Declarative Loops and List Comprehensions for Prolog

1
Declarative Loops and List Comprehensions for
Prolog

• Neng-Fa Zhou
• Brooklyn College
• The City University of New York
• zhou_at_sci.brooklyn.cuny.edu

2
Motivation

• Second ASP-solver Competition
• 38 problems were used
• The B-Prolog team was the only CLP(FD) team
• We had to develop solutions from scratch
• We desperately needed loop constructs

3
One of the ProblemsMaze generation

• Conditions
• 1. There must be a path from the entrance to
  every empty cell.
• 2. There must be no 2x2 blocks of empty cells or
  walls.
• 3. No wall can be completely surrounded by empty
  cells.

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The Array Subscript Notation for Structures and
Lists in B-Prolog

• In arithmetic expressions
• In arithmetic constraints
• In calls to _at_ and _at_
• In any other context, XI1,,In is the same as
  XI1,,In

S is X1X2X3
X1X2 X3
X1,2 _at_ 100X1,2 _at_ 100
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foreach(E1 in D1, . . ., En in Dn, Goal)

• Ei in Di
• Ei is a pattern (usually a variable but can be a
  compound term)
• Di is a collection (a list or a range of integers
  l..u)
• Semantics
• For each combination of values E1?D1, . . ., En ?
  Dn, execute Goal

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Example-1

• Using foreach
• Using recursion

?- L1,2,3,foreach(X in L, write(X)).

• ?- L1,2,3,write_list(L).
• write_list().
• write_list(XT) - write(X), write_list(T).

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Example-2

• Using foreach
• Using recursion

?- foreach(X in 1..3, write(X)).
?- write_int_range(1,3). write_int_range(I,N)-IgtN
,!. write_int_range(I,N) - write(I), I1
is I1, write_int_range(I1,N).
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Example-3 (compound patterns)

• Using foreach
• Using recursion (matching clauses)

?-L(a,1),(b,2),foreach((A,I) in L,
writeln(AI)).
?- L(a,1),(b,2),write_pairs(L). write_pairs()
gt true. write_pairs((A,I)L) gt
writeln(AI), write_pairs(L).
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foreach(E1 in D1, . . ., En in Dn, LVars, Goal)
Variables in LVars are local to each iteration.

• Using foreach
• Using recursion

?-Sf(1,2,3), foreach(I in 1..Slengh,E, (E
_at_ SI, write(E))).
?-Sf(1,2,3),functor(S,_,N),write_args(S,1,N).wri
te_args(S,I,N)-IgtN,!.write_args(S,I,N)-
arg(I,S,E), write(E), I1 is I1,
write_args(S,I1,N).
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List Comprehension
T E1 in D1, . . ., En in Dn, LVars, Goal

• Calls to _at_/2
• Arithmetic constraints

?- L _at_ X X in 1..5.L1,2,3,4,5?-L _at_
(A,I) A in a,b, I in 1..2.L
(a,1),(a,2),(b,1),(b,2)
sum(AI,J I in 1..N, J in 1..N) NN
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foreach With Accumulators
L _at_ (A,I) A in a,b, I in 1..2 foreach(A
in a,b, I in 1..2, ac1(L,),
L0(A,I)L1)
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Application Examples

• Quick sort
• Permutations
• N-Queens
• Non-boolean constraints
• Boolean constraints
• Gaussian elimination
• No-Three-in-a-Line Problem
• Maze generation

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Quick Sort
qsort(,). qsort(HT,S)- L1 _at_ X X
in T, XltH, L2 _at_ X X in T, XgtH,
qsort(L1,S1), qsort(L2,S2),
append(S1,HS2,S).
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Permutations
perms(,). perms(XXs,Ps)-
perms(Xs,Ps1), Ps _at_ P P1 in Ps1, I in
0..Xslength,P,
insert(X,I,P1,P).   insert(X,0,L,XL). insert(X
,I,YL1,YL)- Igt0, I1 is I-1,
insert(X,I1,L1,L).
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The N-Queens Problem

• Qi the number of the row for the ith queen.
• queens(N)-
• length(Qs,N),
• Qs 1..N,
• foreach(I in 1..N-1, J in I1..N,
• (QsI \ QsJ,
• abs(QsI-QsJ) \ J-I)),
• labeling(ff,Qs),
• writeln(Qs).

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The N-Queens Problem (Boolean Constraints)

• Qij1 iff the cell at (i,j) has a queen.
• bqueens(N)-
• new_array(Qs,N,N),
• Vars _at_ QsI,J I in 1..N, J in 1..N,
• Vars 0..1,
• foreach(I in 1..N,
• sum(QsI,J J in 1..N) 1),
• foreach(J in 1..N,
• sum(QsI,J I in 1..N) 1),
• foreach(K in 1-N..N-1,
• sum(QsI,J I in 1..N, J in 1..N,
  I-JK) lt 1),
• foreach(K in 2..2N,
• sum(QsI,J I in 1..N, J in 1..N,
  IJK) lt 1),
• labeling(Vars).

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Gaussian Elimination
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Gaussian Elimination
triangle_matrix(Matrix)-     foreach(I in
1..Matrixlength-1,           Row,J,          
                              (select_nonzero_row
(I,I,Matrix,J)-gt               (I\J-gt
(Row _at_ MatrixI,                     
     MatrixI _at_ MatrixJ,                  
 MatrixJ _at_ Row)                          
        true                ),          
 foreach(K in I1..Matrixlength,
trans_row(I,K,Matrix))              
          true       ) ).
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No-Three-in-a-Line Problem
no3_build(N)- new_array(Board,N,N),
Vars _at_ BoardI,J I in 1..N, J in 1..N,
Vars 0..1, Sum sum(Vars), Sum lt
2N, foreach(X1 in 1..N, Y1 in 1..N, L,SL,
(L _at_ Slope X2 in 1..N, Y2 in
1..N, Slope,
(X2\X1,
Slope is (Y2-Y1)/(X2-X1)),
sort(L,SL), eliminate duplicates
foreach(Slope in SL,
sum(BoardX,Y X in
1..N, Y in 1..N,
(X\X1,Slope(Y-Y1)/(X-X1)))
lt 3))), foreach(X in 1..N, sum(BoardX,Y
Y in 1..N)lt3), labeling(SumVars),
outputBoard(Board,Sum,N).
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Maze generation
There must be no 2x2 blocks of empty cells or
walls foreach(I in 1..M-1, J in 1..N-1,
(MazeI,JMazeI,J1MazeI1,JMazeI1,J1gt
0, MazeI,JMazeI,J1MazeI1,JMazeI1
,J1lt4)),
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Demo

• B-Prolog version 7.4
• N-queens problem
• www.probp.com/examples.htm

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Conclusion

• Foreach and list comprehension constitute a
  better alternative for writing loops
• Recursion
• Failure-driven loops
• Higher-order predicates (e.g., findall,maplist)
• The do construct in ECLiPSe Prolog
• Very useful when used on arrays
• Significantly enhance the modeling power of
  CLP(FD)