Searching a graph or a state space

1. Searching a graphor a state space

2. Searching a tree % ?- path(a,X). % X = a ; X = b ; X = e ; X = f ; X = c ; X = g ; X = h ; % X = d ; no % Order of generated answers is given by the SLD strategy, ie. „depth first search“ h(a,b). h(a,c). h(a,d). h(b,e). h(b,f). h(c,g). h(c,h). path(X,X). path(Z,D):-h(Z,N), path(N,D).

3. Search through a general graph h(a,b). h(a,c). h(a,d). h(b,e). h(b,f). h(c,g). h(c,h). h(e,a). path(X,X). path(Z,D):-h(Z,N), path(N,D). % ?- path(a,X). % X = a ; X = b ; X = e ; X = a ; X = b ; X = e ; X = a ; ... Where is the problem? One should avoid repeated treatment of a node, which has been considered already.

4. Searching a graph with history record /* Used notation:facts listing all the edgesh(Initial_node,End_node)and all the goal nodesg(Node) */ h(a,b). h(a,c). h(a,d). h(b,e). h(b,f).h(c,g). h(c,h). h(c,a). h(g,k). h(g,l).h(d,i). h(d,j). h(j,m). h(j,n). g(d). g(f). g(i). g(l). g(n). % c_hl(+Starting_node,-Goal_node,-Path) c_hl(X,X,[]) :- g(X). c_hl(X,Y,[M|L]) :- h(X,M),c_hl(M,Y,L).

5. Searching a graph with check of cycles % more effectivesolution – elimination of cycles solve(FromN, Solution) :- dfs([],FromN,Solution). dfs(BeginnigPartS,Node,[Node| BeginnigPartS]):- g(Node). dfs(BeginnigPartS, Node, Solution):- h(Node,NodeNext), not member(NodeNext, BeginnigPartS), dfs([NodeNext| BeginnigPartS], NodeNext , Solution). % limited depth dfs % l_dfs(+FromNode, -Solution, +Max_depth) l_dfs(Node,[Node], _ ):- g(Node). l_dfs(Node,[Node|PartSolution], Max_depth ):- Max_Hl > 0,h(Node,NextN),Max1 is Max_Hl –1, l_dfs(NextN, PartSolution, Max1 ).

6. 2. Order predicatessetof (+Z,+Goal(Z,X),-S) It creates the list S of all instances for Z, such that the goal Goal(Z, X) holds (value of X remains fixed) : • The list S is presented as an ordered set, ie. No repetition of elements • In the case there is no solution for Goal(Z,X), the answer of setof is false Příklad: h(a,b). h(a,c). h(a,d). h(b,e). h(b,f). h(c,g). h(c,h). h(c,a). ?- setof(Z,h(a,Z),S). S = [b,c,d] ?- setof(Z,h(X,Z),S). X = a, S = [b, c, d] ; More? X = b, S = [e, f] ; X = c, S = [a, g, h]

7. 2. Order predicatesbagof (+Z,+Goal(Z,X),-S) The variables in the Goal +Goal(Z,X) can be existentially quantifiedusing the symbol^ h(a,b). h(a,c). h(a,d). h(b,e). h(b,f). h(c,g). h(c,h). h(c,a). ?- setof(Z,X^h(X,Z),S). S = [b, c, d, e, f, g, h, a] bagof(+Z,+Goal(Z,_),-S) finds the listS of all instances for Z solving the goal Goal(Z,_). An element can have several occurences in S (it is not a set) --> quicker thansetof ?- bagof(Z, h(a,Z), S). S = [ b, c, d ] % Caution ! There is no edge from d: ?- bagof(Z, h(d,Z), S). fail

8. 2. Order Predicatesfindall (+Term(Z),+Goal(Z),-S) Simmilar to bagof, but it is ever successfull : It creates the list S (with repetition of elements) of instances of the term Term using all instances for Z which solve the Goal(Z) – If there is no solution for Goal(Z), the resulting S is [ ] Příklad: h(a,b). h(a,c). h(a,d). h(b,e). h(b,f). h(c,g). h(c,h). h(c,a). ?- findall(symetricky(X,Y), (h(X,Y),h(Y,X)) ,S). S = [symetricky(a,c)] ?- findall(symetricky(b,Y), (h(b,Y),h(Y,b)) ,S). S = [ ] ?- findall(Z, h(f,Z), S). S = [ ]

9. Breadth first search add_back(+List,+Node,-Result) finds all direct successors of the nodeNodeandadds this list at the back ofthe List, the resulting list is denoted Result. wave(+List,Z) searches for a path, starting from the first element of List and leading to Z (simmilar to OPEN in procedural solution) bfs(X,X) :- goal(X). bfs(X,Y) :- wave([X],Y). wave([],_) :- write(‚No more goal nodes in the graph.'),nl. wave([H|T],H):- goal(H). wave([Y|L],Z):- add_back(L,Y,N),wave(N,Z). add_back(L,Y,N) :-direct_successors(Y,S),append(L,S,N). direct_successors(Y,S) :- bagof(Z,h(Y,Z),S),!. direct_successors(Y,[]).

10. Searching a graph and means of Prolog • trees and acyclic graphs - pure Prolog • finite graphs where cycle can appear - Prolog + negation * • infinite (but localy finite) graphs - Prolog + negation* + 2nd order predicates ** * needed to check the cycle ** breadth first search

11. Searching in the state space • State space = implicitely defined graph, its edges correspond to allowed moves. The problem is described using following predicates: • move(State, Aplicable_action), specifies all actions which can be accomplished in the given State • update(State, Aplicable_action, Resulting_state_after_action), generates new state • init_state(S), final_state(S) • Solution is very close to graph search.

12. The problem g-w-c Reprezentation of states wgc(Pozition_boat, Who_left, Who_right) Reprezentation of actions: action is characterized by cargo of the boat % Description of the initial and goal state init_state(wgc, wgc(left,[w,g,c],[])). final_state(wgc, wgc(right,[],[w,g,c])). % Conditions specifying actions available in the given state move(wgc(left,L,R),Cargo):-element(Cargo,L). move(wgc(right,L,R),Cargo):-element(Cargo,R). move(wgc(B,L,R),alone).

13. % State resulting from the considered action update(wgc(B,L,R),Cargo,wgc(B1,L1,R1)):- update_boat(B,B1), update_banks(Cargo,B,L,R,L1,R1). update_boat(left,right). update_boat(right,left). update_banks(alone,B,L,R,L,R). update_banks(Cargo,left,L,R,L1,R1):- select(Cargo,L,L1), insert(Cargo,R,R1). update_banks(Cargo,right,L,R,L1,R1):-… %Checking for danger legal(wgc(left,L,R)):- not illegal(R). … illegal(L):- member(w,L), member(g,L). …

14. General programmefor state space problems % Alg. Generating and searching through the state space starting from initial_state % and aiming for final_stateapplying following description of the task: % move(State, Applicable_ation) % update(Stav, Applicable_action, State_Result_of_action) solve_dfs(State,History,[]):-final_state(State). solve_dfs(State,History,[Move|Moves]):- move(State,Move), update(State,Move,State1), legal(State1), not member(State1,History), solve_dfs(State1,[State1|History],Moves). %Testování na konkrétní úloze test_dfs(Problem,Moves):- initial_state(Problem,State), solve_dfs(State,[State],Moves).