depth-first strategy: the deepest depth node

21 Implementa/on (Tree Search) func(on TREE-SEARCH(problem) return a solu/on or failure fron/er MAKE-QUEUE(MAKE-NODE(problem.INITIAL-STATE)) loop do if EMPTY?(fron/er) then return failure node CHOOSE-NODE-By-Strategy(fron/er) if problem.goal-test applied to node.state succeeds then return SOLUTION(node) fron/er INSERT-ALL(EXPAND(node, problem), fron/er) The fron/er is the set of nodes that have been generated but not yet expanded

22 Strategy: Breadth-First Search Expand shallowest nodes first Leo to right at any depth 22

23 State Graph Breath-first Search example

24 Depth-First Search Expand deepest nodes first Leo to right at any depth 24

25 Depth-First Search Expand deepest nodes first Leo to right at any depth Backtrack when hit terminal

26 Depth-first Search example State Graph Ques/on: Will the depth grow forever? Where is the terminal?

27 Loop Problems

28 Graph Search (Preven/ng Loop) func(on GRAPH-SEARCH(problem) return a solu/on or failure fron/er MAKE-QUEUE(MAKE-NODE(problem.INITIAL-STATE)) explored_set empty loop do if EMPTY?(fron/er) then return failure node CHOOSE-NODE-By-Strategy(fron/er) if problem.goal-test applied to node.state succeeds then return SOLUTION(node) explored_set INSERT(node, explored_set) for each new_node in EXPAND(node, problem) do if NOT(MEMBER?(new_node, fron/er)) and NOT(MEMBER?(new_node, explored_set)) then fron/er INSERT(new_node, fron/er)

29 Solving Problems by Search How to represent a problem? Examples (e.g., 8puzzle, TOH, Roomba, Travel),, states, ac/ons, ini/als, goals. How to solve a problem? Search: from here to there, from the ini/als to the goals Depth-first, breadth-first How good is your solu/on? (fig 6.1, ALFE) How good is your state? How costly is an ac/on? Best-first, Dynamic Programming, A*, etc. Can you guarantee anything? (op/mal vs heuris/c) How much do you want to pay for your solu/on? How deep/wide can you go? Predetermined vs dynamic (e.g., itera/ve deepening) One way or many ways (bi-direc/onal)? How big is a problem? Can you put the whole world in your head? Tower of Hanoi, chess, robot-and-world, HM: state space for TOH, assign values for state and ac/ons.

30 Best-First search algorithms func(on TREE-SEARCH(problem, strategy) returns a solu/on, or failure ini/alize the search tree using the ini/al state problem loop do if there are no candidates for expansion, then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solu/on else expand the node and add resul/ng nodes to the search tree end Best-first strategy: choose the best node: goodness = (past cost + future cost) When do we use this strategy?

31 Traveling from Arad To Bucharest We have the knowledge about the actual driving distances

32 Best-first Search example Choose the best node to expand: (past cost) + (the best future cost) Sibiu: (140) + ( ) =? Timisoanra: (118) + ( ) =? Zerind: (75) + ( ) =?

33 Two Big Ques/ons about Best? What if the best future cost is unknown to us? What if the best future cost is too expensive to compute?

34 What if we don t know the future? Goodness = past cost + the best future cost What if the best future cost is unknown to us? Es/mate Goodness: uniform cost (naïve) Es/mate Goodness = past cost + es/mated future cost Evalua/on func/on f(n) = g(n) + h(n) g(n) the cost (so far) to reach the current node n h(n) es/mated cost to get from the current node n to the goal f(n) es/mated total cost of path through n to the goal

35 Traveling from Arad To Bucharest Suppose the future driving distances to the goal is unknown to us Sensor:? But we know the straight-line distances to the goal Sensor:?

36 Using the es/mated future cost Choose the best node to expand: (past cost + es/mated future cost) Sibiu: Timisoanra: Zerind:

37 What if our es/ma/on is wrong? Examples are plenty My father told me so I did drive that road before I have an iphone Google map How wrong can we tolerate? Over-es/mate? Under-es/mate? By how much?

38 Es/mate by Admissible Heuris/cs A heuris/c is admissible if it never overes/mates the future cost to reach the goal Admissible heuris/cs are op/mis/c Formally: 1. h(n) h*(n) where h*(n) is the true cost from n to the goal 2. h(n) 0, with h(g)=0 for any goal G e.g. h SLD (n) (GPS) is admissible because it never overes/mates the actual road distance to the goal 38

39 A* search algorithm func(on TREE-SEARCH(problem, strategy) returns a solu/on, or failure ini/alize the search tree using the ini/al state problem loop do if there are no candidates for expansion, then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solu/on else expand the node and add resul/ng nodes to the search tree end Best-first strategy: choose the best node: goodness = (past cost + future cost) A* strategy: choose the admissibly es/mated best node: goodness = (past cost + admissibly es/mated future cost)

40 What if future is too hard to compute? Compute the best future cost for a node must consider all possible paths from the node to the goal This is too expensive because there are so many such paths Solu/on Use Dynamic Programming Not compute path by path But compute stage by stage (breath-first)

41 Dynamic Programming (ALFE 6.1.1) Backward recursion: compute the future cost by back from the goal* stage by stage Backward recursion equa/on:

