% ----------------------------------------------------------------------
% $Id: acfsfgs.red,v 1.7 2003/05/19 10:38:31 dolzmann Exp $
% ----------------------------------------------------------------------
% Copyright (c) 1995-1999 Andreas Dolzmann and Thomas Sturm
% ----------------------------------------------------------------------
% $Log: acfsfgs.red,v $
% Revision 1.7 2003/05/19 10:38:31 dolzmann
% The Groebner Simplifier now uses the gb package.
%
% Revision 1.6 1999/04/13 13:05:33 sturm
% Minor corrections in comments.
%
% Revision 1.5 1999/04/12 09:25:58 sturm
% Updated comments for exported procedures.
%
% Revision 1.4 1999/03/23 08:14:18 dolzmann
% Changed copyright information.
% Added fluids for the rcsid of the file and for the copyright information.
%
% Revision 1.3 1997/10/02 13:49:49 dolzmann
% In procedure acfsf_qe1: Remove atomic formulas containing bound variables
% from the theory.
%
% Revision 1.2 1997/08/24 16:17:41 sturm
% Call cl_sitheo instead of acfsf_gssimpltheo.
% Added service rl_surep with black box rl_multsurep.
% Added service rl_siaddatl.
%
% Revision 1.1 1997/08/22 17:30:39 sturm
% Created an acfsf context based on ofsf.
%
% ----------------------------------------------------------------------
lisp <<
fluid '(acfsf_gs_rcsid!* acfsf_gs_copyright!*);
acfsf_gs_rcsid!* :=
"$Id: acfsfgs.red,v 1.7 2003/05/19 10:38:31 dolzmann Exp $";
acfsf_gs_copyright!* := "Copyright (c) 1995-1999 A. Dolzmann and T. Sturm"
>>;
module acfsfgs;
% Algebraically closed field standard form Groebner simplifier.
% Submodule of [acfsf].
%DS
% <CIMPL> ::= (<GP>, <PROD1>, <PROD2>, <OTHER>)
% <GP> ::= ((<GB> . <PROD>) . <OTHER>)
% <GB> ::= (<SF>,...)
% <PROD> ::= <SF>
% <PROD1> ::= <SF>
% <PROD2> ::= <SF>
% <OTHER> ::= (<ATOMIC_FORMULA>,...)
procedure acfsf_gsc(f,atl);
% Algebraically closed field Groebner simplification via
% conjunctive normal form. [f] is a formula; [atl] is a theory.
% Returns [inconsistent] or a formula equivalent to [f]. The
% returned formula is somehow simpler than [f]. This procedure
% temporarily turns off the switches [groebopt] and [rlsiexpla].
% Accesses the switches [rlverbose], [rlgsvb], [rlgsbnf],
% [rlgsrad], [rlgssub], [rlgsred], [rlgsprod], [rlgserf],
% [rlgsutord], and the fluid [rlradmemv!*].
begin scalar w,svrlgsvb;
svrlgsvb := !*rlgsvb;
if !*rlverbose and !*rlgsvb then on1 'rlgsvb else off1 'rlgsvb;
w := acfsf_gsc1(f,atl);
onoff('rlgsvb,svrlgsvb);
return w
end;
procedure acfsf_gsc1(f,atl);
% Algebraically closed field Groebner simplification via
% conjunctive normal form subroutine. [f] is a formula; [atl] is a
% theory. Returns [inconsistent] or a formula equivalent to [f].
% The returned formula is somehow simpler than [f]. This procedure
% temporarily turns off the switches [groebopt] and [rlsiexpla].
% Accesses the switches [rlgsvb], [rlgsbnf], [rlgsrad], [rlgssub],
% [rlgsred], [rlgsprod], [rlgserf], [rlgsutord], and the fluid
% [rlradmemv!*].
begin scalar phi,!*rlsiexpla; % Hack, but otherwise phi is not a bnf!
if !*rlgsbnf then <<
if !*rlgsvb then ioto_prin2 "[CNF";
phi := cl_simpl(cl_cnf cl_nnf f,atl,-1);
if !*rlgsvb then ioto_prin2 "] "
>> else
phi := cl_simpl(f,atl,-1);
if phi eq 'inctheo then return 'inctheo;
if rl_tvalp phi then
return phi;
phi := acfsf_gssimplify0(phi,atl);
if phi eq 'inctheo then return 'inctheo;
return cl_simpl(phi,atl,-1)
end;
procedure acfsf_gsd(f,atl);
% Algebraically closed field Groebner simplification via
% disjunctive normal form. [f] is a formula; [atl] is a theory.
