Problem solving Basic search. Example: Romania. Example: Romania. Problem types. Intelligent Systems and HCI D7023E. Single-state problem formulation

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1 Intelligent Systems and HI 7023 Lecture 3: asic Search Paweł Pietrzak Problem solving asic search 1 2 xample: Romania On holiday in Romania; currently in rad. light leaves tomorrow from ucharest ormulate goal: - be in ucharest ormulate problem: - states: various cities - actions: drive between cities ind solution: - sequence of cities, e.g., rad, Sibiu, agaras, ucharest 3 xample: Romania 4 Single-state problem formulation problem is defined by four items: 1. initial state e.g., "at rad" 2. actions or successor function S(x) = set of action state pairs! e.g., S(rad) = {<rad " Zerind, Zerind>, } 3. goal test, can be! explicit, e.g., x = "at ucharest"! implicit, e.g., heckmate(x) 4. path cost (additive)! e.g., sum of distances, number of actions executed, etc.! c(x,a,y) is the step cost, assumed to be! 0! solution is a sequence of actions leading from the initial state to a goal state 5 Problem types eterministic, fully observable " single-state problem - gent knows exactly which state it will be in; solution is a sequence Non-observable " sensorless problem (conformant problem) - gent may have no idea where it is; solution is a sequence Nondeterministic and/or partially observable " contingency problem - percepts provide new information about current state - often interleave search, execution Unknown state space " exploration problem 6

2 xample: vacuum world Single-state, start in #5. xample: vacuum world Single-state, start in #5. [Right, Suck] Sensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} 7 8 xample: vacuum world Sensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} [Right,Suck,Left,Suck] xample: vacuum world Sensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} [Right,Suck,Left,Suck] ontingency - Nondeterministic: Suck may dirty a clean carpet - Partially observable: location, dirt at current location. - Percept: [L, lean], i.e., start in #5 or #7 9 ontingency - Nondeterministic: Suck may dirty a clean carpet - Partially observable: location, dirt at current location. - Percept: [L, lean], i.e., start in #5 or #7 [Right, if dirt then Suck] 10 Vacuum world state space graph Vacuum world state space graph states? actions? goal test? path cost? states? integer dirt and robot location actions? Left, Right, Suck goal test? no dirt at all locations path cost? 1 per action 11 12

3 Selecting a state space xample: The 8-puzzle Real world is absurdly complex " state space must be abstracted for problem solving (bstract) state = set of real states (bstract) action = complex combination of real actions (bstract) solution = - set of real paths that are solutions in the real world ach abstract action should be "easier" than the original problem states? actions? goal test? path cost? xample: The 8-puzzle n example problem: Searching a graph Initial State Goal State states? locations of tiles actions? move blank left, right, up, down goal test? = goal state (given) path cost? 1 per move [Note: optimal solution of n-puzzle family is NP-hard] State-Space (explicit) go(x,y,[x T]):-! link(x,z),! go(z,y,t). Simple search algorithm G?- go(a,c,x). X = [a,e,f,c]? ; X = [a,b,f,c]? ; X = [a,b,c]? ; no onsultation H State-Space Representation n abstract representation of a state-space is a downwards growing tree. onnected nodes represent states in the domain. The branching factor denotes how many new states you can move to from any state. This problem has an average of 2. The depth of a node denotes how many moves away from the initial state it is. has two depths, 2 or Initial state is the root Implementing To implement state-space search in Prolog, we need: 1. way of representing a state e.g. the board configuration 2. way of generating all of the next states reachable from a given state; go(x,y,[x T]):- link(x,z), go(z,y,t). 3. way of determining whether a given state is the one we're looking for. Sometimes this might be the goal state (a finished puzzle, a completed route, a checkmate position); other times it might simply be the state we estimate is the best, using some evaluation function; 4. mechanism for controlling how we search the space. 18

