Matrix Multiplication Prolog

Question

I found this code online for matrix multiplication in Prolog, can anyone who has experience in Prolog explain it to me?

% SWI-Prolog has transpose/2 in its clpfd library

:- use_module(library(clpfd)).
 
% N is the dot product of lists V1 and V2.

dot(V1, V2, N) :- 
    maplist(product, V1, V2, P),  
    sumlist(P, N).

product(N1,N2, N3) :- 
    N3 is N1 * N2.
 
% Matrix multiplication with matrices represented
% as lists of lists. M3 is the product of M1 and M2

mmult(M1, M2, M3) :- 
    transpose(M2, MT), 
    maplist(mm_helper(MT), M1, M3).

mm_helper(M2, I1, M3) :- 
   maplist(dot(I1), M2, M3).

James Gaunt · Answer 1 · 2019-04-15T11:58:55.973

It looks horrendous! But it kind of makes sense.

Obviously % is the start of a comment (just in case that's not clear)

:- is defining a function

a function is a series of other functions, separated by commas.

parameters are passed by reference (so no return values), in the code above the last parameter is where the result of each function is stored. I'd guess this is some sort of convention in prolog as it also seems to be the case with the library functions.

maplist sounds like it applies a function to each element in the passed list (i.e. map on a list), stores the values in the result list (last parameter). You can google this function to confirm. But that would make sense with how it has been used to define the dot product function.

transpose is presumably just normal matrix transposition (being imported from a library - as per the top comment).

That covers all the syntax I can see. So ultimately it defines mmult() function, which takes 3 inputs, M1 & M2 are lists of lists of numbers (i.e. matricies) and M3 is the output.

Guy Coder · Accepted Answer · 2019-04-15T18:55:17.043

Lines that start with % are comments

Most programming langues have methods and functions. Prolog has predicates which succeed or fail and can pass values along in variables.

Predicates are denoted by their name and arity as such, transpose/2.

Most programming languages use assignment, but Prolog uses unification. For the special case of solving math expressions Prolog has is/2

:- use_module(library(clpfd)).

Is a directive :- that brings in a library named clpfd. clpfd is used for constraints normally but in this case is used to access the transpose/2 predicate.

dot(V1, V2, N) :- 
    maplist(product,V1,V2,P), 
    sumlist(P,N).

dot/3 is a predicate that takes in two vectors, I am guessing implemented as Prolog list in this case, and binds the dot product as N.
maplist/4 applies the product/3 predicate to the values in V1 and V2 to create the list P.
sumlist/2 adds up the list of values in P and binds N.

product(N1,N2,N3) :- N3 is N1*N2.

product/3 is a helper predicate to take two numbers N1 and N2 and multiply them.

N3 is N1*N2 can be thought of as N3 = (N1 * N2)

mmult(M1, M2, M3) :- 
    transpose(M2,MT), 
    maplist(mm_helper(MT), M1, M3).

transpose/2 is the typical array transpose.
maplist/3 applies the helper predicate mm_helper/3 using MT and M1 to bind M3.

mm_helper(M2, I1, M3) :- 
    maplist(dot(I1), M2, M3).

maplist/3 applies dot/3 to I1 and M2 to bind M3.

I take it you took the code from RosettaCode or it crossed paths with such.

Example run:

?- mmult([[1,2],[3,4]],[[5,6],[7,8]],R).
R = [[19, 22], [43, 50]] ;
true.

Verification of example.

From comment.

What does I1 represents in mm_helper predicate?

See trace below for details

The signature of mm_helper/3 is mm_helper(M2, I1, M3) so the second parameter contains I1. From the lines in the trace

Exit: (11) mm_helper([[5, 7], [6, 8]], [3, 4], [43, 50])

the second parameter is [3,4].

Call: (10) mm_helper([[5, 7], [6, 8]], [1, 2], _3596)

the second parameter is [1,2].

So I2 is a row slice of the first matrix.

