Examples in XSB Prolog

1
Examples in XSB Prolog

• ?pe?e??as?a ?a? ??apa??stas? G??s??
• ?????? 2010
• ?µ?µa ?p?st?µ?? ?p?????st?? ?a?ep?st?µ??? ???t??

2
Typing a program on the XSB command line

• ? ???st?? t?? XSB ??e? t? d??at?t?ta ?a e?s??e?
  t?? p??t?se?? (?e????ta ?a? ?a???e?) t??
  p?????µµat?? t?? ???s?µ?p????ta? t? ??aµµ?
  e?t???? t?? XSB, p.?.
• ?- user. Enter mode to type predicates
  rules.
• Compiling user
• edge(1,2).
• edge(2,3).
• edge(2,4).
• reachable(X,Y) - edge(X,Y).
• reachable(X,Y) - edge(X,Z), reachable(Z, Y).
• end_of_file. Done with terminal mode
• user compiled, cpu time used 0.0500 seconds
• user loaded
• yes

3
Loading a program in XSB

• ??e? ep?s?? t? d??at?t?ta ?a f??t?se? ??a
  a??e?? p?? pe????e? t? ?????? p????aµµa. ??t? ?a
  p??pe? ?a ??e? ?at????? .P
• ?- edges. Load and run edges.P
• edges loaded
• yes
• G?a ?a f??t?se? ??a a??e?? p?? ß??s?eta? se
  d?af??et??? directory ???fete
• ?- source/edges. Load and run edges.P
  which is saved in directory source
• edges loaded
• yes
• To default directory ??a t?? XSB e??a? t?
• xsb-3.1-win32/config/x86-pc-windows/bin
• G?a ?a a???sete t? µ??e??? t?? ???s??a??de?? ????
  st?? ??aµµ? e??a?e??? ? ???ep?????? ? ???ta?? ?
  ???e??? pa?a?????

4
Making queries and termination

• ?- reachable(X,Y).X 1Y 2  Type a
  semi-colon repeatedlyX 2Y 3X 2Y
  4X 1Y 3X 1Y 4no
• ?- halt.  Command to Exit XSBEnd XSB (cputime
  0.15 secs, elapsetime 96.23 secs)

5
Prolog Program Semantics

• ?a p?????µµata st?? Prolog µp????? ?a ??????
  ?ata???t? µe d?? t??p???
• d???t??? (declaratively)
• d?ad??ast??? (procedurally)
• ?ts? a? ?e???s??µe t?? p??tas?
• P - Q, R.
• a?t? µp??e? ?a e?µ??e?te? ?? e???
• ????t??? t? P e??a? a????? a? t? Q e??a? a?????
  ?a? t? R e??a? a?????
• ??ad??ast??? G?a ?a ap?de???e? ?t? t? P e??a?
  a?????, p??pe? p??ta ?a ap?de??? p??ta ?t? to Q
  e??a? a????? ?a? st? s????e?a ?t? ?a? t? R e??a?
  a?????.

6
Loading a program in XSB

• ??a a??e?? µp??e? ?a pe????e?
• export declarations ??????? ta s?µß??a p??
  µp????? ?a ???s?µ?p??????? ap? ???a a??e?a
• local declarations ??????? ta s?µß??a µe t?p???
  µ??? eµß??e?a
• import declarations ep?t??p??? t? ???s? s?µß????
  p?? ????? e?a??e? ap? ???a a??e?a
• ?? d???se?? a?t?? µp????? ?a ?????? se
  ?p???d?p?te s?µe?? t?? p?????µµat?? ?a? ????? t??
  a??????? µ??f?
• - export sym1, ...,syml.
• - local sym1, ...,symm.
• - import sym1, ...,symn from module.
• ?p?? t? symi ??e? t? µ??f? functor/arity.
• p.?.
• - import reachable/2 from edges.

7
Useful examples or , if-then-else

• ?? ?????? or µp??e?te ?a t? e?f??sete ?a? ??
  e???
• or(A)- (A a A b ), write('A is a or b').
• ???s??? ?a µ?? ?e???te t?? pa?e???se??.
• if (X1) then write(b) else write(d)
• a(X)- write(a),
• ( X 1 -gt write(b) write(d) ),
• write(c).

