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added polynomial evaluation at algebraic point

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2012-12-08 20:39:16 -08:00
parent bf2340850a
commit 0d230375be
9 changed files with 266 additions and 26 deletions
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57
src/api/api_algebraic.cpp
View file

@ -22,6 +22,9 @@ Notes:
#include"api_log_macros.h"
#include"api_context.h"
#include"algebraic_numbers.h"
#include"expr2polynomial.h"
#include"cancel_eh.h"
#include"scoped_timer.h"
extern "C" {
@ -318,6 +321,34 @@ extern "C" {
return !Z3_algebraic_eq(c, a, b);
}
static bool to_anum_vector(Z3_context c, unsigned n, Z3_ast a[], scoped_anum_vector & as) {
algebraic_numbers::manager & _am = am(c);
scoped_anum tmp(_am);
for (unsigned i = 0; i < n; i++) {
if (is_rational(c, a[i])) {
_am.set(tmp, get_rational(c, a[i]).to_mpq());
as.push_back(tmp);
}
else if (is_irrational(c, a[i])) {
as.push_back(get_irrational(c, a[i]));
}
else {
return false;
}
}
return true;
}
class vector_var2anum : public polynomial::var2anum {
scoped_anum_vector const & m_as;
public:
vector_var2anum(scoped_anum_vector & as):m_as(as) {}
virtual ~vector_var2anum() {}
virtual algebraic_numbers::manager & m() const { return m_as.m(); }
virtual bool contains(polynomial::var x) const { return static_cast<unsigned>(x) < m_as.size(); }
virtual algebraic_numbers::anum const & operator()(polynomial::var x) const { return m_as.get(x); }
};
Z3_ast_vector Z3_API Z3_algebraic_roots(Z3_context c, Z3_ast p, unsigned n, Z3_ast a[]) {
Z3_TRY;
LOG_Z3_algebraic_roots(c, p, n, a);
@ -331,8 +362,30 @@ extern "C" {
Z3_TRY;
LOG_Z3_algebraic_eval(c, p, n, a);
RESET_ERROR_CODE();
// TODO
return 0;
polynomial::manager & pm = mk_c(c)->pm();
polynomial_ref _p(pm);
polynomial::scoped_numeral d(pm.m());
expr2polynomial converter(mk_c(c)->m(), pm, 0, true);
if (!converter.to_polynomial(to_expr(p), _p, d)) {
SET_ERROR_CODE(Z3_INVALID_ARG);
return 0;
}
algebraic_numbers::manager & _am = am(c);
scoped_anum_vector as(_am);
if (!to_anum_vector(c, n, a, as)) {
SET_ERROR_CODE(Z3_INVALID_ARG);
return 0;
}
{
cancel_eh<algebraic_numbers::manager> eh(_am);
api::context::set_interruptable(*(mk_c(c)), eh);
scoped_timer timer(mk_c(c)->params().m_timeout, &eh);
vector_var2anum v2a(as);
int r = _am.eval_sign_at(_p, v2a);
if (r > 0) return 1;
else if (r < 0) return -1;
else return 0;
}
Z3_CATCH_RETURN(0);
}

10
src/api/api_context.h
View file

@ -32,6 +32,7 @@ Revision History:
#include"event_handler.h"
#include"tactic_manager.h"
#include"context_params.h"
#include"api_polynomial.h"
namespace smtlib {
class parser;
@ -81,6 +82,8 @@ namespace api {
Z3_ast_print_mode m_print_mode;
event_handler * m_interruptable; // Reference to an object that can be interrupted by Z3_interrupt
pmanager m_pmanager;
public:
// Scoped obj for setting m_interruptable
class set_interruptable {
@ -167,6 +170,13 @@ namespace api {
void check_sorts(ast * n);
// ------------------------
//
// Polynomial manager & caches
//
// -----------------------
polynomial::manager & pm() { return m_pmanager.pm(); }
// ------------------------
//
// Solver interface for backward compatibility

34
src/api/api_polynomial.cpp Normal file
View file

@ -0,0 +1,34 @@
/*++
Copyright (c) 2012 Microsoft Corporation
Module Name:
api_polynomial.cpp
Abstract:
Polynomial manager and caches for the external API.
Author:
Leonardo de Moura (leonardo) 2012-12-08
Notes:
--*/
#include"api_polynomial.h"
namespace api {
pmanager::pmanager():
m_pm(m_nm) {
}
pmanager::~pmanager() {
}
void pmanager::set_cancel(bool f) {
m_pm.set_cancel(f);
}
};