% Returns [inconsistent] or a formula equivalent to [f]. The
% returned formula is somehow simpler than [f]. This procedure
% temporarily turns off the switches [groebopt] and [rlsiexpla].
% Accesses the switches [rlverbose], [rlgsvb], [rlgsbnf],
% [rlgsrad], [rlgssub], [rlgsred], [rlgsprod], [rlgserf],
% [rlgsutord], and the fluid [rlradmemv!*].
begin scalar w,svrlgsvb;
svrlgsvb := !*rlgsvb;
if !*rlverbose and !*rlgsvb then on1 'rlgsvb else off1 'rlgsvb;
w := acfsf_gsd1(f,atl);
onoff('rlgsvb,svrlgsvb);
return w
end;
procedure acfsf_gsd1(f,atl);
% Algebraically closed field Groebner simplification via
% disjunctive normal form subroutine. [f] is a formula; [atl] is a
% theory. Returns [inconsistent] or a formula equivalent to [f].
% The returned formula is somehow simpler than [f]. This procedure
% temporarily turns off the switches [groebopt] and [rlsiexpla].
% Accesses the switches [rlgsvb], [rlgsbnf], [rlgsrad], [rlgssub],
% [rlgsred], [rlgsprod], [rlgserf], [rlgsutord], and the fluid
% [rlradmemv!*].
begin scalar phi,!*rlsiexpla; % Hack, but otherwise phi is not a bnf!
if !*rlgsbnf then <<
if !*rlgsvb then ioto_prin2 "[DNF";
phi := cl_simpl(cl_nnfnot cl_dnf f,atl,-1);
if !*rlgsvb then ioto_prin2 "] ";
>> else
phi := cl_simpl(cl_nnfnot f,atl,-1);
if phi eq 'inctheo then return 'inctheo;
if rl_tvalp phi then
return cl_nnfnot phi;
phi := acfsf_gssimplify0(phi,atl);
if phi eq 'inctheo then return 'inctheo;
return cl_simpl(cl_nnfnot phi,atl,-1)
end;
procedure acfsf_gsn(f,atl);
% Algebraically closed field Groebner simplification via normal
% form. [f] is a formula; [atl] is a theory. Returns [inconsistent]
% or a formula equivalent to [f]. The returned formula is somehow
% simpler than [f]. The normal form used depends on the toplevel
% operator of [f]. This procedure temporarily turns off the
% switches [groebopt] and [rlsiexpla]. Accesses the switches
% [rlverbose], [rlgsvb], [rlgsbnf], [rlgsrad], [rlgssub],
% [rlgsred], [rlgsprod], [rlgserf], [rlgsutord], and the fluid
% [rlradmemv!*].
if rl_tvalp f then
f
else if cl_atflp(rl_argn f) then
if rl_op(f) eq 'and then acfsf_gsd(f,atl) else acfsf_gsc(f,atl)
else
if rl_op(f) eq 'and then acfsf_gsc(f,atl) else acfsf_gsd(f,atl);
procedure acfsf_gssimplify0(f,atl);
% Algebraically closed field Groebner simplify. [f] is a
% conjunction of disjunctions of atomic formulas, a disjunction of
% atomic formulas, or an atomic formula; [atl] is a theory. Returns
% [inctheo] or a formula. Accesses the switches [rlgsvb], [rlgsrad],
% [rlgssub], [rlgsred], [rlgsprod], [rlgserf], [rlgsutord], and the
% fluid [rlradmemv!*].
begin scalar acfsf_gstv!*,!*cgbverbose,!*groebopt;
return acfsf_gssimplify(f,atl)
end;
procedure acfsf_gssimplify(f,atl);
% Algebraically closed field Groebner simplify. [f] is a
% conjunction of disjunctions of atomic formulas, a disjunction of
% atomic formulas, or an atomic formula; [atl] is a theory. Returns
% [inctheo] or a formula. Accesses the switches [rlgsvb],
% [rlgsrad], [rlgssub], [rlgsred], [rlgsprod], [rlgserf], and the
% fluid [rlradmemv!*].
