Fsm exercise: solution
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1 Fsm exercise: solution accept(,final,q,[]) :- member(q,final). member(x,[x ]). member(x,[ L]):- member(x,l). 1 / 65
2 Fsm exercise: solution accept(,final,q,[]) :- member(q,final). accept(trans,final,q,[h T]) :- member([q,h,qn],trans), accept(trans,final,qn,t). member(x,[x ]). member(x,[ L]):- member(x,l). 2 / 65
3 Fsm exercise: solution accept(,final,q,[]) :- member(q,final). accept(trans,final,q,[h T]) :- member([q,h,qn],trans), accept(trans,final,qn,t). member(x,[x ]). member(x,[ L]):- member(x,l). Abilities Goals/Preferences Prior Knowledge Agent Stimuli Past Experiences Actions Environment 3 / 65
4 Graph modeling Western Australia Northern Territory South Australia Queensland New South Wales Victoria Tasmania 4 / 65
5 Graph modeling Western Australia Northern Territory Queensland WA NT Q South Australia New South Wales SA NSW Victoria V Victoria Tasmania T Russell & Norvig 5 / 65
6 Graph modeling Western Australia Northern Territory Queensland WA NT Q South Australia New South Wales SA NSW Victoria V Victoria Tasmania T Russell & Norvig arc(wa,nt). arc(nt,q). arc(q,nsw). arc(wa,sa). arc(nt,sa). arc(sa,q). arc(sa,nsw). arc(sa,v). arc(v,nsw). arc2(x,y) :- arc(x,y) ; arc(y,x). 6 / 65
7 Search (in Prolog) Given goal, arc search(node) :- goal(node). search(node) :- arc(node,next), search(next). 7 / 65
8 Search (in Prolog) Given goal, arc search(node) :- goal(node). search(node) :- arc(node,next), search(next). Example: accept(trans,final,q0,string) Node as [Q,UnseenString] 8 / 65
9 Search (in Prolog) Given goal, arc search(node) :- goal(node). search(node) :- arc(node,next), search(next). Example: accept(trans,final,q0,string) Node as [Q,UnseenString] goal(q,[],final) :- member(q,final). 9 / 65
10 Search (in Prolog) Given goal, arc search(node) :- goal(node). search(node) :- arc(node,next), search(next). Example: accept(trans,final,q0,string) Node as [Q,UnseenString] goal(q,[],final) :- member(q,final). arc([q,[h T]],[Qn,T],Trans) :- member([q,h,qn],trans). 10 / 65
11 Search (in Prolog) Given goal, arc search(node) :- goal(node). search(node) :- arc(node,next), search(next). Example: accept(trans,final,q0,string) Node as [Q,UnseenString] goal(q,[],final) :- member(q,final). arc([q,[h T]],[Qn,T],Trans) :- member([q,h,qn],trans). search(q,s,f, ) :- goal(q,s,f). search(q,s,f,t) :- arc([q,s],[qn,sn],t), search(qn,sn,f,t). 11 / 65
12 Search (in Prolog) Given goal, arc search(node) :- goal(node). search(node) :- arc(node,next), search(next). Example: accept(trans,final,q0,string) Node as [Q,UnseenString] goal(q,[],final) :- member(q,final). arc([q,[h T]],[Qn,T],Trans) :- member([q,h,qn],trans). search(q,s,f, ) :- goal(q,s,f). search(q,s,f,t) :- arc([q,s],[qn,sn],t), search(qn,sn,f,t). accept(t,f,q,s) :- search(q,s,f,t). 12 / 65
13 Prolog as search i :- p,q. i :- r. p. r.?- i. 13 / 65
14 Prolog as search i :- p,q. [i] i :- r. p. r.?- i. StartNode = [i] 14 / 65
15 Prolog as search i :- p,q. [i] i :- r. [p,q] [r] p. r.?- i. StartNode = [i] 15 / 65
16 Prolog as search i :- p,q. [i] i :- r. [p,q] [r] p. [q] r.?- i. StartNode = [i] 16 / 65
17 Prolog as search i :- p,q. [i] i :- r. [p,q] [r] p. [q] [] r.?- i. StartNode = [i] 17 / 65