8
Useful examples - for

• ???s??? ?a µ?? ?e???te t?? s?????? te?µat?sµ??.
• Call with i(5) ? Result 4,3,2,1,0
• end condition Ngt0
• i(0).
• i(N) - Ngt0, write(N), nl, N1 is N-1, i(N1).
• Call with for(0,5) ? Result 0,1,2,3,4
• end condition StartltEnd
• for(End , End ).
• for(Start, End )- StartltEnd, write(Start), nl,
• Start1 is Start1, for(Start1,End).

9
Dynamic Predicates

• ?at?????µata
• Stat??? (static) t??p?p?????ta? µ??? µe
  reloading t?? p?????µµat??.
• ???aµ??? (dynamic) µp????? ?a t??p?p??????? t??
  ??a p?? t???e? ??a p????aµµa.
• G?a ?a µp????µe ?a ?e???st??µe µ?a d?µ? d??aµ???,
  st?? a??? t?? p?????µµat?? ?????µe t? d???s?
• - dynamic turn/1 ?
• - dynamic pred/2.
• St? s????e?a µp????µe ?a p??s??s??µe /
  afa???s??µe ded?µ??a ap? t? ß?s? ???s?? µe ???s?
  t??
• assert() e?s??e? ??a ??? ?e????? st? µ??µ?.
• retract() d?a???fe? ??a ?e????? ap? t? µ??µ?.

10
XSB library modules List processing (1/3)

• append(?List1, ?List2, ?List3)
• module basics
• Succeeds if list List3 is the concatenation of
  lists List1 and List2.
• member(?Element, ?List)
• module basics
• Checks whether Element unifies with any element
  of list List, succeeding more than once if there
  are multiple such elements.
• memberchk(?Element, ?List)
• module basics
• Similar to member/2, except that memberchk/2 is
  deterministic, i.e. does not succeed more than
  once for any call.
• delete_ith(Index, List, ?Element, ?RestList)
• module listutil
• Succeeds if the ith element of the list List
  unifies with Element, and RestList is List with
  Element removed.

11
XSB library modules List processing (2/3)

• ith(?Index, ?List, ?Element)
• module basics
• Succeeds if the ith element of the list List
  unifies with Element. Fails if Index is not a
  positive integer or greater than the length of
  List. Either Index and List, or List and
  Element, should be instantiated at the time of
  the call.
• length(?List, ?Length)
• module basics
• Succeeds if the length of the list List is
  Length.
• same_length(?List1, ?List2)
• module basics
• Succeeds if list List1 and List2 are both lists
  of the same number of elements.
• select(?Element, ?L1, ?L2)
• module basics
• List2 derives from List1 by selecting (removing)
  an Element non-deterministically.

12
XSB library modules List processing (3/3)

• reverse(List, ?ReversedList)
• module basics
• Succeeds if ReversedList is the reverse of list
  List. If List is not a proper list, reverse/2
  can succeed arbitrarily many times. It works only
  one way.
• perm(List, ?Perm)
• module basics
• Succeeds when List and Perm are permutations of
  each other. The main use of perm/2 is to
  generate permutations of a given list. Perm may
  be partly instantiated.
• subseq(?Sequence, ?SubSequence, ?Complement)
• module basics
• Succeeds when SubSequence and Complement are
  both subsequences of the list Sequence (the
  order of corresponding elements being preserved)
  and every element of Sequence which is not in
  SubSequence is in the Complement and vice versa.
  
• merge(List1, List2, ?List3)
• module listutil
• Succeeds if List3 is the list resulting from
  merging lists List1 and List2

13
Append example
append(,L,L). append(XL, M, XN) -
append(L,M,N).
append(1,2,3,4,X)?
14
Append example
append(,L,L). append(XL,M,XN) -
append(L,M,N).
append(1,2,3,4,X)?
X1,L2,M3,4,AXN
15
Append example
append(,L,L). append(XL,M,XN) -
append(L,M,N).
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,AXN
16
Append example
append(,L,L). append(XL,M,XN) -
append(L,M,N).
X2,L,M3,4,N2N
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,A1N
17
Append example
append(,L,L). append(XL,M,XN) -
append(L,M,N).
append(,3,4,N)?
X2,L,M3,4,N2N
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,A1N
18
Append example
append(,L,L). append(XL,M,XN) -
append(L,M,N).
append(,3,4,N)?
L 3,4, N L
X2,L,M3,4,N2N
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,A1N
19
Append example
append(,L,L). append(XL,M,XN) -
append(L,M,N).
A 1N N 2N
N L L 3,4 Answer A 1,2,3,4
append(,3,4,N)?
L 3,4, N L
X2,L,M3,4,N2N
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,A1N
20
Building new predicates - Range