begin scalar al,gp,ipart,npart,w,gprem,gprodal,gatl;
atl := cl_sitheo atl;
if atl eq 'inctheo or acfsf_gsinctheop(atl) then
return 'inctheo;
if (cl_atfp f) or (rl_op f eq 'or) then % degenerated cnf
al := acfsf_gssplit!-cnf {f}
else
al := acfsf_gssplit!-cnf rl_argn f;
if w := lto_catsoc('gprem,al) then <<
gp := acfsf_gsextract!-gp atl;
gprem := acfsf_gsgprem(w,gp);
if gprem eq 'false then return 'false;
>>;
gatl := append(atl,lto_catsoc('gprem,al));
gp := acfsf_gsextract!-gp(gatl);
caar gp := acfsf_gsgbf caar gp;
ipart := lto_catsoc('impl,al);
npart := lto_catsoc('noneq,al);
if ipart then
ipart := acfsf_gspart(ipart,gp);
if npart and gatl then
npart := acfsf_gspart(npart,gp);
if gprem then <<
if null !*rlgsprod then <<
gprodal := lto_catsoc('gprodal,al);
gprem := acfsf_gssimulateprod(gprem,gprodal)
>>;
return rl_smkn('and,gprem . nconc(ipart,npart))
>>;
return rl_smkn('and,nconc(ipart,npart))
end;
procedure acfsf_gspreducef(f,gl);
numr gb_reducef(f,gl,acfsf_gsvl(),acfsf_gssm(),acfsf_gssx());
procedure acfsf_gsgreducef(f,gl);
acfsf_gspreducef(f,gb_gbf(gl,acfsf_gsvl(),acfsf_gssm(),acfsf_gssx()));
procedure acfsf_gsgbf(fl);
gb_gbf(fl,acfsf_gsvl(),acfsf_gssm(),acfsf_gssx());
procedure acfsf_gsvl();
if !*rlgsutord then append(td_vars(),{acfsf_gstv!*}) else nil;
procedure acfsf_gssm();
if !*rlgsutord then td_sortmode() else 'revgradlex;
procedure acfsf_gssx();
if !*rlgsutord then td_sortextension() else nil;
procedure acfsf_gsinctheop(atl);
% Algebraically closed field standard form Groebner simplifier
% inconsistent theory predicate. [atl] is a list of atomic
% formulas. [T] or [nil] is returned.
begin scalar w;
if null atl then
return nil;
if !*rlgsvb then ioto_prin2 "Inctheop... ";
w := cl_nnfnot acfsf_gsimplication(
cl_nnfnot rl_smkn('and,atl),'((nil . 1) . nil));
if !*rlgsvb then ioto_prin2t "done.";
return w eq 'false
end;
procedure acfsf_gssplit!-cnf(f);
% Algebraically closed field standard form Groebner simplifier
% split conjunctive normal form. [f] is an list of disjunctions of
% atomic formulas. An assoc list is returned. The returned assoc
% list have the following items. [('impl . imp)] where [imp] is the
% list off all disjunctions containing at least one inequation,
% [('gprem . gprem)] where [gprem] is the list of all atomic
% formulas occuring in [f] and atomic formulas equivalent to
% disjunctions of inequalities occuring in [f], [('noneq . noneq)]
% where [noneq] is a list of disjunctions of atomic formulas
% containing no inequations, and [('gprodal . gprodal)]. The value
% [gprodal] is a assoc list containing to each equation the product
% representation, if the equation was extracted from a disjunction.
begin scalar noneq,imp,prod,gprodal,gprem,w,x;
for each phi in f do
if rl_op phi memq '(and or) then % [phi] is not an atomic formula
if (w := acfsf_gsdis!-type rl_argn phi) eq 'impl then
imp := phi . imp
else if w eq 'noneq then
noneq := phi . noneq
else << % [if w eq 'equal then]
prod := 1;
for each atf in rl_argn phi do
prod := multf(prod,acfsf_arg2l atf);
x := acfsf_0mk2('equal,prod);
gprem := x . gprem;
gprodal := (x . phi) . gprodal
>>
else
gprem := phi . gprem;
if !*rlgsvb then <<
ioto_tprin2t {"global: ",length gprem,"; impl: ",length imp,
"; no neq: ",length noneq, "; glob-prod-al: ",length gprodal,"."}
>>;
return { 'impl . imp, 'noneq . noneq, 'gprem . gprem, 'gprodal . gprodal}
end;
procedure acfsf_gsdis!-type(atl);
% Algebraically closed field standard form Groebner simplifier
% disjunction type. [atl] is a non null list of atomic formulas.