Be careful of what you ask for, you might get more than expected? :)

Full trace of run:

This is only here because of a question in the comment.

[trace]  ?- mmult([[1,2],[3,4]],[[5,6],[7,8]],R).


   Call: (8) mmult([[1, 2], [3, 4]], [[5, 6], [7, 8]], _3238)
   Unify: (8) mmult([[1, 2], [3, 4]], [[5, 6], [7, 8]], _3238)
   Call: (9) clpfd:transpose([[5, 6], [7, 8]], _3536)
   Unify: (9) clpfd:transpose([[5, 6], [7, 8]], _3536)
   Call: (10) error:must_be(list(list), [[5, 6], [7, 8]])
   Unify: (10) error:must_be(list(list), [[5, 6], [7, 8]])
   Exit: (10) error:must_be(list(list), [[5, 6], [7, 8]])
   Call: (10) clpfd:lists_transpose([[5, 6], [7, 8]], _3540)
   Unify: (10) clpfd:lists_transpose([[5, 6], [7, 8]], _3540)
   Call: (11) clpfd:'__aux_maplist/2_same_length+1'([[7, 8]], [5, 6])
   Unify: (11) clpfd:'__aux_maplist/2_same_length+1'([[7, 8]], [5, 6])
   Call: (12) lists:same_length([5, 6], [7, 8])
   Unify: (12) lists:same_length([5, 6], [7, 8])
   Exit: (12) lists:same_length([5, 6], [7, 8])
   Call: (12) clpfd:'__aux_maplist/2_same_length+1'([], [5, 6])
   Unify: (12) clpfd:'__aux_maplist/2_same_length+1'([], [5, 6])
   Exit: (12) clpfd:'__aux_maplist/2_same_length+1'([], [5, 6])
   Exit: (11) clpfd:'__aux_maplist/2_same_length+1'([[7, 8]], [5, 6])
^  Call: (11) apply:foldl(transpose_, [5, 6], _3548, [[5, 6], [7, 8]], _3552)
^  Unify: (11) apply:foldl(clpfd:transpose_, [5, 6], _3554, [[5, 6], [7, 8]], _3558)
   Call: (12) apply:foldl_([5, 6], _3552, clpfd:transpose_, [[5, 6], [7, 8]], _3558)
   Unify: (12) apply:foldl_([5, 6], [_3536|_3538], clpfd:transpose_, [[5, 6], [7, 8]], _3564)
   Call: (13) clpfd:transpose_(5, _3536, [[5, 6], [7, 8]], _3562)
   Unify: (13) clpfd:transpose_(5, _3536, [[5, 6], [7, 8]], _3562)
   Call: (14) clpfd:'__aux_maplist/4_list_first_rest+0'([[5, 6], [7, 8]], _3536, _3560)
   Unify: (14) clpfd:'__aux_maplist/4_list_first_rest+0'([[5, 6], [7, 8]], [_3542|_3544], [_3548|_3550])
   Call: (15) clpfd:list_first_rest([5, 6], _3542, _3548)
   Unify: (15) clpfd:list_first_rest([5, 6], 5, [6])
   Exit: (15) clpfd:list_first_rest([5, 6], 5, [6])
   Call: (15) clpfd:'__aux_maplist/4_list_first_rest+0'([[7, 8]], _3544, _3550)
   Unify: (15) clpfd:'__aux_maplist/4_list_first_rest+0'([[7, 8]], [_3554|_3556], [_3560|_3562])
   Call: (16) clpfd:list_first_rest([7, 8], _3554, _3560)
   Unify: (16) clpfd:list_first_rest([7, 8], 7, [8])
   Exit: (16) clpfd:list_first_rest([7, 8], 7, [8])
   Call: (16) clpfd:'__aux_maplist/4_list_first_rest+0'([], _3556, _3562)
   Unify: (16) clpfd:'__aux_maplist/4_list_first_rest+0'([], [], [])
   Exit: (16) clpfd:'__aux_maplist/4_list_first_rest+0'([], [], [])
