Mercio’s Algorithm for Rational Tree Compare in Prolog

From	Mild Shock <janburse@fastmail.fm>
Newsgroups	comp.lang.prolog
Subject	Mercio’s Algorithm for Rational Tree Compare in Prolog
Date	2025-08-04 02:52 +0200
Message-ID	<106p08a$3b6se$1@solani.org> (permalink)

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Mercio’s Algorithm (2012) for Rational
Tree Compare is specified here mathematically.
It is based on computing truncations A' = (A_0,
A_1, etc..) of a rational tree A:

A < B ⟺ A′ <_lex B′

https://math.stackexchange.com/a/210730

Here is an implementation in Prolog.
First the truncation:

trunc(_, T, T) :- var(T), !.
trunc(0, T, F) :- !, functor(T, F, _).
trunc(N, T, S) :-
    M is N-1,
    T =.. [F|L],
    maplist(trunc(M), L, R),
    S =.. [F|R].

And then the iterative deepening:

mercio(N, X, Y, C) :-
    trunc(N, X, A),
    trunc(N, Y, B),
    compare(D, A, B),
    D \== (=), !, C = D.
mercio(N, X, Y, C) :-
    M is N + 1,
    mercio(M, X, Y, C).

The main entry first uses (==)/2 for a
terminating equality check and if the
rational trees are not equal, falls back
to the iterative deepening:

mercio(C, X, Y) :- X == Y, !, C = (=).
mercio(C, X, Y) :- mercio(0, X, Y, C).

I couldn’t find yet a triple that violates
transitivity. But I am also not much happy
with the code. Looks a little bit expensive
to create a truncation copy iteratively.

Provided there is really no counter example,
maybe we can do mit more smart and faster? It
might also stand the test of conservativity?

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Mercio’s Algorithm for Rational Tree Compare in Prolog Mild Shock <janburse@fastmail.fm> - 2025-08-04 02:52 +0200
  The Original Ganster (OG) of Gameification: IEEE 1044.1-1995 (Was: Mercio’s Algorithm for Rational Tree Compare in Prolog) Mild Shock <janburse@fastmail.fm> - 2025-08-04 13:48 +0200
    The Bitrot called Math Stack Exchange (Re: The Original Ganster (OG) of Gameification: IEEE 1044.1-1995) Mild Shock <janburse@fastmail.fm> - 2025-08-04 14:00 +0200
      I guess its back to Hopcroft and Karp.Re: The Bitrot called Math Stack Exchange) Mild Shock <janburse@fastmail.fm> - 2025-08-04 14:11 +0200
  Szpilrajn Theorem and Suzumura Consistency (Was: Mercio’s Algorithm for Rational Tree Compare in Prolog) Mild Shock <janburse@fastmail.fm> - 2025-08-06 01:50 +0200
    The good thing is we have at least Mercio’s Algorithm (Re: Szpilrajn Theorem and Suzumura Consistency) Mild Shock <janburse@fastmail.fm> - 2025-08-06 08:10 +0200
      Hopcroft and Karp’s is just Contraction (Was: The good thing is we have at least Mercio’s Algorithm) Mild Shock <janburse@fastmail.fm> - 2025-08-06 08:13 +0200
        Re: Hopcroft and Karp’s is just Contraction (Was: The good thing is we have at least Mercio’s Algorithm) Mild Shock <janburse@fastmail.fm> - 2025-08-06 08:23 +0200
      Should we use minimum_coa/3 ? (Was: The good thing is we have at least Mercio’s Algorithm) Mild Shock <janburse@fastmail.fm> - 2025-08-08 23:45 +0200
        Perfectly balanced , as all things should be! (Was: Should we use minimum_coa/3 ?) Mild Shock <janburse@fastmail.fm> - 2025-08-08 23:49 +0200
  Mercios decidability was already attested in 2012 (Was: Mercio’s Algorithm for Rational Tree Compare in Prolog) Mild Shock <janburse@fastmail.fm> - 2025-08-14 20:26 +0200
    Performance of Mercio’s Total Order (Was: Mercios decidability was already attested in 2012) Mild Shock <janburse@fastmail.fm> - 2025-08-15 23:49 +0200
      Fuzzy Testing is your Swiss Knife (Re: Performance of Mercio’s Total Order) Mild Shock <janburse@fastmail.fm> - 2025-08-15 23:55 +0200
        Yeah, we have another name! (Was: Fuzzy Testing is your Swiss Knife) Mild Shock <janburse@fastmail.fm> - 2025-08-16 12:38 +0200
        Monte Carlo sampling the frontier version (Was: Yeah, we have another name!) Mild Shock <janburse@fastmail.fm> - 2025-08-16 12:42 +0200
        AI most hated by formal verification (Was: Fuzzy Testing is your Swiss Knife) Mild Shock <janburse@fastmail.fm> - 2025-11-26 17:02 +0100
  An NPU could give 1000x more LIPS (Was: Mercio’s Algorithm for Rational Tree Compare in Prolog) Mild Shock <janburse@fastmail.fm> - 2025-11-27 14:19 +0100
    Neural Network based dif/2 respectively (#\=)/2 (Re: An NPU could give 1000x more LIPS) Mild Shock <janburse@fastmail.fm> - 2025-11-27 14:51 +0100
      Zeus: A Language for Expressing Algorithms in Hardware (Re: Neural Network based dif/2 respectively (#\=)/2) Mild Shock <janburse@fastmail.fm> - 2025-11-27 15:03 +0100
        Googles TPU muscle in 2017 [Prolog Community is Sleepy Joe] (Re: Zeus: A Language for Expressing Algorithms in Hardware) Mild Shock <janburse@fastmail.fm> - 2025-11-27 15:23 +0100
    100% serious Giga Logical Inferences per Second (GLIPS) (Was: An NPU could give 1000x more LIPS) Mild Shock <janburse@fastmail.fm> - 2025-11-28 14:50 +0100
      The tables have turned: GigaLIP on the Laptop? (Re: 100% serious Giga Logical Inferences per Second (GLIPS) Mild Shock <janburse@fastmail.fm> - 2025-11-28 15:12 +0100

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