42 Best-First by Dynamic Programming func(on TREE-SEARCH(problem, strategy) returns a solu/on, or failure ini/alize the search tree using the ini/al state problem loop do if there are no candidates for expansion, then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solu/on else expand the node and add resul/ng nodes to the search tree end Best-first strategy: choose the best node: goodness = (past cost + future cost) A* strategy: choose the best node: goodness = (past cost + admissibly es/mated future cost) Dynamic programming: choose the best node: goodness = (past cost + V(s))

43 Evalua/on of search strategies A search strategy is defined by picking the order of node expansion. Search algorithms are commonly evaluated according to the following four criteria: Completeness: does it always find a solu/on if one exists? Time complexity: how long does it take as func/on of number of nodes? Space complexity: how much memory does it require? Op(mality: does it guarantee the least-cost solu/on? Time and space complexity are measured in terms of: b max branching factor of the search tree d depth of the least-cost solu/on m max depth of the search tree (may be infinity)

44 Solving Problems by Search How to represent a problem? Examples (e.g., 8puzzle, TOH, Roomba, Travel),, states, ac/ons, ini/als, goals. How to solve a problem? Search: from here to there, from the ini/als to the goals Depth-first, breadth-first How good is your solu/on? (fig 6.1, ALFE) How good is your state? How costly is an ac/on? Best-first, Dynamic Programming, A*, etc. Can you guarantee anything? (op/mal vs heuris/c) How much do you want to pay for your solu/on? How deep/wide can you go? Predetermined vs dynamic (e.g., itera/ve deepening) One way or many ways (bi-direc/onal)? How big is a problem? Can you put the whole world in your head? Tower of Hanoi, chess, robot-and-world, HM: state space for TOH, assign values for state and ac/ons.

45 Other Search-related Techniques Itera/ve deepening Limit the search depth, incrementally increase it Bi-direc/onal search Search from the ini/al to the goal, as well as from the goal to the ini/al

46 Solving Problems by Search How to represent a problem? Examples (e.g., 8puzzle, TOH, Roomba, Travel),, states, ac/ons, ini/als, goals. How to solve a problem? Search: from here to there, from the ini/als to the goals Depth-first, breadth-first How good is your solu/on? (fig 6.1, ALFE) How good is your state? How costly is an ac/on? Best-first, Dynamic Programming, A*, etc. Can you guarantee anything? (op/mal vs heuris/c) How much do you want to pay for your solu/on? How deep/wide can you go? Predetermined vs dynamic (e.g., itera/ve deepening, fixed depth) One way or many ways (bi-direc/onal)? How big is a problem? Can you put the whole world in your head? Tower of Hanoi, chess, robot-and-world, HM: state space for TOH, assign values for state and ac/ons.

47 Complexity: How big is a problem? When you solve N=64, it would be the end of the world!

48 Time complexity of depth-first In the worst case: the (only) goal node may be on the right-most branch, b 0 b 1 b d b d-1 G b d Time complexity = b d + b d = b d+1-1 Thus: O(b d ) b - 1 CS561 - Lectures Macskassy - Spring

49 Space complexity of depth-first Largest number of nodes in QUEUE is reached in bottom leftmost node. Example: m = 3, b = 3 : G... QUEUE contains all nodes. Thus: 7. In General: ((b-1) * m) + b Order: O(m*b) 49

50 Comparing uninformed search strategies Criterion Breadth- Uniform Depth- Depth- Itera/ve Bidirec/onal first cost first limited deepening (if applicable) Time b^d b^d b^m b^l b^d b^(d/2) Space b^d b^d bm bl bd b^(d/2) Op/mal? Yes Yes No No Yes Yes Complete? Yes Yes No Yes, Yes Yes if l d b max branching factor of the search tree d depth of the least-cost solu/on m max depth of the state-space (may be infinity) l depth cutoff 50

51 Summary Problem formula/on usually requires abstrac/ng away real-world details to define a state space that can be explored using computer algorithms. Once problem is formulated in abstract form, complexity analysis helps us picking out best algorithm to solve problem. Variety of uninformed search strategies; difference lies in method used to pick node that will be further expanded. Itera/ve deepening search only uses linear space and not much more /me than other uniformed search strategies. CS 561, Lectures

52 Solving Problems by Search How to represent a problem? Examples (e.g., 8puzzle, TOH, Roomba, Travel),, states, ac/ons, ini/als, goals. How to solve a problem? Search: from here to there, from the ini/als to the goals Depth-first, breadth-first How good is your solu/on? (fig 6.1, ALFE) How good is your state? How costly is an ac/on? Best-first, Dynamic Programming, A*, etc. Can you guarantee anything? (op/mal vs heuris/c) How much do you want to pay for your solu/on? How deep/wide can you go? Predetermined vs dynamic (e.g., itera/ve deepening) One way or many ways (bi-direc/onal)? How big is a problem? Can you put the whole world in your head? Tower of Hanoi, chess, robot-and-world, HM: state space for TOH, assign values for state and ac/ons.

53 Home Work Tower of Hanio (n=3) 1. Write the state space, states, ac/ons 2. Assume the cost of an ac/on to move a disk is the weight of the disk, compute the best future cost V(s 0 ) for the start state using dynamic programming Start state s 0 goal The weight of the disk: disk1=1, disk2=2, disk3=3