Prolog Tribute to Hao Wang

[Mild Shock]

This code here doesn’t make much sense:

prove(L --> R):-
member(A => B,L),
del(A => B,L,NewL),!,

One can combine member/2 and del/3 into select/3. select/3
together with member/2 is part of the Prologue to Prolog:

*A Prologue for Prolog (working draft)*
https://www.complang.tuwien.ac.at/ulrich/iso-prolog/prologue

So if I further strip away using a two sided sequent,
I can implement Hoa Wangs implication fagment:

P1. Initial rule: if λ, ζ are strings of atomic
formulae, then λ -> ζ is a theorem if some atomic
formula occurs an both sides of the arrow.

P5a. Rule —> => If ζ, φ -> λ, ψ, ρ, then ζ -> λ, φ => ψ, ρ
P5b. Rule => -> If λ, ψ, ρ -> π and λ, ρ -> π, φ then λ, φ => ψ, ρ -> π

(Hao Wang. Toward Mechanical Mathematics. IBM
Journal of Research and Development 4:1 (1960), 15.)

as follows in 3 lines:

prove(L) :- select((A->B),L,R), !, prove([-A,B|R]).
prove(L) :- select(-(A->B),L,R), !, prove([A|R]), prove([-B|R]).
prove(L) :- select(-A,L,R), member(A,R), !.

Seems to work, I can prove Peirce Law:

?- prove([(((p->q)->p)->p)]).
true.
See also:

*Hao Wang on the formalisation of mathematics*
Lawrence C. Paulson 26 Jul 2023
https://lawrencecpaulson.github.io/2023/07/26/Wang.html

[Mild Shock]

Hi,

But its funny that people still work on UNSAT,
because its known that SAT is NP complete.

But don't worry, I sometimes do the same.
Imogen seems to chocke on SYJ202:

SYJ202+1.005.imo Provable. Time: 2.129
SYJ202+1.006.imo Provable. Time: 3.790
SYJ202+1.007.imo Provable. Time: 16.222
SYJ202+1.008.imo Provable. Time: 143.802

I assume its just the same problem linearly growing,
but the time is exponential or something.

BTW: Complexity for intuitionistic propositional logic
is even worse, its PSPACE complete. Here a recent

attempt featuring intuitRIL and SuperL

Implementing Intermediate Logics
https://iltp.de/ARQNL-2024/download/proceedings_preli/2_ARQNL_2024_paper_8.pdf

Have Fun!

Bye

Post by Mild Shock
prove(L --> R):-
    member(A => B,L),
    del(A => B,L,NewL),!,
One can combine member/2 and del/3 into select/3. select/3
*A Prologue for Prolog (working draft)*
https://www.complang.tuwien.ac.at/ulrich/iso-prolog/prologue
So if I further strip away using a two sided sequent,
P1. Initial rule: if λ, ζ are strings of atomic
formulae, then λ -> ζ is a theorem if some atomic
formula occurs an both sides of the arrow.
P5a. Rule —> =>    If ζ, φ -> λ, ψ, ρ, then ζ -> λ, φ => ψ, ρ
P5b. Rule => -> If λ, ψ, ρ -> π and λ, ρ -> π, φ then λ, φ => ψ, ρ -> π
(Hao Wang. Toward Mechanical Mathematics. IBM
Journal of Research and Development 4:1 (1960), 15.)
prove(L) :- select((A->B),L,R), !, prove([-A,B|R]).
prove(L) :- select(-(A->B),L,R), !, prove([A|R]), prove([-B|R]).
prove(L) :- select(-A,L,R), member(A,R), !.
?- prove([(((p->q)->p)->p)]).
true.
*Hao Wang on the formalisation of mathematics*
Lawrence C. Paulson 26 Jul 2023
https://lawrencecpaulson.github.io/2023/07/26/Wang.html