18 Prolog as search i :- p,q. [i] i :- r. [p,q] [r] p. [q] [] r.?- i. StartNode = [i] yes goal([]). 18 / 65
19 Prolog as search i :- p,q. [i] i :- r. [p,q] [r] p. [q] [] r.?- i. StartNode = [i] yes goal([]). prove(node) :- goal(node). prove(node) :- arc(node,next), prove(next). 19 / 65
20 KB and arc i :- p,q. i :- r. p. r. 20 / 65
21 KB and arc i :- p,q. i :- r. [i,p,q] [i,r] p. [p] r. [r] 21 / 65
22 KB and arc i :- p,q. i :- r. [i,p,q] [i,r] p. [p] r. [r] KB = [[i,p,q],[i,r],[p],[r]] 22 / 65
23 KB and arc i :- p,q. i :- r. [i,p,q] [i,r] p. [p] r. [r] KB = [[i,p,q],[i,r],[p],[r]] arc(node1,node2,kb) :-?? 23 / 65
24 KB and arc i :- p,q. i :- r. [i,p,q] [i,r] p. [p] r. [r] KB = [[i,p,q],[i,r],[p],[r]] arc([h T],N,KB) :- member([h B],KB), append(b,t,n). 24 / 65
25 KB and arc i :- p,q. i :- r. [i,p,q] [i,r] p. [p] r. [r] KB = [[i,p,q],[i,r],[p],[r]] arc([h T],N,KB) :- member([h B],KB), append(b,t,n). prove(node,kb) :- goal(node) ; arc(node,next,kb), prove(next,kb). 25 / 65
26 Non-termination (due to poor choice) i :- p,q. [i] i :- r. p :- i. r.?- i. prove([], ). prove([h T],KB) :- member([h B],KB), append(b,t,next), prove(next,kb).?- prove([i],[[i,p,q],[i,r],[p,i],[r]]). 26 / 65
27 Non-termination (due to poor choice) i :- p,q. [i] i :- r. [p,q] [r] p :- i. r.?- i. prove([], ). prove([h T],KB) :- member([h B],KB), append(b,t,next), prove(next,kb).?- prove([i],[[i,p,q],[i,r],[p,i],[r]]). 27 / 65
28 Non-termination (due to poor choice) i :- p,q. [i] i :- r. [p,q] [r] p :- i. [i,q] [] r.?- i. prove([], ). prove([h T],KB) :- member([h B],KB), append(b,t,next), prove(next,kb).?- prove([i],[[i,p,q],[i,r],[p,i],[r]]). 28 / 65
29 Non-termination (due to poor choice) i :- p,q. [i] i :- r. [p,q] [r] p :- i. [i,q] [] r. [p,q,q] [r,q]?- i. prove([], ). prove([h T],KB) :- member([h B],KB), append(b,t,next), prove(next,kb).?- prove([i],[[i,p,q],[i,r],[p,i],[r]]). 29 / 65
30 Non-termination (due to poor choice) i :- p,q. [i] i :- r. [p,q] [r] p :- i. [i,q] [] r. [p,q,q] [r,q]?- i. [i,q,q] [q] prove([], ). prove([h T],KB) :- member([h B],KB), append(b,t,next), prove(next,kb).?- prove([i],[[i,p,q],[i,r],[p,i],[r]]). 30 / 65
31 Non-termination (due to poor choice) i :- p,q. [i] i :- r. [p,q] [r] p :- i. [i,q] [] r. [p,q,q] [r,q]?- i. [i,q,q] [q] prove([], ). prove([h T],KB) :- member([h B],KB), append(b,t,next), prove(next,kb).?- prove([i],[[i,p,q],[i,r],[p,i],[r]]). 31 / 65
32 Determinization (do all) A fsm [Trans, Final, Q0] such that for all [Q,X,Qn] and [Q,X,Qn ] in Trans, Qn = Qn is a deterministic finite automaton (DFA). 32 / 65
33 Determinization (do all) A fsm [Trans, Final, Q0] such that for all [Q,X,Qn] and [Q,X,Qn ] in Trans, Qn = Qn is a deterministic finite automaton (DFA). Fact. Every fsm has a DFA accepting the same language. 33 / 65
34 Determinization (do all) A fsm [Trans, Final, Q0] such that for all [Q,X,Qn] and [Q,X,Qn ] in Trans, Qn = Qn is a deterministic finite automaton (DFA). Fact. Every fsm has a DFA accepting the same language. Proof: Subset (powerset) construction arcd(nodelist,nextlist) :- setof(next, arcln(nodelist,next), NextList). arcln(nodelist,next) :- member(node,nodelist), arc(node,next). 34 / 65