% ['equal], ['impl], or ['noneq] is returned. ['equal] is returned
% if and only if all atomic formulas have the relation [equal];
% [impl] is returned, if and only if one of the atomic formula is
% an equality, otherwise [noneq] is returned.
begin scalar op,w;
if null atl then return 'equal;
op := acfsf_op car atl;
if op eq 'neq then return 'impl;
w := acfsf_gsdis!-type cdr atl;
if w eq 'impl then return 'impl;
if op eq 'equal and w eq 'equal then return 'equal;
return 'noneq
end;
procedure acfsf_gsextract!-gp(atl);
% Algebraically closed field standard form extract global premise.
% [atl] is a list of atomic formulas. A GP is returned.
begin scalar w;
w := acfsf_gsdis2impl(for each at in atl collect acfsf_negateat(at));
return ( (car w . multf(cadr w, caddr w)) . cadddr w)
end;
procedure acfsf_gsgprem(atl,gp);
% Algebraically closed field standard form Groebner simplifier
% simplify global premise. [atl] is a list of atomic formulas; [gp]
% is a GP. A formula is returned.
begin scalar w;
if !*rlgsvb then ioto_prin2 "[GP";
w := cl_nnfnot acfsf_gsimplication(cl_nnfnot rl_smkn('and,atl),gp);
if !*rlgsvb then ioto_prin2 "] ";
return w
end;
procedure acfsf_gspart(part,gp);
% Algebraically closed field standard form Groebner simplify
% simplify part. [part] is a list of disjunctions of atomic
% formulas and atomic formulas. [gp] is a GP. A list [l] of
% disjunctions of atomic formulas and atomic formulas is returned.
% The formula on position $i$ in [l] is somehow simpler than the
% formula on the position $i$ in part. Supposed that the formula
% $\bigwedge(g_i=0)$ is true where $g_i$ are the terms in [gp] then
% the positional corresponding fomulas in the two lists [part] and
% [l] are equivalent.
begin scalar w,curlen,res;
if !*rlgsvb then curlen := length part;
res := for each phi in part collect <<
if !*rlgsvb then ioto_prin2 {"[",curlen};
w := acfsf_gsimplication(phi,gp);
if !*rlgsvb then << curlen := curlen - 1; ioto_prin2 {"] "} >>;
w
>>;
if !*rlgsvb then ioto_cterpri();
return res
end;
procedure acfsf_gsimplication(f,gp);
% Algebraically closed field standard form Groebner simplification
% implication. [f] is a disjunction of atomic formulas or an atomic
% formula. [gp] is a GP. Returns a formula. It is a truth value, an
% atomic formula or a disjunction of atomic formulas, unless the
% simplification of an atomic formula yields a complex formula.
begin scalar prem,prod1,prod2,gprod,rprod,iprem,w,z,atl,natl;
if cl_cxfp f then atl := rl_argn f else atl := {f};
w := acfsf_gsdis2impl atl;
iprem := car w;
prod1 := cadr w;
prod2 := caddr w;
gprod := cdar gp;
prem := append(iprem,caar gp);
if null prem then return f;
prem := acfsf_gsgbf prem;
z := numr simp acfsf_gsmkradvar();
rprod := acfsf_gseqprod(prod1,prod2,gprod,prem,z);
if rprod eq 'true then <<
if !*rlgsvb then ioto_prin2 "!";
return 'true
>>;
w := acfsf_gsusepremise(cdr gp,prem,z);
if w eq 'true then <<
if !*rlgsvb then ioto_prin2 "!";
return 'true
>>;
natl := acfsf_gsredatl(atl,prem,z,rprod);
if natl eq 'true then <<
if !*rlgsvb then ioto_prin2 "!";
return 'true
>>;
if rprod and rprod neq 'false then natl := rprod . natl;
natl := nconc(natl,acfsf_gspremise(iprem,caar gp));
return rl_smkn('or,natl)
end;
procedure acfsf_gsredatl(atl,prem,z,rprod);
% Algebraically closed field standard form reduce atomic formula
% list. [atl] is a list of SF's; [prem] is a Groebner basis; [z] is
% a kernel; [rprod] is a flag. Returns ['true] or a list of atomic
% formulas.