   Exit: (15) clpfd:'__aux_maplist/4_list_first_rest+0'([[7, 8]], [7], [[8]])
   Exit: (14) clpfd:'__aux_maplist/4_list_first_rest+0'([[5, 6], [7, 8]], [5, 7], [[6], [8]])
   Exit: (13) clpfd:transpose_(5, [5, 7], [[5, 6], [7, 8]], [[6], [8]])
   Call: (13) apply:foldl_([6], _3538, clpfd:transpose_, [[6], [8]], _3588)
   Unify: (13) apply:foldl_([6], [_3566|_3568], clpfd:transpose_, [[6], [8]], _3594)
   Call: (14) clpfd:transpose_(6, _3566, [[6], [8]], _3592)
   Unify: (14) clpfd:transpose_(6, _3566, [[6], [8]], _3592)
   Call: (15) clpfd:'__aux_maplist/4_list_first_rest+0'([[6], [8]], _3566, _3590)
   Unify: (15) clpfd:'__aux_maplist/4_list_first_rest+0'([[6], [8]], [_3572|_3574], [_3578|_3580])
   Call: (16) clpfd:list_first_rest([6], _3572, _3578)
   Unify: (16) clpfd:list_first_rest([6], 6, [])
   Exit: (16) clpfd:list_first_rest([6], 6, [])
   Call: (16) clpfd:'__aux_maplist/4_list_first_rest+0'([[8]], _3574, _3580)
   Unify: (16) clpfd:'__aux_maplist/4_list_first_rest+0'([[8]], [_3584|_3586], [_3590|_3592])
   Call: (17) clpfd:list_first_rest([8], _3584, _3590)
   Unify: (17) clpfd:list_first_rest([8], 8, [])
   Exit: (17) clpfd:list_first_rest([8], 8, [])
   Call: (17) clpfd:'__aux_maplist/4_list_first_rest+0'([], _3586, _3592)
   Unify: (17) clpfd:'__aux_maplist/4_list_first_rest+0'([], [], [])
   Exit: (17) clpfd:'__aux_maplist/4_list_first_rest+0'([], [], [])
   Exit: (16) clpfd:'__aux_maplist/4_list_first_rest+0'([[8]], [8], [[]])
   Exit: (15) clpfd:'__aux_maplist/4_list_first_rest+0'([[6], [8]], [6, 8], [[], []])
   Exit: (14) clpfd:transpose_(6, [6, 8], [[6], [8]], [[], []])
   Call: (14) apply:foldl_([], _3568, clpfd:transpose_, [[], []], _3618)
   Unify: (14) apply:foldl_([], [], clpfd:transpose_, [[], []], [[], []])
   Exit: (14) apply:foldl_([], [], clpfd:transpose_, [[], []], [[], []])
   Exit: (13) apply:foldl_([6], [[6, 8]], clpfd:transpose_, [[6], [8]], [[], []])
   Exit: (12) apply:foldl_([5, 6], [[5, 7], [6, 8]], clpfd:transpose_, [[5, 6], [7, 8]], [[], []])
^  Exit: (11) apply:foldl(clpfd:transpose_, [5, 6], [[5, 7], [6, 8]], [[5, 6], [7, 8]], [[], []])
   Exit: (10) clpfd:lists_transpose([[5, 6], [7, 8]], [[5, 7], [6, 8]])
   Exit: (9) clpfd:transpose([[5, 6], [7, 8]], [[5, 7], [6, 8]])
   Call: (9) '__aux_maplist/3_mm_helper+1'([[1, 2], [3, 4]], _3238, [[5, 7], [6, 8]])
   Unify: (9) '__aux_maplist/3_mm_helper+1'([[1, 2], [3, 4]], [_3596|_3598], [[5, 7], [6, 8]])
   Call: (10) mm_helper([[5, 7], [6, 8]], [1, 2], _3596)
   Unify: (10) mm_helper([[5, 7], [6, 8]], [1, 2], _3596)
   Call: (11) '__aux_maplist/3_dot+1'([[5, 7], [6, 8]], _3596, [1, 2])