• Create a list containing all integers within a
  given range.
• - export range/3.
• range(I,I,I).
• range(I,K,IL) - I lt K, I1 is I 1,
  range(I1,K,L).
• range(I,K,L) - I lt K, and L is the list
  containing all
• consecutive integers from I to K.
• ?.?.
• ?- range(4, 7, L).
• L 4,5,6,7
• no

21
Building new predicates - Slice

• Extract a slice from a list.
• - export slice/4.
• slice(X_,1,1,X).
• slice(XXs,1,K,XYs) - K gt 1, K1 is K - 1,
• slice(Xs,1,K1,Ys).
• slice(_Xs,I,K,Ys) - I gt 1, I1 is I - 1, K1
  is K - 1,
• slice(Xs,I1,K1,Ys).
• slice(L1,I,K,L2) - L2 is the list of the
  elements of L1
• between index I and index K (both included).
• ?.?.
• ?- slice(19,21,35,43,56,68,77,89.90, 3, 5,
  L2).
• L2 35,43,56
• no

22
Building new predicates - Split

• Split a list into two parts.
• - export split/4.
• split(L,0,,L).
• split(XXs,N,XYs,Zs) - N gt 0, N1 is N -
  1, split(Xs,N1,Ys,Zs).
• split(L,N,L1,L2) - the list L1 contains the
  first N elements
• of the list L, the list L2 contains the
  remaining elements.
• ?.?.
• ?- split(1,2,3,4,5,6,7,8,9,10, 3, L1, L2).
• L1 1,2,3
• L2 4,5,6,7,8,9,10
• no

23
Building new predicates - Rotate

• Rotate a list N places to the left.
• - import append/3 from basics.
• - import length/2 from basics.
• - import split/4 from split.
• - export rotate/3.
• rotate(L1,N,L2) - N gt 0, length(L1,NL1),
• N1 is N mod NL1, rotate_left(L1,N1,L2).
• rotate(L1,N,L2) - N lt 0, length(L1,NL1),
• N1 is NL1 (N mod NL1),
  rotate_left(L1,N1,L2).
• rotate_left(L,0,L).
• rotate_left(L1,N,L2) - N gt 0,
  split(L1,N,S1,S2),append(S2,S1,L2).
• rotate(L1,N,L2) - the list L2 is obtained
  from the list L1
• by rotating the elements of L1 N places to the
  left.
• ?.?.

24
Building new predicates Random Selection

• Extract a given number of randomly selected
  elements from a list.
• - import random/3 from random.
• - import length/2 from basics.
• - import delete_ith/4 from listutil.
• - export rnd_select/3.
• rnd_select(_,0,).
• rnd_select(X,1,X).
• rnd_select(Xs,N,XZs) - N gt 0, length(Xs,L),
  random(1,L,I),
• delete_ith(I,Xs,X,Ys),
• N1 is N-1, rnd_select(Ys,N1,Zs).
• rnd_select(L,N,R) - the list R contains N
  randomly selected
• items taken from the list L.
• ?.?.
• ?- rnd_select(1,2,3,4,5,6,7,8,9, 5, R).
• R 7,2,8,1,4

25
Building new predicates Random Permutation

• Generate a random permutation of the elements of
  a list.
• - import length/2 from basics.
• - import rnd_select/3 from random_selection.
• - export rnd_permu/2.
• rnd_permu(L1,L2) - length(L1,N),
  rnd_select(L1,N,L2).
• rnd_permu(L1,L2) - the list L2 is a random
  permutation of
• the elements of the list L1.
• ?.?.
• ?- rnd_permu(1,2,3,4,5,6,7,8,9, L).
• L 7,3,5,1,8,2,4,6,9.
• no

