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Sections 9.2-3
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flach committed Mar 27, 2017
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% Prolog programs from Section 9.2 of the book %
% SIMPLY LOGICAL: Intelligent reasoning by example %
% (c) Peter A. Flach/John Wiley & Sons, 1994. %
% %
% Predicates: induce_rlgg/2 %
% rlgg/4 %
% %
% NB. This file needs predicates defined in %
% the file 'library'. %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%:-consult(library).


%%% 9.2 Bottom-up induction %%%

induce_rlgg(Exs,Clauses):-
writeln('%%% Bottom-up induction %%%'),
writeln('Examples'),
writelns(Exs),
pos_neg(Exs,Poss,Negs),
bg_model(BG),
append(Poss,BG,Model),
induce_rlgg(Poss,Negs,Model,Clauses).

induce_rlgg(Poss,Negs,Model,Clauses):-
covering(Poss,Negs,Model,[],Clauses).

% split positive and negative examples
pos_neg([],[],[]).
pos_neg([+E|Exs],[E|Poss],Negs):-
pos_neg(Exs,Poss,Negs).
pos_neg([-E|Exs],Poss,[E|Negs]):-
pos_neg(Exs,Poss,Negs).

% covering algorithm
covering(Poss,Negs,Model,H0,H):-
construct_hypothesis(Poss,Negs,Model,Hyp),!,
remove_pos(Poss,Model,Hyp,NewPoss),
nl,writeln('<return> to continue'),
readln(_),nl,nl,
covering(NewPoss,Negs,Model,[Hyp|H0],H).
covering(P,_N,_M,H0,H):-
append(H0,P,H). % add uncovered examples to hypothesis

% remove covered positive examples
remove_pos([],_M,_H,[]).
remove_pos([P|Ps],Model,Hyp,NewP):-
covers_ex(Hyp,P,Model),!,
write('Covered example: '),write(P),nl,
remove_pos(Ps,Model,Hyp,NewP).
remove_pos([P|Ps],Model,Hyp,[P|NewP]):-
remove_pos(Ps,Model,Hyp,NewP).


% extensional coverage, relative to a ground model
covers_ex((Head:-Body),Example,Model):-
try((Head=Example,forall(element(L,Body),element(L,Model)))).

% construct a clause by means of RLGG
construct_hypothesis([E1,E2|_Es],Negs,Model,Clause):-
write('RLGG of '),write(E1),write(' and '),write(E2),write(' is'),
rlgg(E1,E2,Model,Cl),
reduce(Cl,Negs,Model,Clause),!,
nl,portray_clause(Clause),nl.
construct_hypothesis([_E1,E2|Es],Negs,Model,Clause):-
write(' too general'),nl,
construct_hypothesis([E2|Es],Negs,Model,Clause).


% rlgg(E1,E2,M,C) <- C is RLGG of E1 and E2 relative to M
rlgg(E1,E2,M,(H:-B)):-
anti_unify(E1,E2,H,[],S10,[],S20),
varsin(H,V), % determine variables in head of clause
rlgg_bodies(M,M,[],B,S10,_S1,S20,_S2,V).

rlgg_bodies([],_B2,B,B,S1,S1,S2,S2,_V).
rlgg_bodies([L|B1],B2,B0,B,S10,S1,S20,S2,V):-
rlgg_literal(L,B2,B0,B00,S10,S11,S20,S21,V),
rlgg_bodies(B1,B2,B00,B,S11,S1,S21,S2,V).

rlgg_literal(_L1,[],B,B,S1,S1,S2,S2,_V).
rlgg_literal(L1,[L2|B2],B0,B,S10,S1,S20,S2,V):-
same_predicate(L1,L2),
anti_unify(L1,L2,L,S10,S11,S20,S21),
varsin(L,Vars),var_proper_subset(Vars,V), % no new variables in literal
!,rlgg_literal(L1,B2,[L|B0],B,S11,S1,S21,S2,V).
rlgg_literal(L1,[_L2|B2],B0,B,S10,S1,S20,S2,V):-
rlgg_literal(L1,B2,B0,B,S10,S1,S20,S2,V).


:-op(600,xfx,'<-').

anti_unify(Term1,Term2,Term):-
anti_unify(Term1,Term2,Term,[],_S1,[],_S2).

anti_unify(Term1,Term2,Term1,S1,S1,S2,S2):-
Term1 == Term2,!.
anti_unify(Term1,Term2,V,S1,S1,S2,S2):-
subs_lookup(S1,S2,Term1,Term2,V),!.
anti_unify(Term1,Term2,Term,S10,S1,S20,S2):-
nonvar(Term1),nonvar(Term2),
functor(Term1,F,N),functor(Term2,F,N),!,
functor(Term,F,N),
anti_unify_args(N,Term1,Term2,Term,S10,S1,S20,S2).
anti_unify(Term1,Term2,V,S10,[Term1<-V|S10],S20,[Term2<-V|S20]).

anti_unify_args(0,_Term1,_Term2,_Term,S1,S1,S2,S2).
anti_unify_args(N,Term1,Term2,Term,S10,S1,S20,S2):-
N>0,N1 is N-1,
arg(N,Term1,Arg1),
arg(N,Term2,Arg2),
arg(N,Term,Arg),
anti_unify(Arg1,Arg2,Arg,S10,S11,S20,S21),
anti_unify_args(N1,Term1,Term2,Term,S11,S1,S21,S2).

subs_lookup([T1<-V|_Subs1],[T2<-V|_Subs2],Term1,Term2,V):-
T1 == Term1,
T2 == Term2,!.
subs_lookup([_S1|Subs1],[_S2|Subs2],Term1,Term2,V):-
subs_lookup(Subs1,Subs2,Term1,Term2,V).