35 Determinization (do all) A fsm [Trans, Final, Q0] such that for all [Q,X,Qn] and [Q,X,Qn ] in Trans, Qn = Qn is a deterministic finite automaton (DFA). Fact. Every fsm has a DFA accepting the same language. Proof: Subset (powerset) construction arcd(nodelist,nextlist) :- setof(next, arcln(nodelist,next), NextList). arcln(nodelist,next) :- member(node,nodelist), arc(node,next). goald(nodelist) :- member(node,nodelist), goal(node). 35 / 65
36 Determinization (do all) A fsm [Trans, Final, Q0] such that for all [Q,X,Qn] and [Q,X,Qn ] in Trans, Qn = Qn is a deterministic finite automaton (DFA). Fact. Every fsm has a DFA accepting the same language. Proof: Subset (powerset) construction arcd(nodelist,nextlist) :- setof(next, arcln(nodelist,next), NextList). arcln(nodelist,next) :- member(node,nodelist), arc(node,next). goald(nodelist) :- member(node,nodelist), goal(node). searchd(nl) :- goald(nl); (arcd(nl,nl2), searchd(nl,nl2)). 36 / 65
37 Determinization (do all) A fsm [Trans, Final, Q0] such that for all [Q,X,Qn] and [Q,X,Qn ] in Trans, Qn = Qn is a deterministic finite automaton (DFA). Fact. Every fsm has a DFA accepting the same language. Proof: Subset (powerset) construction arcd(nodelist,nextlist) :- setof(next, arcln(nodelist,next), NextList). arcln(nodelist,next) :- member(node,nodelist), arc(node,next). goald(nodelist) :- member(node,nodelist), goal(node). searchd(nl) :- goald(nl); (arcd(nl,nl2), searchd(nl,nl2)). search(node) :- searchd([node]). 37 / 65
38 Frontier search (manage choices) start node ends of paths on frontier explored nodes unexplored nodes Poole & Mackworth search(node) :- frontiersearch([node]). frontiersearch([node ]) :- goal(node). frontiersearch([node Rest]) :- findall(next, arc(node,next), Children), add2frontier(children,rest,newfrontier), frontiersearch(newfrontier). 38 / 65
39 Breadth-first: queue (FIFO) [1] [2,3] [3,4,5] [4,5,6,7] 39 / 65
40 Breadth-first: queue (FIFO) [1] [2,3] [3,4,5] [4,5,6,7] add2frontier(children,[],children). add2frontier(children,[h T],[H More]) :- add2frontier(children,t,more). 40 / 65
41 Depth-first: stack (LIFO) [1] [2, ] [3,13, ] [4,12,13, ] 41 / 65
42 Depth-first: stack (LIFO) [1] [2, ] [3,13, ] [4,12,13, ] add2frontier([],rest,rest). add2frontier([h T],Rest,[H TRest]) :- add2frontier(t,rest,trest). 42 / 65
43 If-then-else and cut! i :- p,!,q. i :- r. p. r.?- i. 43 / 65
44 If-then-else and cut! i :- p,!,q. [i] i :- r. p. r.?- i. 44 / 65
45 If-then-else and cut! i :- p,!,q. [i] i :- r. [p,!,q] [r] p. r.?- i. 45 / 65
46 If-then-else and cut! i :- p,!,q. [i] i :- r. [p,!,q] [r] p. [!,q] r.?- i. Cut! is true but destroys backtracking. 46 / 65
47 If-then-else and cut! i :- p,!,q. i :- r. [i] [p,!,q] p. [!,q] r. [q]?- i. Cut! is true but destroys backtracking. 47 / 65
48 If-then-else and cut! i :- p,!,q. i :- r. [i] [p,!,q] p. [!,q] r. [q]?- i. no Cut! is true but destroys backtracking. 48 / 65
49 Review: Depth-first as frontier search prove([], ). % goal([]). prove(node,kb) :- arc(node,next,kb), prove(next,kb). 49 / 65
50 Review: Depth-first as frontier search prove([], ). % goal([]). prove(node,kb) :- arc(node,next,kb), prove(next,kb). fs([[] ], ). fs([node More],KB) :- findall(x,arc(node,x),l), append(l,more,newfrontier), fs(newfrontier,kb). 50 / 65
