/
nested_horn.pro
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nested_horn.pro
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:-op(600,xfx,(<-)).
:-op(500,fy,(~)).
% prover for nested Horn + false + true - compiled from all except disjunction
holds(A):-A<-[].
%true<-_:-!.
A<-Vs:-memberchk(A,Vs),!.
_<-Vs:-memberchk(false,Vs),!.
(B<-As)<-Vs1:-!,append(As,Vs1,Vs2),B<-Vs2.
G<-Vs1:- % atomic(G), G not on Vs1
membtest(G,Vs1),
select((B<-As),Vs1,Vs2), % outer select loop
select(A,As,Bs), % inner select loop
holds_imp(A,B,Vs2), % A element of the body of B
!,
trimmed((B<-Bs),NewB), % trim empty bodies
G<-[NewB|Vs2].
holds_imp((D<-Cs),B,Vs):-!,(D<-Cs)<-[(B<-[D])|Vs].
%holds_imp(true,_B,_Vs):-!.
holds_imp(A,_B,Vs):-memberchk(A,Vs).
membtest(G,Vs):-memberchk((G<-_),Vs),!. % if not, we just fail
membtest(_,Vs):-memberchk((false<-_),Vs). % could still be infered from false
trimmed((B<-[]),R):-!,R=B.
trimmed(BBs,BBs).
nholds(F):-neg_expand(F,NF),holds(NF).
holds_classicaly(F):-
glivenko(F,G),
holds(G).
nholds_classicaly(F):-
glivenko(F,G),
nholds(G).
% abductive implicational IPC
subset_of(Xs,Ss):-
length(Xs,L),
between(0,L,K),
ksubset(K,Xs,Ss).
% subsets of K elements of a set with N elements
ksubset(0,_,[]).
ksubset(K,[X|Xs],[X|Rs]):-K>0,K1 is K-1,ksubset(K1,Xs,Rs).
ksubset(K,[_|Xs],Rs):-K>0,ksubset(K,Xs,Rs).
leaves_of(Form,Leaves):-
leaves_of(Form,Xs,[]),
sort(Xs,Leaves).
leaves_of(H<-Bs)-->!,leaves_of(H),map_leaves(Bs).
leaves_of(X)-->[X].
map_leaves([])-->[].
map_leaves([X|Xs])-->leaves_of(X),map_leaves(Xs).
leaves_of_all([])-->[].
leaves_of_all([X|Xs])-->leaves_of(X),leaves_of_all(Xs).
%% abductive inference of atomic formulas in a formula
%% that must hold for the formula to hold in IPC
needed_for(Prover,Formula,Hypos):-
% ex: Prover = (<-)
Formula=(H<-_),
leaves_of(Formula,Ls),
selectchk(H,Ls,Others),
subset_of(Others,Hypos),
call(Prover,Formula,Hypos).
% nested Horn clause formula generators
nested_horn(N,T):-gen_formula(N,gen_horn,T).
trimmed_horn(N,T):-gen_formula(N,gen_trimmed_horn,T).
gen_formula(N,TreeGen,T):-
length(Vs,N),
natpartitions(Vs),
call(TreeGen,T,Vs).
% set partition generator - counted by Bell numbers
% computes set partitions seen as distinct logic variables
% second arg has the unique variables
mpart_of([],[]).
mpart_of([U|Xs],[U|Us]):-
mcomplement_of(U,Xs,Rs),
mpart_of(Rs,Us).
% mimic computing the complement
% but just fuse logic variables
% representing equivalence classes
mcomplement_of(_,[],[]).
mcomplement_of(U,[X|Xs],NewZs):-
mcomplement_of(U,Xs,Zs),
mplace_element(U,X,Zs,NewZs).
mplace_element(U,U,Zs,Zs).
mplace_element(_,X,Zs,[X|Zs]).
% from set partitions, with 0..N marking distinct variables
natpartitions(Vs):-
mpart_of(Vs,Ns),
length(Ns,SL),
succ(L,SL),
numlist(0,L,Ns).
% sorted nested Horn clauses, with no repetitions
% OEIS A000108 Catalan 1,2,5,14,42,132,429,1430,4862,16796
gen_horn(Tree,Leaves):-gen_horn(Tree,Leaves,[]).
gen_horn(V)-->[V].
gen_horn((A<-[B|Bs]))-->
[A],
gen_horn(B),
gen_horns(Bs).
gen_horns([])-->[].
gen_horns([B|Bs])-->
gen_horn(B),
gen_horns(Bs).
% A105633: [1,2,4,9,22,57,154,429,1223,3550,10455,31160,93802,284789]
%% canonicalized Horn clauses
%% one repres. of each equiv. class
gen_trimmed_horn(Tree,Leaves):-
generate_trimmed_horn(Tree,Leaves,[]).
generate_trimmed_horn(V)-->[V].
generate_trimmed_horn((A<-[B|Bs]))-->
[A],
generate_trimmed_horn(B),
generate_trimmed_horns(B,Bs).
generate_trimmed_horns(_,[])-->[].
generate_trimmed_horns(B,[C|Bs])-->
generate_trimmed_horn(C),
{B@<C},
generate_trimmed_horns(C,Bs).
