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% ---------------------------------------------------------------------- % $Id: ofsfgs.red,v 1.10 2003/05/19 10:39:00 dolzmann Exp $ % ---------------------------------------------------------------------- % Copyright (c) 1995-2003 Andreas Dolzmann % ---------------------------------------------------------------------- % $Log: ofsfgs.red,v $ % Revision 1.10 2003/05/19 10:39:00 dolzmann % The Groebner package now uses the GB package. % % Revision 1.9 1999/03/23 07:41:37 dolzmann % Changed copyright information. % % Revision 1.8 1997/08/24 16:16:59 sturm % Call cl_sitheo instead of ofsf_gssimpltheo. % Added service rl_surep with black box rl_multsurep. % Added service rl_siaddatl. % % Revision 1.7 1996/10/07 12:03:24 sturm % Added fluids for CVS and copyright information. % % Revision 1.6 1996/09/26 11:55:42 dolzmann % Reformated source code. % % Revision 1.5 1996/09/05 11:37:47 dolzmann % Removed unused variable vl in procedure ofsf_gsmkradvar. % % Revision 1.4 1996/09/05 11:14:57 dolzmann % Removed unused variable curtorder in procedure ofsf_gsmkradvar. % % Revision 1.3 1996/07/13 11:19:09 dolzmann % Introduced new switches !*rlgsbnf, !*rlgsutord and related code. % % Revision 1.2 1996/07/07 14:44:10 sturm % Call cl_nnfnot instead of cl_nnf1. % % Revision 1.1 1996/03/22 12:14:07 sturm % Moved and split. % % ---------------------------------------------------------------------- lisp << fluid '(ofsf_gs_rcsid!* ofsf_gs_copyright!*); ofsf_gs_rcsid!* := "$Id: ofsfgs.red,v 1.10 2003/05/19 10:39:00 dolzmann Exp $"; ofsf_gs_copyright!* := "Copyright (c) 1995-2003 by A. Dolzmann" >>; module ofsfgs; % Ordered field standard form groebner simplifier. Submodule of [ofsf]. %DS % <cimpl> ::= (<gp>, <prod1>, <prod2>, <other>) % <gp> ::= ((<gb> . <prod>) . <other>) % <gb> ::= (<sf>,...) % <prod> ::= <sf> % <prod1> ::= <sf> % <prod2> ::= <sf> % <other> ::= (<atomic_formula>,...) procedure ofsf_gsc(f,atl); % Ordered field standard form groebner simplification via % conjunctive normal form. [f] is an formula; [atl] is a list of % atomic formulas, which are considered to describe a theory. An % formula equivalent to [f] is returned. The returned formula is % somehow simpler than [f]. begin scalar w,svrlgsvb; svrlgsvb := !*rlgsvb; if !*rlverbose and !*rlgsvb then on1 'rlgsvb else off1 'rlgsvb; w := ofsf_gsc1(f,atl); onoff('rlgsvb,svrlgsvb); return w end; procedure ofsf_gsc1(f,atl); % Ordered field standard form groebner simplification via % conjunctive normal form. [f] is an formula; [atl] is a list of % atomic formulas, which are considered to describe a theory. An % formula equivalent to [f] or ['inctheo] is returned. The % returned formula is somehow simpler than [f]. 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 := ofsf_gssimplify0(phi,atl); if phi eq 'inctheo then return 'inctheo; return cl_simpl(phi,atl,-1) end; procedure ofsf_gsd(f,atl); % Ordered field standard form groebner simplification via % disjunctive normal form. [f] is an formula; [atl] is a list of % atomic formulas, which are considered to describe a theory. An % formula equivalent to [f] or ['inctheo] is returned. The % returned formula is somehow simpler than [f]. begin scalar w,svrlgsvb; svrlgsvb := !*rlgsvb; if !*rlverbose and !*rlgsvb then on1 'rlgsvb else off1 'rlgsvb; w := ofsf_gsd1(f,atl); onoff('rlgsvb,svrlgsvb); return w end; procedure ofsf_gsd1(f,atl); % Ordered field standard form groebner simplification via % disjunctive normal form. [f] is an formula; [atl] is a list of % atomic formulas, which are considered to describe a theory. An % formula equivalent to [f] or ['inctheo] is returned. The % returned formula is somehow simpler than [f]. 