26
Eight Queens Problem

• ? st???? e??a? ?a t?p??et????? ??t? ßas???sse? se
  µ?a s?a????a ????? ?a µp??e? ?a ape??e?ta? ??p??a
  ap? ?p??ad?p?te ????. Ta p??pe? ep?µ???? ? ???e
  ßas???ssa ?a ß??s?eta? se d?af??et??? ??aµµ?,
  st??? ?a? d?a????? t?? s?a????a?. ? p??te???µe??
  ??s? ap?te?e? µ?a ?e???? ??s? ??a s?a????e? (?a?
  a?t?st???? p????? ßas???ss??) ?p????d?p?te
  µe??????.
• ??apa??st??µe t?? ??se?? t?? ßas???ss?? µe µ?a
  ??sta t?? a???µ?? 1-?. G?a pa??de??µa ? ??sta
  4,2,7,3,6,8,5,1 s?µa??e? ?t? ? ßas???ssa st??
  p??t? st??? ß??s?eta? st? ??aµµ? 4, ? ßas???ssa
  st? de?te?? st??? st? ??aµµ? 2. ?.?.?. ?e a?t??
  t?? t??p? e?asfa?????µe ?t? ? ???e ßas???ssa
  ß??s?eta? se d?af??et??? st??? ?a? ??aµµ?. ??
  µ??? p?? ap?µ??e? e??a? ?a ???e? ? ??e???? t??
  d?a??????. ??a ßas???ssa t?p??et?µ??? st? st??? ?
  ?a? ??aµµ? ? ?ata?aµß??e? d?? d?a???????, µ?a µe
  a???µ? C XY ?a? µ?a µe a???µ? DX-Y.

27
Eight Queens The Solution

• - import memberchk/2 from basics.
• - import range/3 from range.
• - import select/3 from basics.
• queens(N,Qs) - Qs is a solution of the
  N-queens problem
• queens(N,Qs) - range(1,N,Rs), permu(Rs,Qs),
  test(Qs,1,,).
• permu(,).
• permu(Qs,YYs) - select(Y,Qs,Rs),
  permu(Rs,Ys).
• test(Qs,X,Cs,Ds) - the queens in Qs,
  representing columns X to
• N, are not in conflict with the diagonals Cs
  and Ds Cs and Ds keep
• track of the already occupied diagonals.
• test(,_,_,_).
• test(YYs,X,Cs,Ds) -
• C is X-Y, \ memberchk(C,Cs),
• D is XY, \ memberchk(D,Ds),
• X1 is X 1,

28
Knights Tour Problem

• ? st???? e??a? ?a ß???µe µ?a p??e?a p?? µp??e? ?a
  d?a????e? ??a ????? se ??e? t?? ??se?? µ?a
  s?a????a? a??a??et?? µe??????, ????? ?a
  ep?s?efte? t?? ?d?a ??s? pa?ap??? ap? µ?a f????.

29
Knights Tour The Solution (1/2)

• - import memberchk/2 from basics.
• knights(N,Knights) - Knights is a knight's
  tour on a NxN chessboard
• knights(N,Knights) - M is NN-1,
  knights(N,M,1/1,Knights).
• closed_knights(N,Knights) - Knights is a
  knight's tour on a NxN
• chessboard which ends at the same square where
  it begun.
• closed_knights(N,Knights) - knights(N,Knights),
  Knights X/Y_, jump(N,X/Y,1/1).
• knights(N,M,Visited,Knights) - the list of
  squares Visited must be
• extended by M further squares to give the
  solution Knights of the
• NxN chessboard knight's tour problem.
• knights(_,0,Knights,Knights).
• knights(N,M,Visited,Knights) -
• Visited X/Y_,
• jump(N,X/Y,U/V),
• \ memberchk(U/V,Visited),

30
Knights Tour The Solution (2/2)

• jumps on an NxN chessboard from square A/B to
  C/D
• jump(N,A/B,C/D) -
• jump_dist(X,Y),
• C is AX, C gt 0, C lt N,
• D is BY, D gt 0, D lt N.
• jump distances
• jump_dist(1,2).
• jump_dist(2,1).
• jump_dist(2,-1).
• jump_dist(1,-2).
• jump_dist(-1,-2).
• jump_dist(-2,-1).
• jump_dist(-2,1).
• jump_dist(-1,2).

31
?? p??ß??µa t?? ß?s???

