CS 236501 Introduction to AI

1. CS 236501Introduction to AI Tutorial 2 Heuristic Search

2. Blind vs. Heuristic Search • Blind Search • Do not use any domain specific knowledge. • Requires: initial state, operators and a goal predicate. • Informed (Heuristic) Search • Requires in addition a function for evaluating states. This function, called also a heuristic function, estimates the cost of the optimal path from a state to a goal state. • Can improve the search in two ways: • Leading the algorithm towards a goal state • Pruning off branches that do not lie on any (optimal) solution paths. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

3. Problem Definition • Additional knowledge is added to problem definition • Problem • Initial state (InitState) • Successor function (Succ) • Goal predicate (Goal-p) • Heuristic Function (h, h: states -> scores) • The heuristic function estimates the distance of a state to the goal state • Lower heuristic value means we are closer to a solution Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

4. Example – Binary Tree • The problem is to find the number 12 in a binary tree • A state is represented by a number • Initial State: 1 • Successor Function: Succ(x) = {2*x, 2*x + 1} • Goal predicate: Goal-p(x) == true iff x == 12 • Possible Heuristic function: h(x) = |x – 12| Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

5. General Search Routine • SearchFunction (States, Problem, Combiner) • { • If States == {} • return ‘Search Failed’ • CurrState <- Get and Remove first state in States • if (Problem.Goal-p(CurrState) == true) • return CurrState //or any other solution form • else • { • successors <- Problem.Succ(CurrState) • States <-Combiner(successors,States) • SearchFunction(States, Problem, Combiner) • } • } Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

6. Priority Combiner • Priority-Combiner(states1, states2) • { • Result <- {} • For s in states1 • Result.Insert(s, h(s)) • For s in states2 • Result.Insert(s, h(s)) • } • Priority-Combiner is aware of the heuristic function that characterizes the problem • Insert() inserts a state to a sorted list according to the value associated with the state Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

7. Best First Search • Priority-Combiner <- Problem.h • States <- Problem.InitState • Solution = SearchFunction(States, Problem, • Priority-Combiner) • States <-(1) • States <-(3 2) • States <-(7 6 2) • States <-(14 15 6 2) • States <-(15 6 2 28 29) • States <-(6 2 28 29 30 31) • States <-(12 13 2 28 29 30 31) • 12 is found! Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

8. Beam Search • Beam-Priority-Combiner(states1, states2) • { • Result <- {} • For s in states1 • Result.Insert(s, h(s)) • For s in states2 • Result.Insert(s, h(s)) • If |Result| > BeamWidth • Result <- First BeamWidth examples from Result • } • BeamWidth is a parameter of Beam-Priority-Combiner Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

9. Beam Search (cont.) • Beam-Priority-Combiner <- Problem.h • Beam-Priority-Combiner.BeamWidth <- 2 • States <- Problem.InitState • Solution = SearchFunction(States, Problem, • Beam-Priority-Combiner) • Is beam search complete? Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

10. Beam Search - Example • States <-(1) • States <-(3 2) • States <-(7 6) • States <-(14 15) // 6 is out • States <-(15 28) • States <-(28 30) • ... 1 2 3 6 7 14 15 28 30 56 57 • How can we solve this problem? Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

11. Iterative Widening Beam Search • Iter-Wide-Search(Problem, MaxWidth) • { • Beam-Priority-Combiner <- Problem.h • for i from 1 to MaxWidth • Beam-Priority-Combiner.BeamWidth <- i • States <- Problem.InitState • Solution = SearchFunction(States, Problem, • Beam-Priority-Combiner) • If Solution != {} • return Solution • } Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

12. Iterative Widening Beam Search- Example • States <-(1) Beam Width = 2 • States <-(3 2) • States <-(7 6) • States <-(14 15) • ...(additional limit necessary to prevent infinite search) • ----------------- • States <-(1) Beam Width = 3 • States <-(3 2) • States <-(7 6 2) • States <-(14 15 6) • States <-(15 6 28) • States <-(6 28 30) • States <-(12 13 28) • 12 is found! Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

13. Local search • Keep one state in memory • At each step move to one of the neighbors of a state • Heuristic function is used to choose the next step • Used in search spaces with high branching factors, in optimization problems, or when the solution path is not required (just the goal state) Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