% remove redundant literals
reduce((H:-B0),Negs,M,(H:-B)):-
setof0(L,(element(L,B0),not(var_element(L,M))),B1),
reduce_negs(H,B1,[],B,Negs,M).

% reduce_negs(H,B1,B0,B,N,M) <- B is a subsequence of B1
% such that H:-B does not
% cover elements of N
reduce_negs(H,[_L|B0],In,B,Negs,M):-
append(In,B0,Body),
not(covers_neg((H:-Body),Negs,M,_N)),!,
reduce_negs(H,B0,In,B,Negs,M).
reduce_negs(H,[L|B0],In,B,Negs,M):-
reduce_negs(H,B0,[L|In],B,Negs,M).
reduce_negs(H,[],Body,Body,Negs,M):-
not(covers_neg((H:-Body),Negs,M,_N)).

covers_neg(Clause,Negs,Model,N):-
element(N,Negs),
covers_ex(Clause,N,Model).


%%% From library %%%

%%% Lists and sets

% element(X,Ys) <- X is an element of the list Ys
element(X,[X|_Ys]).
element(X,[_Y|Ys]):-
element(X,Ys).

var_element(X,[Y|_Ys]):-
X == Y. % syntactic identity
var_element(X,[_Y|Ys]):-
var_element(X,Ys).

var_remove_one(X,[Y|Ys],Ys):-
X == Y. % syntactic identity
var_remove_one(X,[Y|Ys],[Y|Zs]):-
var_remove_one(X,Ys,Zs).

var_proper_subset([],Ys):-
Ys \= [].
var_proper_subset([X|Xs],Ys):-
var_remove_one(X,Ys,Zs),
var_proper_subset(Xs,Zs).


% try(Goal) <- Goal succeeds, but variables are not instantiated
try(Goal):-
not(not(Goal)).


%%% Various.

% variant of setof/3 which succeeds with the empty list
% if no solutions can be found
setof0(X,G,L):-
setof(X,G,L),!.
setof0(_X,_G,[]).

% same_predicate(L1,L2) <- literals L1 and L2 have
% the same predicate and arity
same_predicate(L1,L2):-
functor(L1,P,N),functor(L2,P,N).

varsin(Term,Vars):-
varsin(Term,[],V),
sort(V,Vars).

varsin(V,Vars,[V|Vars]):-
var(V),!.
varsin(Term,V0,V):-
functor(Term,_,N),
varsin_args(N,Term,V0,V).

varsin_args(0,_,Vars,Vars).
varsin_args(N,Term,V0,V):-
N>0, N1 is N-1,
arg(N,Term,ArgN),
varsin(ArgN,V0,V1),
varsin_args(N1,Term,V1,V).


%%% For lectures

writelns(X):-
( X=[] -> nl
; X=[H|T] -> writelns(H),writelns(T)
; otherwise -> write(X),nl
).


%%% Queries %%%

%%%%%%%%%%%%%%%%%% element/2 %%%%%%%%%%%%%%%%%%%%%%%%

% bg_model([]).

query1(Clauses):-
induce_rlgg([+element(b,[b]),
+element(2,[2,3]),
+element(3,[1,2,3]),
+element(b,[a,b]),
+element(3,[2,3]),
+element(3,[3]),
-element(3,[a,b]),
-element(a,[])
],Clauses).

%%%%%%%%%%%%%%%%%% append/3 %%%%%%%%%%%%%%%%%%%%%%%

% bg_model([]).

query2(Clauses):-
induce_rlgg([+append([1,2],[3,4],[1,2,3,4]),
+append([a],[],[a]),
+append([],[],[]),
+append([],[1,2,3],[1,2,3]),
+append([2],[3,4],[2,3,4]),
+append([],[3,4],[3,4]),
-append([a],[b],[b]),
-append([c],[b],[c,a]),
-append([1,2],[],[1,3])
],Clauses).

%%%%%%%%%%%%%%%%%% num/2 %%%%%%%%%%%%%%%%%%%%%%%

bg_model([num(1,one),
num(2,two),
num(3,three),
num(4,four),
num(5,five)
]).

query3(Clauses):-
induce_rlgg([+listnum([],[]),
+listnum([2,three,4],[two,3,four]),
+listnum([4],[four]),
+listnum([three,4],[3,four]),
+listnum([two],[2]),
-listnum([1,4],[1,four]),
-listnum([2,three,4],[two]),
-listnum([five],[5,5])
],Clauses).

%%%%%%%%%%%%%%%%%% names/2 %%%%%%%%%%%%%%%%%%%%%%%

/*
bg_model([person(mick,jagger),
person(david,bowie),
person(tina,turner),
person(johann,sebastian),
person(ludwig,van)
]).
*/

query4(Clauses):-
induce_rlgg([+names([],[]),
+names([david,turner,johann],[p(david,bowie),p(tina,turner),p(johann,sebastian)]),
+names([johann],[p(johann,sebastian)]),
+names([turner,johann],[p(tina,turner),p(johann,sebastian)]),
+names([bowie],[p(david,bowie)]),
-names([mick,johann],[p(mick,mick),p(johann,sebastian)]),
-names([david,turner,johann],[p(david,bowie)]),
-names([van],[p(ludwig,van),p(ludwig,van)])
],Clauses).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





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