   Unify: (11) '__aux_maplist/3_dot+1'([[5, 7], [6, 8]], [_3602|_3604], [1, 2])
   Call: (12) dot([1, 2], [5, 7], _3602)
   Unify: (12) dot([1, 2], [5, 7], _3602)
   Call: (13) '__aux_maplist/4_product+0'([1, 2], [5, 7], _3626)
   Unify: (13) '__aux_maplist/4_product+0'([1, 2], [5, 7], [_3608|_3610])
   Call: (14) product(1, 5, _3608)
   Unify: (14) product(1, 5, _3608)
   Call: (15) _3608 is 1*5
   Exit: (15) 5 is 1*5
   Exit: (14) product(1, 5, 5)
   Call: (14) '__aux_maplist/4_product+0'([2], [7], _3610)
   Unify: (14) '__aux_maplist/4_product+0'([2], [7], [_3620|_3622])
   Call: (15) product(2, 7, _3620)
   Unify: (15) product(2, 7, _3620)
   Call: (16) _3620 is 2*7
   Exit: (16) 14 is 2*7
   Exit: (15) product(2, 7, 14)
   Call: (15) '__aux_maplist/4_product+0'([], [], _3622)
   Unify: (15) '__aux_maplist/4_product+0'([], [], [])
   Exit: (15) '__aux_maplist/4_product+0'([], [], [])
   Exit: (14) '__aux_maplist/4_product+0'([2], [7], [14])
   Exit: (13) '__aux_maplist/4_product+0'([1, 2], [5, 7], [5, 14])
   Call: (13) backward_compatibility:sumlist([5, 14], _3602)
   Unify: (13) backward_compatibility:sumlist([5, 14], _3602)
   Call: (14) lists:sum_list([5, 14], _3602)
   Unify: (14) lists:sum_list([5, 14], _3602)
   Exit: (14) lists:sum_list([5, 14], 19)
   Exit: (13) backward_compatibility:sumlist([5, 14], 19)
   Exit: (12) dot([1, 2], [5, 7], 19)
   Call: (12) '__aux_maplist/3_dot+1'([[6, 8]], _3604, [1, 2])
   Unify: (12) '__aux_maplist/3_dot+1'([[6, 8]], [_3644|_3646], [1, 2])
   Call: (13) dot([1, 2], [6, 8], _3644)
   Unify: (13) dot([1, 2], [6, 8], _3644)
   Call: (14) '__aux_maplist/4_product+0'([1, 2], [6, 8], _3668)
   Unify: (14) '__aux_maplist/4_product+0'([1, 2], [6, 8], [_3650|_3652])
   Call: (15) product(1, 6, _3650)
   Unify: (15) product(1, 6, _3650)
   Call: (16) _3650 is 1*6
   Exit: (16) 6 is 1*6
   Exit: (15) product(1, 6, 6)
   Call: (15) '__aux_maplist/4_product+0'([2], [8], _3652)
   Unify: (15) '__aux_maplist/4_product+0'([2], [8], [_3662|_3664])
   Call: (16) product(2, 8, _3662)
   Unify: (16) product(2, 8, _3662)
   Call: (17) _3662 is 2*8
   Exit: (17) 16 is 2*8
   Exit: (16) product(2, 8, 16)
   Call: (16) '__aux_maplist/4_product+0'([], [], _3664)
   Unify: (16) '__aux_maplist/4_product+0'([], [], [])
   Exit: (16) '__aux_maplist/4_product+0'([], [], [])
   Exit: (15) '__aux_maplist/4_product+0'([2], [8], [16])
   Exit: (14) '__aux_maplist/4_product+0'([1, 2], [6, 8], [6, 16])
   Call: (14) backward_compatibility:sumlist([6, 16], _3644)
   Unify: (14) backward_compatibility:sumlist([6, 16], _3644)
   Call: (15) lists:sum_list([6, 16], _3644)
   Unify: (15) lists:sum_list([6, 16], _3644)