% transformers
% from/to implcational formulas
iform_horn(I,((H<-Cs)<-Bs)):-var(I),!,append(Cs,Bs,Ds),iform_horn(I,(H<-Ds)).
iform_horn((A->B),(H<-Bs)):-!,iform_horns((A->B),Bs,H).
iform_horn(H,H).
iform_horns((A->B),[HA|Bs],H):-!,iform_horn(A,HA),iform_horns(B,Bs,H).
iform_horns(H,[],H).
neg_expand(~H<-Bs,R):-!,maplist(neg_expand,[H|Bs],Cs),R = false<-Cs.
neg_expand(~(H<-Bs),R):-!,maplist(neg_expand,[H|Bs],[NH|NBs]),R = false<-(NH<-NBs).
neg_expand(H<-Bs,R):-!,maplist(neg_expand,[H|Bs],[NH|NBs]),R=(NH<-NBs).
neg_expand(~X,R):-!,neg_expand(X,NX),R=(false<-[NX]).
neg_expand(A,A).
% turns formula into equivalent to being classically provable
%glivenko(H<-Bs,R):-!,maplist(glivenko,[H|Bs],[G|Cs]),R = (G<-Cs).
%glivenko(false,R):-!,R=false.
glivenko(X,R):-R=(false<-[false<-[X]]).
% natural number -> hereditatily finite set
% with urelments smaller than M
nat2uhfs(M,U,U):-U<M.
nat2uhfs(M,N,Hs):-N>=M,
nat2nats(N,Ns),
maplist(nat2uhfs(M),Ns,Hs).
% hereditatily finite set -> natural number
uhfs2nat(U,U):-atomic(U).
uhfs2nat(Ns,N):-compound(Ns),
maplist(uhfs2nat,Ns,Ms),
nats2nat(Ms,N).
% natural number to nested Horn clause
nat2uhorn(M,U,U):-U<M.
nat2uhorn(M,N,HBs):-N>=M,
nat2nats(N,Ns),
maplist(nat2uhorn(M),Ns,Hs),
hornify(Hs,HBs).
hornify([X],(false<-[X])):-!.
hornify([X,Y|Xs],(X<-[Y|Xs])).
% nested Horn clause -> natural number
uhorn2nat(U,U):-atomic(U).
uhorn2nat(HBs,N):-compound(HBs),
hornify(Hs,HBs),
maplist(uhorn2nat,Hs,Ns),
nats2nat(Ns,N).
% hereditarily finite sets generators
nat2nats(0,[]).
nat2nats(N,Ns):-N>0,findall(I,nat2bit(N,I),Ns).
nat2bit(N,I):-
M is msb(N),
between(0,M,I),
1 is getbit(N,I).
nat2hfs(0,[]).
nat2hfs(N,Hs):-N>0,
nat2nats(N,Ns),
maplist(nat2hfs,Ns,Hs).
nats2nat([],0).
nats2nat([I|Ns],N):-
E is 1<<I,
nats2nat(Ns,N1),
N is N1+E.
hfs2nat(Ns,N):-
maplist(hfs2nat,Ns,Ms),
nats2nat(Ms,N).
% canonical equiprovable formula of depth at most 3
to_hcan(T,C):-
reset_gensym(v),
hcan(T,C).
nvar(V):-gensym(v,V).
simple(X):-atomic(X),!.
simple(X):-var(X).
hcan(A,R):-simple(A),!,R=A.
hcan((H<-Bs),(H<-Ds)):-hcans(Bs,Cs,Ds,Cs). %,append(Cs,Ds,Rs).
hcans([],[])-->[].
hcans([B|Bs],[C|Cs])-->
heq(B,C),
hcans(Bs,Cs).
heq(A,A)-->{simple(A)},!.
heq((H<-Bs),N)-->
{nvar(N)},
hcans(Bs,Cs),
[(N<-(H<-Cs)),(H<-[N|Cs])].
% counts nb. of solutions of Goal
sols_count(Goal, Times) :-
Counter = counter(0),
( Goal,
inc(Counter),
fail
; arg(1, Counter, Times)
).
counts_for(M,Generator,Ks):-
findall(K,
(between(0,M,L),sols_count(call(Generator,L,_),K)),
Ks).
ppp(X):-portray_clause(X).
inc(Ctr):-arg(1,Ctr,X),succ(X,Y),nb_setarg(1,Ctr,Y).
go1:-
Ctr=proven(0),
NCtr=unprovable(0),
M=3000,
( between(0,M,N),
N1 is floor(log(1+N)),
between(0,N1,U),
nat2uhorn(U,N,HBs),
uhorn2nat(HBs,NN),
assertion(N=:=NN),
( holds(HBs)->
R=proven,inc(Ctr)
; R=unprovable,inc(NCtr)
),
0=:=random(M),0=:=random(U),ppp((N=NN):'------>':HBs:R),
fail
; Ctr=proven(K),
NCtr=unprovable(NK),
D is K/(K+NK),
writeln([Ctr,NCtr,density=D])
).
go:-
trimmed_horn(10,T),
holds_classicaly(T),not(holds(T)),
once(needed_for((<-),T,Ls)),
Ls=[_,_|_],
ppp(T),
ppp(Ls),nl,fail.
go2:-
T=f<-[a,g<-[b<-[c,d<-[e<-[i,j],f<-[k]]],m],r<-[p,x]],
to_hcan(T,C),
ppp(T),
ppp(C).
err:-
trimmed_horn(4,T),
to_hcan(T,C),
(holds(T)->Orig='YES';Orig='NO'),
(holds(C)->Can='YES';Can='NO'),
Orig\==Can,
ppp((orig(Orig):-T)),
ppp((can(Can):-C)),nl,
fail.