begin scalar a,w,natl;
while atl do <<
a := car atl;
atl := cdr atl;
w := acfsf_gsredat(a,prem,z,rprod);
if w eq 'true then
atl := nil
else if w and w neq 'false then
natl := w . natl
>>;
if w eq 'true then return 'true;
return natl
end;
procedure acfsf_gsusepremise(atl,prem,z);
% Algebraically closed field standard form use premise. [atl] is a
% list of atomic formulas; [prem] is a Groebner basis; [z] is a
% kernel. returns [nil] or ['true].
begin scalar w;
while atl do <<
w := acfsf_gsredat(car atl,prem,z,nil);
if w eq 'true then
atl := nil
else
atl := cdr atl;
>>;
if w eq 'true then return 'true;
end;
procedure acfsf_gseqprod(iprod1,iprod2,gprod,prem,z);
% Algebraically closed field standard form equation product.
% [iprod1], [iprod2], and [prem] are SF's; [prem] is a list of
% SF's; [z] is a kernel. Returns [nil] or a formula.
begin scalar p,w;
p := multf(iprod1,multf(iprod2,gprod));
% Comment the test on [!*rlgsrad] out if the radical membership
% test should always be performed for the equation product.
if !*rlgsrad and
(null acfsf_gsgreducef(1,addf(1,negf multf(p,z)) . prem))
then
return 'true;
w := acfsf_gstryeval('equal,acfsf_gspreducef(p,prem));
if rl_tvalp w then return w;
if null !*rlgsprod then return nil;
if !*rlgsred then
return acfsf_0mk2('equal,acfsf_gspreducef(iprod1,prem));
return acfsf_0mk2('equal,iprod1);
end;
procedure acfsf_gsmkradvar();
% Algebraically closed field standard form Groebner simplifier make
% radical memebership test variable. Returns an identifier that is
% not used as an algebraic mode variable.
begin scalar w; integer n;
w := 'rlgsradmemv!*;
while get(w,'avalue) do
w := mkid(w,n := n+1);
if !*rlgsutord then
acfsf_gsupdtorder w;
return w;
end;
procedure acfsf_gsupdtorder(v);
% Algebraically closed field standard form Groebner simplifier
% update term order. [v] is a kernel. Inserts the main variable [v]
% into the variable list of the global fixed term order of the
% [groebner] package. Not all torders are supported, if a variable
% list is present. To get over this problem one can insert the tag
% variable [v] in the variable list before calling the Groebner
% simplifier.
if td_vars() and v memq td_vars() then % vl needs update
if not(td_sortmode() memq '(lex gradlex revgradlex gradlexgradlex
gradlexrevgradlex lexgradlex lexrevgradlex weighted))
then
rederr {"term order",td_sortmode(), "not supported"}
else
acfsf_gstv!* := v;
procedure acfsf_gstryeval(rel,lhs);
% Algebraically closed field standard form try evaluation. [rel] is
% an acfsf-relation; [lhs] a SF. returns [nil], a truth value or an
% atomic formula. In the first case the atomic formula $([lhs]
% [rel] 0)$ cannot be evaluated or should be ignored. In the other
% case the returned value is equivalent to the the atomic formula.
begin scalar w,!*rlsiexpla;
if !*rlgserf then <<
w := cl_simplat(acfsf_0mk2(rel,lhs),nil);
return if rl_tvalp w then w
>>;
if domainp lhs then
return cl_simplat(acfsf_0mk2(rel,lhs),nil)
end;
procedure acfsf_gsdis2impl(atl);
% Algebraically closed field standard form Groebner simplifier
% disjunction to implication. [atl] is a list of atomic formulas. A
% CIMPL is returned. The classification of the atomic formulas in
% [atl] is done by [acfsf_attype].