14. Local Search • LocalSearch(state, NextStepFunction) • { • if Goal-p(state) • return state • else • { • nextState <- NextStepFunction(state) • if nextState == NIL • return ‘Failure’ • else • LocalSearch(nextState, NextStepFunction) • } • } Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

15. Steepest Ascent Hill-Climbing • BestSuccessor(state) • { • Candidates <- {} • bestH <- infinity • succ <- Problem.Succ(state) • for s in succ • { • if h(s) < bestH • bestH <- h(s), Candidates <- {s} • if h(s) == bestH • add s to Candidates • } • if bestH < h(state) • return a random member in Candidates • else // no better successor • return NIL • } Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

16. Steepest Acsent Hill-Climbing • state <- Problem.InitState • Solution = LocalSearch(state, BestSuccessor) • * Failure indicates local minimum Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

17. Steepest Ascent Hill-Climbingwith sideways steps • BestOrEqual_Successor(state) • { • Same as BestSuccessor, but allows returning a successor with • heuristic value equal to h(state) • } • state <- Problem.InitState • Solution = LocalSearch(state, BestOrEqual_Successor) Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

18. Hill-Climbing: more variations • Steepest Ascent Hill-Climbing with local minimum avoidance • When in a local minimum, make k random steps, and continue hill-climbing • Stochastic Hill-Climbing • Among better successors, choose one at random with probability proportional to the improvement • First-Choice Hill-Climbing • Choose the first successor that improves the heuristic value • Restart Hill-Climbing • When in a local minima, restart the search with a random node Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

19. Heuristic Function • An accurate heuristic function will help to guide the heuristic search quickly to the solution • In rare cases we can define a heuristic function that is exact • In most cases we try to define an accurate estimation • Considerations when choosing a heuristic function: • Quality • Computational efficiency Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

20. Example: LightzOut • http://www.gamesgnome.com/logicpuzzles/lights/ • http://games.hughesclan.com/lights/ • http://javaboutique.internet.com/LightzOut/ • 5x5 board • Each cell can be “On” or “Off” • Click on a cell inverts the states of its 4 neighbors • The goal is to turn off the lights Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

21. Example: LightzOut Click on the central cell Goal: Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

22. LightzOut – solution example Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

23. LightzOut: Heuristic Function 1 • H(s) = the number of cells that are “On” • Motivation: the less lights there are “On”, the closer we are to the solution • Advantages: • Easy to implement • Fast (when using some optimization tricks) • Disadvantages: • Misleading, sometimes we need to pass through a “worse” state than the current state in order to solve the puzzle • Does not estimate the correct distance to the goal Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

24. LightzOut: Heuristic Function 1 h = 3 h = 5 h = 5 Disadvantage #1: Here, first step to the solution goes through a worse state Disadvantage #2: High heuristic value, although one step from the solution Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

25. LightzOut: Heuristic Function 2 • H(s) = the sum of Manhattan distances of each “On” cell from its nearest neighbor that is “On” • H(board with one cell on) = some arbitrary number (e.g., 10) • Motivation: adjacent “On” cells are easier to solve h = 5 h = 8 Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

26. LightzOut: Heuristic Function 2 • Advantages: • Gives advantage to boards with adjacent cells, which are indeed easier to solve • Disadvantages • Time complexity is O(N4), very heavy • Does not distinguish between an easy-to-solve (cross) and hard-to-solve (almost cross) adjacent groups • Still not exact Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

27. Example: LightzOut • Which search algorithm should we use? • Note: the suggested heuristics are not exact and sometimes misleading • Best-First Search has high memory requirements • Beam search is a possibility • For low beam width may fail to find a solution • Higher beam width improves the chances to solve Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

28. Example: LightzOut • Steepest-Ascent Hill-Climbing is very likely to get stuck in a local minima • Example: Heuristics #1 for the first board in slide 22 • Hill-Climbing with sideways steps, Stochastic Hill-Climbing and First-Choice Hill-Climbing may have the same problem Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

29. Example: LightzOut • Hill-Climbing with local minimum avoidance is a good option • Local minima will not terminate the search, and random steps might lead to a state close to the solution • Will find long solutions (if we are interested in a solution path) • Restart Hill-Climbing is not possible • No way to generate a random state Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

30. Summary • Heuristic searches are guided by the knowledge of the problem domain • Good heuristic function is important for an efficient heuristic search • It is hard to choose the “best” heuristic search that suits a problem. Sometimes experiments help us to understand an algorithm behavior on a certain problem Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich