Generating Prolog source code. 
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 Generating Prolog source code.

From time to time I want to generate a range of versions of an algorithm,
in order to benchmark them.

In a great many cases, composition from higher order fragments followed by
partial execution does the trick nicely.

However, some of the things I want to try involve
    - variable numbers of clauses
    - variable numbers of conjuncts/disjuncts/if-then-parts
    - variable numbers of arguments in goals
or the like.

For example, I wanted to try out radix sort with various radices.
If you want to do a radix-K sort, you need a distribution predicate
with two input arguments and K accumulator pairs, and it needs K clauses.

Here is radix sort for K=4, from which you may be able to see the pattern.
(To follow Z, U, D, T, think en Francais; too many digits have the same
initial in English.)

sort2(Input, Bits, Output) :-
    sort2(Input, Bits, Output, 0).

sort2(List0, Bits, List, Offset) :-
    (   Offset < Bits ->
        radb2(List0, Offset, List1),
        Offset1 is Offset + 2,
        sort2(List1, Bits, List, Offset1)
    ;   Offset >= Bits,
        List = List0
    ).

radb2(Input, Offset, Output) :-
    radb2(Input, Offset, Output, Z, Z, U, U, D, D, []).

radb2([], _, Z, Z, U, U, D, D, T, T).
radb2([X|Xs], Offset, Z0, Z, U0, U, D0, D, T0, T) :-
    Digit is (X >> Offset) /\ 3,
    radb2(Digit, Offset, Z0, Z, U0, U, D0, D, T0, T, X, Xs).

radb2(0, Offset, [X|Z0], Z, U0, U, D0, D, T0, T, X, Xs) :-
    radb2(Xs, Offset, Z0, Z, U0, U, D0, D, T0, T).
radb2(1, Offset, Z0, Z, [X|U0], U, D0, D, T0, T, X, Xs) :-
    radb2(Xs, Offset, Z0, Z, U0, U, D0, D, T0, T).
radb2(2, Offset, Z0, Z, U0, U, [X|D0], D, T0, T) :-
    radb2(Xs, Offset, Z0, Z, U0, U, D0, D, T0, T).
radb2(3, Offset, Z0, Z, U0, U, D0, D, [X|T0], T) :-
    radb2(Xs, Offset, Z0, Z, U0, U, D0, D, T0, T).

To generate versions of this, I have a fairly small (and incredibly ugly)
`predicate' called expand/2.  Basically:
        F(X1,...,Xm)`G(Y1,...,Yn)
expands to FG(X1',...,Xm',Y1',...,Yn')
where Z' is the expansion of Z;
         [...,for(Gen,[T1,...,Tn],...]
or      F(...,for(Gen,[T1,...,Tn],...)
lets you generate a sequence of list elements or compound term arguments.
        for(Gen,T,S,E)
generates
        T1 S T2 S ... S Tn S E
where T1 ... Tn are the expansions of T.

For example,
        (radb`K`''([], Offset, for(memb(Ys, Y), [Y,Y])))
turns into
        radb2([], Offset, Y0,Y0, Y1,Y1, Y2,Y2, Y3,Y3)

%   memb(Xs, X)         if X is an element of Xs
%   memb(Xs, X, Ys, Y)  if X, Y are elements of Xs, Ys in the same position.
%   inx0(I, Xs, X)      if X is element I of Xs (zero-origin)
%   inx0(I, Xs,X, Ys,Y) if X, Y are elements of Xs, Ys at position I.

mk_sort(K) :-
    integer(K), K > 0,
    N is 1 << K,
    M is N - 1,
    length(Xs, N),
    length(Ys, N),
    [_|Zs] = Xs,

    numberVars([Input,Bits,Output,List,List0,List1,
                Offset,Offset1,Digit,A,As,Xs,Ys], 0, T1),

    expand([

        (sort`K`''(Input, Bits, Output) :-
            sort`K`''(Input, Bits, Output, 0)),

        (sort`K`''(List0, Bits, List, Offset) :-
            (   Offset < Bits ->
                radb`K`''(List0, Offset, List1),
                Offset1 is Offset + K,
                sort`K`''(List1, Bits, List, Offset1)
            ;   Offset >= Bits,
                List = List0
        )),

        (radb`K`''(Input, Offset, Output) :-
            radb`K`''(Input, Offset, Output, for(memb(Zs, Z), [Z, Z]), [])),

        (radb`K`''([], Offset, for(memb(Ys, Y), [Y,Y]))),
        (radb`K`''([A|As], Offset, for(memb(Xs,X, Ys,Y), [X,Y])) :-
            Digit is (A >> Offset) /\ M,
            radb`K`''(Digit, Offset, for(memb(Xs,X, Ys,Y), [X,Y]), A, As)),

        for(inx0(I, Xs, _), [
            (radb`K`''(I, Offset,
                for((inx0(J,Xs,X, Ys,Y), (J =:= I -> Q = [A|X] ; Q = X)),
                    [Q,Y]), A, As) :-
                radb`K`''(As, Offset, for(memb(Xs,X, Ys,Y), [X,Y])) )
        ])
    ], Clauses),

    numberVars(Clauses, T1, _),
    !,
    (   memb(Clauses, Clause),
        portraycl(Clause), put(46), nl, nl,
        fail
    ;   true
    ).

In developing this I ran into some problems:  would you believe a Prolog
system where the predicate for getting the name of an atom FAILS when the
atom in question is ''?  (Jeff, can I be the first to notice this?)

However, I have to say that it is *much* easier to get this right than
direct Prolog code for generating the clauses.  For example, to generate
a version of map_integer_list/2+K we'd do

mk_map_integer_list(K, Clauses) :-
    length(Xs, K),
    length(Xss, K),
    numbervars([Pred,N,N0,N1,Xss,Xs], 0, _),

    expand([
        (map_integer_list(Pred, N, for(memb(Xss,As), [As])) :-
            ( integer(N) -> N >= 0 ; var(N) ),
            map_integer_list_(for(memb(Xss,As), [As]), 0, N, Pred)),

        (map_integer_list_(for(memb(Xs,A), [[]]), N, N, Pred) :- !),
        (map_integer_list_(for(memb(Xs,A, Xss,As), [[A|As]]),
                           N0, N, Pred) :-
            ( var(N) -> true ; N0 < N ),
            N1 is 1 + N0,
            call(Pred, N1, for(memb(Xs,A), [A])),
            map_integer_list_(for(memb(Xss,As), [As]), N1, N, Pred))
      ], Clauses).

Efficiency is not a concern, because this takes place at program generation
time, not compile time, not run time.  What I principally care about here is
clarity.  Something like Goedel's meta-level stuff would be more elegant, I
suppose, but it is even bulkier than the Prolog code I'm replacing.

Does anyone have a better notation for this kind of thing that I can switch
to?  I would be very happy to junk my expand/2 implementation.

--
"conventional orthography is ... a near optimal system for the
 lexical representation of English words." Chomsky & Halle, S.P.E.
Richard A. O'Keefe; http://www.*-*-*.com/ ~ok; RMIT Comp.Sci.



Mon, 13 Jul 1998 03:00:00 GMT  
 Generating Prolog source code.

Quote:
>In a great many cases, composition from higher order fragments followed by
>partial execution does the trick nicely.

>However, some of the things I want to try involve
>    - variable numbers of clauses
>    - variable numbers of conjuncts/disjuncts/if-then-parts
>    - variable numbers of arguments in goals
>or the like.

It seems to me that these can all fit in the framework of higher order
code and partial evaluation.  You just need a sufficiently good
partial evaluator.

For example, variable numbers of arguments in goals can be handled
by passing a list; if the partial evaluator notices that the list
is always the same length, then it can use a separate argument
for each list element.  

If you need variable numbers of clauses, write the code recursively, or
using a higher-order predicate disj/1 that takes a list of goals and
evaluates their disjunction.

        disj([X|Xs]) :- call(X) ; disj(Xs).

If the partial evaluator notices that the argument to disj/1 is
always of the same length, then it can fully unfold such calls.

Now, if your next question is "where can I get such a partial evaluator?",
I'm afraid I don't have a good answer for you.

Quote:
>    integer(K), K > 0,
>    N is 1 << K,
>    M is N - 1,
>    length(Xs, N),
>    length(Ys, N),
>    [_|Zs] = Xs,
[...]
>        (radb`K`''(Input, Offset, Output) :-
>            radb`K`''(Input, Offset, Output, for(memb(Zs, Z), [Z, Z]), [])),

You could write this instead as

        radb(K, Input, Offset, Output) :-
                N is 1 << K,
                length([_|Zs], N),
                for(List, memb(Zs, Z), [Z, Z]),
                radb(Input, Offset, Output, List, []).

and rely on your partial evaluator to optimize this in the case
when K is known at compile time.

--
Fergus Henderson                WWW: http://www.cs.mu.oz.au/~fjh



Fri, 17 Jul 1998 03:00:00 GMT  
 Generating Prolog source code.

Quote:


>>In a great many cases, composition from higher order fragments followed by
>>partial execution does the trick nicely.

>>However, some of the things I want to try involve
>>    - variable numbers of clauses
>>    - variable numbers of conjuncts/disjuncts/if-then-parts
>>    - variable numbers of arguments in goals
>>or the like.
>It seems to me that these can all fit in the framework of higher order
>code and partial evaluation.  You just need a sufficiently good
>partial evaluator.

My question is not "how do I get from a specification to usable Prolog
code".  My question is "what is a good notation to write the original
specification in".

Quote:
>For example, variable numbers of arguments in goals can be handled
>by passing a list; if the partial evaluator notices that the list
>is always the same length, then it can use a separate argument
>for each list element.  

I am in complete agreement with you about that; for several years I've
been calling the transformation "argument N is always passed instantiated
with the same principal functor -> replace argument N by the arguments of
the instance" 'argument unpacking' or 'argument unwrapping'; if you
unwrap p([a,b]) three times the three functors ./2 ./2 []/0 disappear and
you get p(a,b).  Fine.

That's not the problem.  SPECIFYING THE LIST is the problem.

Quote:
>Now, if your next question is "where can I get such a partial evaluator?",
>I'm afraid I don't have a good answer for you.

That was never my question.  I am happy to write any partial executor I want.
My question was and remains solely concerned with NOTATION for the input to
this partial executor.

Quote:
>>    integer(K), K > 0,
>>    N is 1 << K,
>>    M is N - 1,
>>    length(Xs, N),
>>    length(Ys, N),
>>    [_|Zs] = Xs,
>[...]
>>        (radb`K`''(Input, Offset, Output) :-
>>            radb`K`''(Input, Offset, Output, for(memb(Zs, Z), [Z, Z]), [])),
>You could write this instead as
>        radb(K, Input, Offset, Output) :-
>            N is 1 << K,
>            length([_|Zs], N),
>            for(List, memb(Zs, Z), [Z, Z]),
>            radb(Input, Offset, Output, List, []).
>and rely on your partial evaluator to optimize this in the case
>when K is known at compile time.

I am sorry, but that completely misses the point.  Twice, in fact.
(a) I want control over the name of the resulting predicate.
    I want it to be called radb1 or radb2 or whatever; I do NOT want
    the notation taking that bit out of my hands.

    However, I would be willing to put up with that.  The problem is

(b) It is precisely the for(_,_) notation that I am most unhappy with.
    By the way, passing it as a list either will not work or will not
    work easily.  In this very example I wanted to say
        Output, {X<i>, X<i>, for i = 2..n} []
    in one place, but
        {X0<i>, X<i>, for i = 1..n}
    in another place.  The lists, in short, do not always line up, so
    that the unwrapping transformation can't get off the ground.
    To make that work, I would have to use lists _everywhere_, which
    would leave me pretty much where I started.

    In strict point of fact, I have seriously considered writing
        radb`K:[Input, Offset, Output]
    instead of
        radb`K`''(Input, Offset, Output)

I also note that Fergus seems to have elided the bit where I grounded all
the object-level variables.  I assure you all that that bit was not of
minor significance.  I have a very strong preference for amalgamated
object/meta level (as in the traditional Prolog-in-Prolog interpreter)
but found for this kind of stuff it really did NOT pay.  I have also given
consideration to writing

        (radb`K`''(?input, ?offset, ?output) :-
            ...)

and may switch to that, but will still need to cons up ground terms to
stand for the variable lists.

So, can anyone help with my actual question:  what's a good NOTATION for
this, that can handle the fact that the variable parts of a call to
P/(N+K) are not always in the same place, so that unwrapping doesn't always
work?

--
"conventional orthography is ... a near optimal system for the
 lexical representation of English words." Chomsky & Halle, S.P.E.
Richard A. O'Keefe; http://www.cs.rmit.edu.au/~ok; RMIT Comp.Sci.



Fri, 17 Jul 1998 03:00:00 GMT  
 
 [ 3 post ] 

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