   Exit: (15) lists:sum_list([6, 16], 22)
   Exit: (14) backward_compatibility:sumlist([6, 16], 22)
   Exit: (13) dot([1, 2], [6, 8], 22)
   Call: (13) '__aux_maplist/3_dot+1'([], _3646, [1, 2])
   Unify: (13) '__aux_maplist/3_dot+1'([], [], [1, 2])
   Exit: (13) '__aux_maplist/3_dot+1'([], [], [1, 2])
   Exit: (12) '__aux_maplist/3_dot+1'([[6, 8]], [22], [1, 2])
   Exit: (11) '__aux_maplist/3_dot+1'([[5, 7], [6, 8]], [19, 22], [1, 2])
   Exit: (10) mm_helper([[5, 7], [6, 8]], [1, 2], [19, 22])
   Call: (10) '__aux_maplist/3_mm_helper+1'([[3, 4]], _3598, [[5, 7], [6, 8]])
   Unify: (10) '__aux_maplist/3_mm_helper+1'([[3, 4]], [_3686|_3688], [[5, 7], [6, 8]])
   Call: (11) mm_helper([[5, 7], [6, 8]], [3, 4], _3686)
   Unify: (11) mm_helper([[5, 7], [6, 8]], [3, 4], _3686)
   Call: (12) '__aux_maplist/3_dot+1'([[5, 7], [6, 8]], _3686, [3, 4])
   Unify: (12) '__aux_maplist/3_dot+1'([[5, 7], [6, 8]], [_3692|_3694], [3, 4])
   Call: (13) dot([3, 4], [5, 7], _3692)
   Unify: (13) dot([3, 4], [5, 7], _3692)
   Call: (14) '__aux_maplist/4_product+0'([3, 4], [5, 7], _3716)
   Unify: (14) '__aux_maplist/4_product+0'([3, 4], [5, 7], [_3698|_3700])
   Call: (15) product(3, 5, _3698)
   Unify: (15) product(3, 5, _3698)
   Call: (16) _3698 is 3*5
   Exit: (16) 15 is 3*5
   Exit: (15) product(3, 5, 15)
   Call: (15) '__aux_maplist/4_product+0'([4], [7], _3700)
   Unify: (15) '__aux_maplist/4_product+0'([4], [7], [_3710|_3712])
   Call: (16) product(4, 7, _3710)
   Unify: (16) product(4, 7, _3710)
   Call: (17) _3710 is 4*7
   Exit: (17) 28 is 4*7
   Exit: (16) product(4, 7, 28)
   Call: (16) '__aux_maplist/4_product+0'([], [], _3712)
   Unify: (16) '__aux_maplist/4_product+0'([], [], [])
   Exit: (16) '__aux_maplist/4_product+0'([], [], [])
   Exit: (15) '__aux_maplist/4_product+0'([4], [7], [28])
   Exit: (14) '__aux_maplist/4_product+0'([3, 4], [5, 7], [15, 28])
   Call: (14) backward_compatibility:sumlist([15, 28], _3692)
   Unify: (14) backward_compatibility:sumlist([15, 28], _3692)
   Call: (15) lists:sum_list([15, 28], _3692)
   Unify: (15) lists:sum_list([15, 28], _3692)
   Exit: (15) lists:sum_list([15, 28], 43)
   Exit: (14) backward_compatibility:sumlist([15, 28], 43)
   Exit: (13) dot([3, 4], [5, 7], 43)
   Call: (13) '__aux_maplist/3_dot+1'([[6, 8]], _3694, [3, 4])
   Unify: (13) '__aux_maplist/3_dot+1'([[6, 8]], [_3734|_3736], [3, 4])
   Call: (14) dot([3, 4], [6, 8], _3734)
   Unify: (14) dot([3, 4], [6, 8], _3734)
   Call: (15) '__aux_maplist/4_product+0'([3, 4], [6, 8], _3758)
   Unify: (15) '__aux_maplist/4_product+0'([3, 4], [6, 8], [_3740|_3742])
   Call: (16) product(3, 6, _3740)
   Unify: (16) product(3, 6, _3740)