51 Review: Depth-first as frontier search prove([], ). % goal([]). prove(node,kb) :- arc(node,next,kb), prove(next,kb). fs([[] ], ). fs([node More],KB) :- findall(x,arc(node,x),l), append(l,more,newfrontier), fs(newfrontier,kb). Cut? 51 / 65
52 Tracking the frontier [[i]] i :- p,!,q. [i] i :- r. p. r.?- i. 52 / 65
53 Tracking the frontier [[i]] [[p,!,q],[r]] i :- p,!,q. [i] i :- r. [p,!,q] [r] p. r.?- i. 53 / 65
54 Tracking the frontier [[i]] [[p,!,q],[r]] [[!,q],[r]] i :- p,!,q. [i] i :- r. [p,!,q] [r] p. [!,q] r.?- i. 54 / 65
55 Tracking the frontier [[i]] [[p,!,q],[r]] [[!,q],[r]] [[q]] i :- p,!,q. i :- r. [i] [p,!,q] p. [!,q] r. [q]?- i. 55 / 65
56 Tracking the frontier [[i]] [[p,!,q],[r]] [[!,q],[r]] [[q]] [] i :- p,!,q. i :- r. [i] [p,!,q] p. [!,q] r. [q]?- i. 56 / 65
57 Tracking the frontier [[i]] [[p,!,q],[r]] [[!,q],[r]] [[q]] [] i :- p,!,q. i :- r. [i] [p,!,q] p. [!,q] r. [q]?- i. no 57 / 65
58 Cut via frontier depth-first search fs([[] ], ). fs([node More],KB) :- findall(x,arc(node,x),l), append(l,more,newfrontier), fs(newfrontier,kb). 58 / 65
59 Cut via frontier depth-first search fs([[] ], ). fs([[cut T] ],KB)) :- fs([t],kb). fs([node More],KB) :- findall(x,arc(node,x),l), append(l,more,newfrontier), fs(newfrontier,kb). 59 / 65
60 Cut via frontier depth-first search fs([[] ], ). fs([[cut T] ],KB)) :- fs([t],kb). fs([node More],KB) :- Node = [H ], H\== cut, findall(x,arc(node,x),l), append(l,more,newfrontier), fs(newfrontier,kb). 60 / 65
61 Cut via frontier depth-first search fs([[] ], ). fs([[cut T] ],KB)) :- fs([t],kb). fs([node More],KB) :- Node = [H ], H\== cut, findall(x,arc(node,x),l), append(l,more,newfrontier), fs(newfrontier,kb). if(p,q,r) :- (p,!,q); r. % contra (p,q);r 61 / 65
62 Cut via frontier depth-first search fs([[] ], ). fs([[cut T] ],KB)) :- fs([t],kb). fs([node More],KB) :- Node = [H ], H\== cut, findall(x,arc(node,x),l), append(l,more,newfrontier), fs(newfrontier,kb). if(p,q,r) :- (p,!,q); r. % contra (p,q);r negation-as-failure(p) :- (p,!,fail); true. 62 / 65
63 Exercise (Prolog) Suppose a positive integer Seed links nodes 1,2,... in two ways arc(n,m,seed) :- M is N*Seed. arc(n,m,seed) :- M is N*Seed +1. e.g. Seed=3 gives arcs (1,3), (1,4), (3,9), (3, 10) / 65
64 Exercise (Prolog) Suppose a positive integer Seed links nodes 1,2,... in two ways arc(n,m,seed) :- M is N*Seed. arc(n,m,seed) :- M is N*Seed +1. e.g. Seed=3 gives arcs (1,3), (1,4), (3,9), (3, 10)... Goal nodes are multiples of a positive integer Target goal(n,target) :- 0 is N mod Target. e.g. Target=13 gives goals 13, 26, / 65
65 Exercise (Prolog) Suppose a positive integer Seed links nodes 1,2,... in two ways arc(n,m,seed) :- M is N*Seed. arc(n,m,seed) :- M is N*Seed +1. e.g. Seed=3 gives arcs (1,3), (1,4), (3,9), (3, 10)... Goal nodes are multiples of a positive integer Target goal(n,target) :- 0 is N mod Target. e.g. Target=13 gives goals 13, 26, Modify frontier search to define predicates breadth1st(+start,?found, +Seed, +Target) depth1st(+start,?found, +Seed, +Target) that search breadth-first and depth-first respectively for a Target-goal node Found linked to Start by Seed-arcs. 65 / 65
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