Logic Programming: Search Strategies

Transcription

1 O Logic Programming: earch trategies Alan maill ov 5, 2012 Alan maill Logic Programming: earch trategies ov 5, /1

2 oday Problem representation earch epth irst terative eepening readth irst A/O (alternating/game tree) search Alan maill Logic Programming: earch trategies ov 5, /1

3 earch Problems Many classical A/C problems can be formulated as search problems. xamples: raph searching locks world Missionaries and cannibals Planning (e.g. robotics) Alan maill Logic Programming: earch trategies ov 5, /1

4 earch paces iven by: et of states s 1, s 2,... oal predicate goal( ) tep predicate s(, Y ) that says we can go from state to state Y A start state (or states) A solution is a path leading from the to a goal state satisfying goal(). Alan maill Logic Programming: earch trategies ov 5, /1

5 O xample: locks world ake configuration of blocks as a list of three towers, each tower being a list of blocks in a tower from top to bottom. A C [[c,b,a],[],[c]) Alan maill Logic Programming: earch trategies ov 5, /1

6 O xample: locks world Move a block from top of a tower to top of another tower: A C [[b,a],[],[c]) Alan maill Logic Programming: earch trategies ov 5, /1

7 xample: locks world ext move: A C [[a],[b],[c]) Alan maill Logic Programming: earch trategies ov 5, /1

8 xample: locks world hen A C [[],[a,b],[c]) Alan maill Logic Programming: earch trategies ov 5, /1

9 Prolog representation tate is a list of stacks of blocks: [[a,b,c],[],[]] ransitions move a block from the top of one stack to the top of another: s([[a As],s,Cs], [As,[A s],cs]). s([[a As],s,Cs], [As,s,[A Cs]]).... Can specify particular goal position: goal([[],[],[1,b,c]]. Alan maill Logic Programming: earch trategies ov 5, /1

10 An abstract problem space hink of the graph generated by these s(a,b). declarations. s(b,c). s(c,a). n this case: s(c,f(d)). the graph is infinite s(f(),f(g())). there is a loop near the top of the graph s(f(g()),). goal(d). Alan maill Logic Programming: earch trategies ov 5, /1

11 problem 1: cycles We can already see in the blocks world example and in the abstract search space that it is easy to follow actions around in cycles, and not find the goal, even if there is a path to the goal. here are two main approaches to deal with this: remember where you ve been work with depth bound Alan maill Logic Programming: earch trategies ov 5, /1

12 olution 1: remember where you ve been dfs_noloop(path,ode,[ode Path]) :- goal(ode). dfs_noloop(path,ode,path1) :- s(ode,ode1), \+(member(ode1,path)), dfs_noloop([ode Path],ode1,Path1). Alan maill Logic Programming: earch trategies ov 5, /1

13 Problem 2: nfinite tate pace Compare the graph from the abstract search space. epth irst earch has similar problems to Prolog proof search: We may miss solutions because state space is infinite; ven if state space is finite, may wind up finding easy solution only after a long exploration of pointless part of search space Alan maill Logic Programming: earch trategies ov 5, /1

14 olution 2: depth bounding Keep track of depth, stop if bound exceeded ote: does not avoid loops (can do this too) dfs_bound(_,ode,[ode]) :- goal(ode). dfs_bound(,ode,[ode Path]) :- > 0, s(ode,ode1), M is -1, dfs_bound(m,ode1,path) Alan maill Logic Programming: earch trategies ov 5, /1

15 Problem 3: what is a good bound? n general, we just don t know in advance: oo low? Might miss solutions oo high? Might spend a long time searching pointlessly Alan maill Logic Programming: earch trategies ov 5, /1

16 olution 3: iterative deepening se the following with some small start value for dfs_id(,ode,path) :- dfs_bound(,ode,path) ; M is +1, dfs_id(m,ode,path). : if there is no solution, this will not terminate. Alan maill Logic Programming: earch trategies ov 5, /1

17 readth first search Keep track of all possible solutions, try shortest ones first; do this by maintaining a queue of solutions bfs([[ode Path] _], [ode Path]) :- goal(ode). bfs([path Paths], ) :- extend(path,ewpaths), append(paths,ewpaths,paths1), bfs(paths1,). bfs_start(,p) :- bfs([[]],p). Alan maill Logic Programming: earch trategies ov 5, /1

18 extending paths extend([ode Path],ewPaths) :- bagof([ewode,ode Path], (s(ode,ewode), \+ (member(ewode,[ode Path]))), ewpaths),!. %% if there are no next steps, %% bagof will fail and we ll fall through. extend(_path,[]). Alan maill Logic Programming: earch trategies ov 5, /1

19 O Problem: speed Concatenating new paths to end of list is slow Avoid this using difference lists? Alan maill Logic Programming: earch trategies ov 5, /1

20 A/O search o far we ve considered graph search problems Just want to find some path from start to end Other problems have more structure e.g. 2-player games A/O search is a useful abstraction Alan maill Logic Programming: earch trategies ov 5, /1

21 xample: oughts and Crosses MA () M (O) MA () O O O... M (O) O O O MAL tility O O O O O O O O O O Alan maill Logic Programming: earch trategies ov 5, /1

22 epresentation or(,odes) is an O node with possible next states odes Our move and(,odes) is an A node with possible next states odes Opponent moves goal() is a win for us Alan maill Logic Programming: earch trategies ov 5, /1

23 xample: A simple game and(a,[b,c]). or(b,[d,a]). or(c,[d,e]). goal(e). What is the graph here? Alan maill Logic Programming: earch trategies ov 5, /1

24 asic dea andor(ode) :- goal(ode). andor(ode) :- or(ode,odes), member(ode1,odes), andor(ode1). andor(ode) :- and(ode,odes), solveall(odes). solveall(odes) :-... Alan maill Logic Programming: earch trategies ov 5, /1

25 olutions or each A state, we need solutions for all possible next states or each O state, we just need one choice A solution is thus a tree, or strategy Can adapt previous program to produce solution tree; Can also incorporate iterative deepening, loop avoidance,. heuristic measures of good positions leads to algorithms like MiniMax. Alan maill Logic Programming: earch trategies ov 5, /1

26 urther eading ratko, Prolog Programming for Artificial ntelligence ch. 8 (difference lists), ch. 11 (/) also Ch. 12 (est), 13 (A/O) Alan maill Logic Programming: earch trategies ov 5, /1