   Call: (17) _3740 is 3*6
   Exit: (17) 18 is 3*6
   Exit: (16) product(3, 6, 18)
   Call: (16) '__aux_maplist/4_product+0'([4], [8], _3742)
   Unify: (16) '__aux_maplist/4_product+0'([4], [8], [_3752|_3754])
   Call: (17) product(4, 8, _3752)
   Unify: (17) product(4, 8, _3752)
   Call: (18) _3752 is 4*8
   Exit: (18) 32 is 4*8
   Exit: (17) product(4, 8, 32)
   Call: (17) '__aux_maplist/4_product+0'([], [], _3754)
   Unify: (17) '__aux_maplist/4_product+0'([], [], [])
   Exit: (17) '__aux_maplist/4_product+0'([], [], [])
   Exit: (16) '__aux_maplist/4_product+0'([4], [8], [32])
   Exit: (15) '__aux_maplist/4_product+0'([3, 4], [6, 8], [18, 32])
   Call: (15) backward_compatibility:sumlist([18, 32], _3734)
   Unify: (15) backward_compatibility:sumlist([18, 32], _3734)
   Call: (16) lists:sum_list([18, 32], _3734)
   Unify: (16) lists:sum_list([18, 32], _3734)
   Exit: (16) lists:sum_list([18, 32], 50)
   Exit: (15) backward_compatibility:sumlist([18, 32], 50)
   Exit: (14) dot([3, 4], [6, 8], 50)
   Call: (14) '__aux_maplist/3_dot+1'([], _3736, [3, 4])
   Unify: (14) '__aux_maplist/3_dot+1'([], [], [3, 4])
   Exit: (14) '__aux_maplist/3_dot+1'([], [], [3, 4])
   Exit: (13) '__aux_maplist/3_dot+1'([[6, 8]], [50], [3, 4])
   Exit: (12) '__aux_maplist/3_dot+1'([[5, 7], [6, 8]], [43, 50], [3, 4])
   Exit: (11) mm_helper([[5, 7], [6, 8]], [3, 4], [43, 50])
   Call: (11) '__aux_maplist/3_mm_helper+1'([], _3688, [[5, 7], [6, 8]])
   Unify: (11) '__aux_maplist/3_mm_helper+1'([], [], [[5, 7], [6, 8]])
   Exit: (11) '__aux_maplist/3_mm_helper+1'([]

I think it's misleading to say that Prolog predicates return true or false. They can succeed or fail, but there are not even Prolog values for true and false, you cannot assign this return value to a variable, and you cannot place a predicate evaluation in the place of a true or false value in a call. I know you know this, I just worry that this wording could mislead. — Daniel Lyons, Apr 15 '19 at 15:57
@DanielLyons Thanks. As always feel free to edit either my questions or answers. It is all creative commons. — Guy Coder, Apr 15 '19 at 15:59
Thank you for the explanation, but did you figure out what does I1 represents in mm_helper predicate? — Bettosama, Apr 15 '19 at 17:06
@Bettosama See updated answer. Now you can peek and poke around with how it runs using the trace output. — Guy Coder, Apr 15 '19 at 18:20
@Bettosama If you ask any questions related to the the trace output you have to post them as a new question. StackOverflow is not a discussion site. — Guy Coder, Apr 15 '19 at 18:21

Matrix Multiplication Prolog

2 Answers2

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