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 := ofsf_gssimplify0(phi,atl); if phi eq 'inctheo then return 'inctheo; return cl_simpl(cl_nnfnot phi,atl,-1) end; procedure ofsf_gsn(f,atl); % Ordered field standard form groebner simplification via boolean % normal form. [f] is an formula; [atl] is a list of atomic % formulas, which are considered to describe a theory. An formula % equivalent to [f] or ['inctheo] is returned. The returned formula % is somehow simpler than [f]. This procedure calls in dependency % of the structure of [f] either [ofsf_gsc] or [ofsf_gsd]. The % following heuristic is used: Is [f] a conjunction of atomic % formulas or a disjunction of formulas with at least one complex % formula then [ofsf_gsd] is called; in other cases [ofsf_gsc] is % called. if rl_tvalp f then f else if cl_atflp(rl_argn f) then if rl_op(f) eq 'and then ofsf_gsd(f,atl) else ofsf_gsc(f,atl) else if rl_op(f) eq 'and then ofsf_gsc(f,atl) else ofsf_gsd(f,atl); procedure ofsf_gssimplify0(f,atl); % Ordered field standard form groebner simplify. [f] is a % conjunction over disjunctions of atomic formulas or a % disjunctions of atomic formulas or an atomic formula; [atl] is a % list of atomic formulas, which are considered to describe a % theory. A formula is returned. begin scalar ofsf_gstv!*,!*cgbverbose,!*groebopt; return ofsf_gssimplify(f,atl) end; procedure ofsf_gssimplify(f,atl); % Ordered field standard form groebner simplify. [f] is a % conjunction over disjunctions of atomic formulas or a % disjunctions of atomic formulas or an atomic formula; [atl] is a % list of atomic formulas, which are considered to describe a % theory. A formula is returned. begin scalar al,gp,ipart,npart,w,gprem,gprodal,gatl; atl := cl_sitheo atl; if atl eq 'inctheo or ofsf_gsinctheop(atl) then return 'inctheo; if (cl_atfp f) or (rl_op f eq 'or) then % degenerated cnf al := ofsf_gssplit!-cnf {f} else al := ofsf_gssplit!-cnf rl_argn f; if w := lto_catsoc('gprem,al) then << gp := ofsf_gsextract!-gp atl; gprem := ofsf_gsgprem(w,gp); if gprem eq 'false then return 'false; >>; gatl := append(atl,lto_catsoc('gprem,al)); gp := ofsf_gsextract!-gp(gatl); caar gp := ofsf_gsgbf caar gp; ipart := lto_catsoc('impl,al); npart := lto_catsoc('noneq,al); if ipart then ipart := ofsf_gspart(ipart,gp); if npart and gatl then npart := ofsf_gspart(npart,gp); if gprem then << if null !*rlgsprod then << gprodal := lto_catsoc('gprodal,al); gprem := ofsf_gssimulateprod(gprem,gprodal) >>; return rl_smkn('and,gprem . nconc(ipart,npart)) >>; return rl_smkn('and,nconc(ipart,npart)) end; procedure ofsf_gspreducef(f,gl); numr gb_reducef(f,gl,ofsf_gsvl(),ofsf_gssm(),ofsf_gssx()); procedure ofsf_gsgreducef(f,gl); ofsf_gspreducef(f,gb_gbf(gl,ofsf_gsvl(),ofsf_gssm(),ofsf_gssx())); procedure ofsf_gsgbf(fl); gb_gbf(fl,ofsf_gsvl(),ofsf_gssm(),ofsf_gssx()); procedure ofsf_gsvl(); if !*rlgsutord then append(td_vars(),{ofsf_gstv!*}) else nil; procedure ofsf_gssm(); if !*rlgsutord then td_sortmode() else 'revgradlex; procedure ofsf_gssx(); if !*rlgsutord then td_sortextension() else nil; %% procedure ofsf_gsvl(); %% td_vars(); %% %% procedure ofsf_gssm(); %% td_sortmode(); %% %% procedure ofsf_gssx(); %% td_sortextension(); procedure ofsf_gsinctheop(atl); % Ordered 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 ofsf_gsimplication( cl_nnfnot rl_smkn('and,atl),'((nil . 1) . nil)); if !*rlgsvb then ioto_prin2t "done."; return w eq 'false end; procedure ofsf_gssplit!-cnf(f); % Ordered 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 := ofsf_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,ofsf_arg2l atf); x := ofsf_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 ofsf_gsdis!-type(atl); % Ordered 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 := ofsf_op car atl; if op eq 'neq then return 'impl; w := ofsf_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 ofsf_gsextract!-gp(atl); % Ordered field standard form extract global premise. [atl] is a % list of atomic formulas. A GP is returned. begin scalar w; w := ofsf_gsdis2impl(for each at in atl collect ofsf_negateat(at)); return ( (car w . multf(cadr w, caddr w)) . cadddr w) end; procedure ofsf_gsgprem(atl,gp); % Ordered 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 ofsf_gsimplication(cl_nnfnot rl_smkn('and,atl),gp); if !