• ?? p??ß??µa t?? ß?s??? ap?te?e? ??a ??ass???
  p??ß??µa ??????? (puzzle).
• Se µ?a ???? e??? p?taµ?? ?p?????? ??a? ß?s???,
  ??a? ????? ??a p??ßat? ?a? ??a deµ?t? sa???. ?
  d?a??s?µ? ß???a ???? µ??? d?? a?t??e?µe?a ???e
  f???. ?? ?p???s??µe ?t? se ???e st??µ? ? ?????
  ?a? t? p??ßat?, ?a??? ?a? t? p??ßat? ?a? t?
  deµ?t? de? µp????? ?a e??a? µ??a t??? se µ?a ????
  (????? t?? ß?s??), p??a e??a? ? a???????a
  ????se?? ? ?p??a p??pe? ?a ???e? ??a ?a pe??s???
  ???? ap??ta?t?
• ?? pa?ap??? p??ß??µa e??a? ??s?ast??? ??a
  p??ß??µa s?ed?asµ?? ????se?? (Planning). Sta
  p??ß??µata t?? ?at?????a? a?t??, µ?a ??s? e??a? ?
  a?a??t?s? st? ???? t?? ?atast?se?? t?? ??sµ?? t??
  p??ß??µat??. ???e t?t??a ?at?stas? pe?????fe? t??
  "??sµ?" t?? p??ß??µat?? t?? s???e???µ??? ???????
  st??µ?. ?? µetaß?se?? ap? µ?a s???e???µ???
  ?at?stas? se µ?a ep?µe?? ?????ta? µ?s? te?est??.

32
?p????? ??apa??stas??

• ???a? p????? ?? d?a??s?µe? ep?????? µe t?? ?p??e?
  µp????µe ?a a?apa?ast?s??µe t? s???e???µ???
  p??ß??µa. ? pa?a??t? ???p???s? ???s?µ?p??e? ??a
  t?? pe????af? t?? ?at?stas?? ??a s???eto ??? t??
  µ??f??
• state(shepherd(A),sheep(B),wolf(C),hey(D))
• ?p?? ta A, B, C, D e??a? ?? ???e? st?? ?p??e?
  ß?????ta? a?t?st???a ? ß?s???, t? p??ßat?, ?
  ????? ?a? t? deµ?t? sa???.
• ?e ß?s? t?? pa?ap??? a?apa??stas? ??????ta? ??
  te?est??, p?? a?t?st?????? ??s?ast??? st??
  ????se?? p?? µp????? ?a ?????? ??a ?a ???e? t?
  p??ß??µa.

33
???sµ?? ?at?????µ?t??

• ?? ?at?????µa move/3 pe?????fe? t?? ep?t?ept??
  ????se?? p?? µp????? ?a ??????. ? µetaß??t?
  State e??a? ? "t?????sa" ?at?stas?, ? µetaß??t?
  Move e??a? ? ????s? p?? ?aµß??e? ???a ??a ?a
  p?????e? ? ??a ?at?stas? p?? e??p??e?ta? µe t??
  µetaß??t? NewState.
• move/3
• move(State,Move,NewState)
• Move the shepherd
• move(state(shepherd(X1),sheep(S),wolf(W),hey(H)),
  go_to_other_shore(X1,X2), state(shepherd(X2),shee
  p(S),wolf(W),hey(H)))- opposite(X1,X2).
• Take the sheep to the other side
• move(state(shepherd(X1),sheep(X1),wolf(W),hey(H))
  , move_sheep(X1,X2), state(shepherd(X2),sheep(X2),
  wolf(W),hey(H)))- opposite(X1,X2).

34
???sµ?? ?at?????µ?t??

• Take the wolf to the other side
• move(state(shepherd(X1),sheep(S),wolf(X1),hey(H))
  , move_wolf(X1,X2), state(shepherd(X2),sheep(S),wo
  lf(X2),hey(H)))- opposite(X1,X2).
• Take the hey to the other side
• move(state(shepherd(X1),sheep(S),wolf(W),hey(X1))
  , move_hey(X1,X2), state(shepherd(X2),sheep(S),wol
  f(W),hey(X2)))- opposite(X1,X2).
• ?? ?at?????µa opposite/2 d????e? t?? d?? ap??a?t?
  ???e?.
• opposite/2
• opposite(X1,X2).
• opposite(a,b).
• opposite(b,a).

35
???sµ?? ?at?????µ?t??

• ?? ?at?????µa is_valid/1 pet??a??e? ?ta? ?
  ?at?stas? State e??a? ep?t?ept?, d??ad? de?
  pa?aß???e? t??? pe?????sµ??? t?? p??ß??µat??..
• is_valid/1
• is_valid(State).
• is_valid(state(shepherd(X),sheep(X),wolf(_),hey(_)
  )).
• is_valid(state(shepherd(X),sheep(_),wolf(X),hey(X)
  )).

36
???sµ?? ?at?????µ?t??

• ?? ?at?????µa end_state/1 pet??a??e? ?ta? ? State
  e??a? ? te???? ?at?stas?, d??ad? ??a ta
  a?t??e?µe?a e??a? st?? ???? b (?e????µe ?t?
  a????? ??a ta a?t??e?µe?a e??a? st?? ???? a).
• end_state/1
• end_state(State).
• end_state(state(shepherd(b),sheep(b),wolf(b),hey(b
  ))).