begin scalar prem,prod1,prod2,other,w,a;
prod1 := prod2 := 1;
for each at in atl do <<
w := acfsf_gsattype at;
if w then <<
a := car w;
if a eq 'equal then
prod1 := multf(cdr w,prod1)
else if a eq 'cequal then
prod2 := multf(cdr w,prod2)
else if a eq 'neq then
prem := cdr w . prem
else
rederr {"BUG IN ACFSF_GSDIS2IMPL",car w}
>>;
if not (w memq '(equal neq)) then
other := at . other
>>;
return {prem, prod1, prod2, other};
end;
procedure acfsf_gsattype(at);
% Algebraically closed field standard form Groebner simplifier
% atomic formula type. [at] is an atomic formula. [nil] or a pair
% $(\rho,p)$ is returned. $\rho$ is either ['equal], ['neq], or
% ['cequal]; $p$ is a SF.
(if w eq 'equal then
('equal . acfsf_arg2l at)
else if w memq '(geq leq) then
('cequal . acfsf_arg2l at)
else if w eq 'neq then
('neq . acfsf_arg2l at)) where w=acfsf_op at;
procedure acfsf_gsredat(at,gb,z,flag);
% Algebraically closed field standard form Groebner simplifier
% reduce atomic formula. [at] is an atomic formula; [gb] is a
% Groebner basis; [z] is a variable; [flag] is a flag. [nil], a
% truth value or an atomic formula is returned. The behavior of
% this procedure depends on the switches [rl_gsred] and [rl_gsrad].
% [nil] is returned if the atomic formula belongs to the premise or
% [flag] is [T] and [at] is an equation. Is [flag] is non [nil]
% then equations can be ignored. In the other cases the returned
% value is equivalent to [at]. The intention of this procedure is
% the reduction of [at] wrt. the radical generated by [gb].
begin scalar w,x,op,arg,nat;
op := acfsf_op at;
if (op eq 'neq) or (flag and op eq 'equal) then return nil;
arg := acfsf_arg2l at;
w := acfsf_gspreducef(arg,gb);
if !*rlgsred then
nat := cl_simplat(acfsf_0mk2(op,w),nil)
else
if x := acfsf_gstryeval(op,w) then
nat := x
else
nat := at;
if (rl_tvalp nat) or (op eq 'equal) or (null !*rlgsrad) then
return nat;
if null acfsf_greducef(1,addf(1,negf multf(w,z)) . gb) then
return cl_simplat(acfsf_0mk2(op,nil),nil);
return nat;
end;
procedure acfsf_gspremise(tl,gp);
% Algebraically closed field standard form Groebner simplify
% premise. [tl] and [gp] are lists of SF's. A list of atomic
% formulas is returned. The behavior of this procedure depends on
% the switches [rl_gsred] and [rl_gssub]. The conjunction over the
% returned formulas is equivalent to the formula $\bigvee(t_i \neq
% 0)$ supposed that $\bigwedge(g_j = 0)$, where $t_i$ are the terms
% in [tl] and $g_j$ are the terms in [gp]. If the switch [rl_gsred]
% is on then all terms $t_i$ are reduced modulo Id([gp]). If the
% switch [!*rl_gssub] is on, the term list is substituted by the
% reduced Groebner base of the term list.
begin scalar gb,rtl,w;
if !*rlgsred then <<
gb := acfsf_gsgbf gp;
for each sf in tl do
if w := acfsf_gspreducef(sf,gb) then
rtl := lto_insert(w,rtl);
>> else
rtl := tl;
if !*rlgssub then
return for each sf in acfsf_gsgbf rtl collect
acfsf_0mk2('neq,sf);
return for each sf in rtl collect
acfsf_0mk2('neq,sf)
end;
procedure acfsf_gssimulateprod(prem,prodal);
% Algebraically closed field standard form simulate rlprod switch.
% [prem] is a quantifier free formula. [prodal] is an assoc list
% containing to some equations its product representation.
begin scalar w,res;
if rl_tvalp prem then return prem;
if cl_atfp prem and (w := lto_cassoc(prem,prodal)) then
return w;
res := for each f in rl_argn prem collect
if cl_atfp f and (w := lto_cassoc(f,prodal)) then w else f;
return rl_mkn(rl_op prem,res)
end;
endmodule; % [acfsfgs]
end; % of file