*rlgsvb then ioto_prin2 "] "; return w end; procedure ofsf_gspart(part,gp); % Ordered 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 := ofsf_gsimplication(phi,gp); if !*rlgsvb then << curlen := curlen - 1; ioto_prin2 {"] "} >>; w >>; if !*rlgsvb then ioto_cterpri(); return res end; procedure ofsf_gsimplication(f,gp); % Ordered 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 := ofsf_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 := ofsf_gsgbf prem; z := numr simp ofsf_gsmkradvar(); rprod := ofsf_gseqprod(prod1,prod2,gprod,prem,z); if rprod eq 'true then << if !*rlgsvb then ioto_prin2 "!"; return 'true >>; w := ofsf_gsusepremise(cdr gp,prem,z); if w eq 'true then << if !*rlgsvb then ioto_prin2 "!"; return 'true >>; natl := ofsf_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,ofsf_gspremise(iprem,caar gp)); return rl_smkn('or,natl) end; procedure ofsf_gsredatl(atl,prem,z,rprod); % Ordered 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 := ofsf_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 ofsf_gsusepremise(atl,prem,z); % Ordered 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 := ofsf_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 ofsf_gseqprod(iprod1,iprod2,gprod,prem,z); % Ordered 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 ofsf_gsgreducef(1,addf(1,negf multf(p,z)) . prem)) then return 'true; w := ofsf_gstryeval('equal,ofsf_gspreducef(p,prem)); if rl_tvalp w then return w; if null !*rlgsprod then return nil; if !*rlgsred then return ofsf_0mk2('equal,ofsf_gspreducef(iprod1,prem)); return ofsf_0mk2('equal,iprod1); end; procedure ofsf_gsmkradvar(); % Ordered 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 ofsf_gsupdtorder w; return w; end; procedure ofsf_gsupdtorder(v); % Ordered 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 ofsf_gstv!* := v; procedure ofsf_gstryeval(rel,lhs); % Ordered field standard form try evaluation. [rel] is an % ofsf-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(ofsf_0mk2(rel,lhs),nil); return if rl_tvalp w then w; >>; if domainp lhs then return cl_simplat(ofsf_0mk2(rel,lhs),nil); end; procedure ofsf_gsdis2impl(atl); % Ordered 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 [ofsf_attype]. begin scalar prem,prod1,prod2,other,w,a; prod1 := prod2 := 1; for each at in atl do << w := ofsf_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 OFSF_GSDIS2IMPL",car w} >>; if not (w memq '(equal neq)) then other := at . other >>; return {prem, prod1, prod2, other}; end; procedure ofsf_gsattype(at); % Ordered 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 . ofsf_arg2l at) else if w memq '(geq leq) then ('cequal . ofsf_arg2l at) else if w eq 'neq then ('neq . ofsf_arg2l at)) where w=ofsf_op at; procedure ofsf_gsredat(at,gb,z,flag); % Ordered 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 := ofsf_op at; if (op eq 'neq) or (flag and op eq 'equal) then return nil; arg := ofsf_arg2l at; w := ofsf_gspreducef(arg,gb); if !*rlgsred then nat := cl_simplat(ofsf_0mk2(op,w),nil) else if x := ofsf_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 ofsf_gsgreducef(1,addf(1,negf multf(w,z)) . gb) then return cl_simplat(ofsf_0mk2(op,nil),nil); return nat; end; procedure ofsf_gspremise(tl,gp); % Ordered 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 := ofsf_gsgbf gp; for each sf in tl do if w := ofsf_gspreducef(sf,gb) then rtl := lto_insert(w,rtl); >> else rtl := tl; if !*rlgssub then return for each sf in ofsf_gsgbf rtl collect ofsf_0mk2('neq,sf); return for each sf in rtl collect ofsf_0mk2('neq,sf) end; procedure ofsf_gssimulateprod(prem,prodal); % Ordered field standard form simulate rlprod switch. [prem] is a % quantifier free formula. [prodal] is an assoc list containing to % some equations its product representation. truth value or an 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; % [ofsfgs] end; % of file