37
???sµ?? ?at?????µ?t??

• ?? ?at?????µa solve/3 ß??s?e? t?? se??? t??
  ????se?? p?? p??pe? ?a ?????? ??a ?a ep????e? t?
  p??ß??µa. ??s?ast??? ? a??????µ?? a?a??t?s?? p??
  ???s?µ?p??e?ta? e??a? ? ?ata-ß???? a?a??t?s? t??
  Prolog.
• ? µetaß??t? State e??a? ? "t?????sa" ?at?stas?, ?
  µetaß??t? Moves ? ??sta t?? ????se?? ?a? ?
  Previous_States ? ??sta t?? ?atast?se?? t??
  ?p??e? ??e? "ep?s?e?te?" µ???? t??a ? d?ad??as?a
  a?a??t?s??.
• ???e ??a ?at?stas? p?? p????pte? (NextState), de?
  p??pe? ?a a???e? st?? ??sta a?t? ??a t?? ap?f???
  ß?????.
• solve/3
• solve(State,Moves,Previous_States)
• solve(State,,_)- end_state(State).
• solve(State,MoveMoves,Previous_States)-
• move(State,Move,NextState),
• \ member(NextState,Previous_States),
• is_valid(NextState),
• solve(NextState,Moves,NextStatePrevious_States
  ).

38
???sµ?? ?at?????µ?t??

• ?? ep?µe?? ?at?????µa t?p??e? st?? ????? ta µ???
  µ?a? ??sta?, ?a? ???s?µ?p??e?ta? ??a ?a t?p??e?
  t?? ????se?? st?? ?????.
• pretty_write/1
• pretty_write(List)
• pretty_write().
• pretty_write(FirstRest)- write(' -gt '),
  write(First), nl,
• pretty_write(Rest).
• ?? ?at?????µa run/1, ???s?µ?p??e?ta? ??a ?a
  e?te??s??µe t? p????aµµa.
• run(Moves)-solve(state(shepherd(a),sheep(a),wolf(
  a),hey(a)),Moves,
• state(shepherd(a),sheep(a),wolf(a),hey(a))),
• pretty_write(Moves),nl.

39
?s??se??

• (as?.1) ??µ??????ste d?ad??as?a factorial p?? ?a
  ?p??????e? t? pa?a???t??? e??? a???µ??. ?.?.
• ?- factorial(4,X).
• X24
• (as?.2) ??µ??????ste d?ad??as?a median p?? ?a
  ?p??????e? t?? µesa?? µ?a? ??sta? p?? pe????e?
  a???µ??? pe??tt?? p??????. ?.?.
• ?- median(2,8,6,4,7,9,3,X).
• X6
• (as?.3) ??µ??????ste d?ad??as?a set_equality p??
  ?a e????e? a? d?? ??ste? pe??????? ta ?d?a
  st???e?a (de? µa? e?d?af??e? ? se??? p?? ?????
  st? ??sta). ?.?.
• ?- set_equality(1,2,3,3,1,2).
• yes
• ?- set_equality(1,2,3,3,1).
• no

40
?s??se??

• (as?.4) ??ste t? p??ß??µa t?? ?e?ap?st???? µe
  t??? ?a??ßa????.
• ?e????af? p??ß??µat??
• St?? ???e? e??? p?taµ?? ?p?????? s??????? t?e??
  ?e?ap?st???? ?a? t?e?? ?a??ßa??? ?ts? ?ste se
  ???e ???? ?a ?p???e? ?s?? a???µ?? ?e?ap?st????
  ?a? ?a??ß????. ?p?s??, ?p???e? ?a? µ?a ß???a se
  µ?a ap? t?? ???e?. T????µe ???? ?a µetafe????? se
  µ?a ????. ?µ??, ? ß???a ??a ?a ?????e? ??e???eta?
  t??????st?? ??a ?t?µ?, ???? µ??? µ???? d?? ?t?µa
  ?a? se ?aµ?a pe??pt?s? de? p??pe? ? a???µ?? t??
  ?a??ß???? ?a e??a? µe?a??te??? ap? a?t?? t??
  ?e?ap?st???? se ??p??a ????.
• nice link
• http//gym-tsepel.ioa.sch.gr/play/problem2.htm