4 epth-irst Search go(x,y,[x T]):-! link(x,z),! go(z,y,t).?- go(a,c,x). X = [a,e,f,c]? ; X = [a,b,f,c]? ; X = [a,b,c]? ; no This simple search algorithm uses Prolog s unification routine to find the first link from the current node then follows it. It always follows the left-most branch of the search tree first; following it down until it either finds the goal state or hits a dead-end. It will then backtrack to find another branch to follow = depth-first search (S). 19 Properties of depth-first search omplete? No: fails in infinite-depth spaces, spaces with loops - Modify to avoid repeated states along path " complete in finite spaces Time? O(b m ): terrible if m is much larger than d - but if solutions are dense, may be much faster than breadthfirst Space? O(bm), i.e., linear space! Optimal? No b - branching factor, m - max depth of any node (can be "!) d - depth of the solution 20 sample problem Start Three missionaries and three cannibals are standing on the bank of a river. There is a boat but it is only big enough to carry two persons. If the cannibals, at any time, outnumber the missionaries on either side of the river the cannibals will eat the missionaries. We now want to find a way to transport all missionaries and cannibals over the river without having any of the missionaries eaten. search space for the cannibals-and-missionaries problem Wanna play? :) nd Iterative eepening If the optimal solution is the shortest path from the initial state to the goal state depth-first search will usually not find this. We need to vary the depth at which we look for a solution; increasing the depth every time we have exhausted all nodes at a particular depth. Z=a go(a,z,[a]). link(z,c). link(a,c). We can take advantage of Prolog s backtracking to implement this very simply. epth-irst go(x,y,[x T]):- link(x,z), go(z,y,t). Iterative eepening heck if current node is goal. ind an intermediate node. heck whether intermediate links with goal

5 go(a,z,sol). link(z,c). go(a,e,[e,a]). link(e,c). go(a,z1,sol). link(z1,z). go(a,z1,[a]). Z1=a link(a,z). Z1=a 25 26?- go(a,c,[c,b,a]). go(a,b,[b,a]). go(a,z,sol). link(z,c). go(a,z1,[a]). go(a,z1,sol). link(z1,z). Z1=a go(a,z2,sol). link(z2,z1). S = [c,b,a]? S = [c,b,a]?; go(a,f,sol). go(a,e,sol). go(a,a,sol). S = [c,b,a]?; S = [c,f,e,a]? Iterative eepening Iterative eepening search is quite useful as: - it is simple; - reaches a solution quickly, and - with minimal memory requirements as at any point in the search it is maintaining only one path back to the initial state. However: - on each iteration it has to re-compute all previous levels and extend them to the new depth; - may not terminate (e.g. loop); - may not be able to handle complex state-spaces; - can t be used in conjunction with problem-specific heuristics as keeps no memory of optional paths. 30

6 Properties of iterative deepening search omplete? Yes Time? (d+1)b 0 + d b 1 + (d-1)b b d = O(b d ) Space? O(bd) Optimal? Yes, if step cost = 1 d - depth of solution, b - branching factor (number of successors) readth-irst Search?- go(a,c,x). X = [a,b,c]? ; X = [a,e,f,c]? ; X = [a,b,f,c]? ; no epth-first =,,,, readth-first =,,, simple, common alternative to depth-first search is: breadth-first search. This checks every node at one level of the space, before moving onto the next level. It is distinct from iterative deepening as it maintains a list (agenda) of alternative candidate nodes that can be expanded at each depth 1st 2nd 3rd Properties of breadth-first search omplete? Yes (if b is finite) Time? 1+b+b 2 +b 3 + +b d + b(b d -1) = O(b d+1 ) Space? O(b d+1 ) (keeps every node in memory) Optimal? Yes (if cost = 1 per step) Space is the bigger problem (more than time) d - depth of solution, b - branching factor (number of successors) 33 Uniform-cost search xpand least-cost unexpanded node Implementation: - agenda = queue ordered by path cost quivalent to breadth-first if step costs all equal omplete? Yes, if step cost! ε Time? # of nodes with g # cost of optimal solution, O(b ceilling(*/! ) where * is the cost of the optimal solution Space? # of nodes with g # cost of optimal solution, O(b ceiling(*/!) ) Optimal? Yes nodes expanded in increasing order of g(n) 34