39
src/api/api_polynomial.h Normal file
View file

@ -0,0 +1,39 @@
/*++
Copyright (c) 2012 Microsoft Corporation
Module Name:
api_polynomial.h
Abstract:
Polynomial manager and caches for the external API.
Author:
Leonardo de Moura (leonardo) 2012-12-08
Notes:
--*/
#ifndef _API_POLYNOMIAL_H_
#define _API_POLYNOMIAL_H_
#include"polynomial.h"
namespace api {
class pmanager {
unsynch_mpz_manager m_nm;
polynomial::manager m_pm;
// TODO: add support for caching expressions -> polynomial and back
public:
pmanager();
virtual ~pmanager();
polynomial::manager & pm() { return m_pm; }
void set_cancel(bool f);
};
};
#endif

21
src/api/python/z3.py
View file

@ -1126,6 +1126,27 @@ def Var(idx, s):
_z3_assert(is_sort(s), "Z3 sort expected")
return _to_expr_ref(Z3_mk_bound(s.ctx_ref(), idx, s.ast), s.ctx)
def RealVar(idx, ctx=None):
"""
Create a real free variable. Free variables are used to create quantified formulas.
They are also used to create polynomials.
>>> RealVar(0)
Var(0)
"""
return Var(idx, RealSort(ctx))
def RealVarVector(n, ctx=None):
"""
Create a list of Real free variables.
The variables have ids: 0, 1, ..., n-1
>>> x0, x1, x2, x3 = RealVarVector(4)
>>> x2
Var(2)
"""
return [ RealVar(i, ctx) for i in range(n) ]
#########################################
#
# Booleans

24
src/api/python/z3num.py
View file

@ -481,6 +481,30 @@ class Numeral:
def ctx_ref(self):
return self.ctx.ref()
def eval_sign_at(p, vs):
"""
Evaluate the sign of the polynomial `p` at `vs`. `p` is a Z3
Expression containing arithmetic operators: +, -, *, ^k where k is
an integer; and free variables x that is_var(x) is True. Moreover,
all variables must be real.
The result is 1 if the polynomial is positive at the given point,
-1 if negative, and 0 if zero.
>>> x0, x1, x2 = RealVarVector(3)
>>> eval_sign_at(x0**2 + x1*x2 + 1, (Numeral(0), Numeral(1), Numeral(2)))
1
>>> eval_sign_at(x0**2 - 2, [ Numeral(Sqrt(2)) ])
0
>>> eval_sign_at((x0 + x1)*(x0 + x2), (Numeral(0), Numeral(Sqrt(2)), Numeral(Sqrt(3))))
1
"""
num = len(vs)
_vs = (Ast * num)()
for i in range(num):
_vs[i] = vs[i].ast
return Z3_algebraic_eval(p.ctx_ref(), p.as_ast(), num, _vs)
if __name__ == "__main__":
import doctest

80
src/ast/expr2polynomial.cpp
View file

@ -49,21 +49,24 @@ struct expr2polynomial::imp {
polynomial::polynomial_ref_vector m_presult_stack;
polynomial::scoped_numeral_vector m_dresult_stack;
bool m_use_var_idxs;
volatile bool m_cancel;
imp(expr2polynomial & w, ast_manager & am, polynomial::manager & pm, expr2var * e2v):
imp(expr2polynomial & w, ast_manager & am, polynomial::manager & pm, expr2var * e2v, bool use_var_idxs):
m_wrapper(w),
m_am(am),
m_autil(am),
m_pm(pm),
m_expr2var(e2v == 0 ? alloc(expr2var, am) : e2v),
m_expr2var_owner(e2v == 0),
m_expr2var(e2v == 0 && !use_var_idxs ? alloc(expr2var, am) : e2v),
m_expr2var_owner(e2v == 0 && !use_var_idxs),
m_var2expr(am),
m_cached_domain(am),
m_cached_polynomials(pm),
m_cached_denominators(pm.m()),
m_presult_stack(pm),
m_dresult_stack(pm.m()),
m_use_var_idxs(use_var_idxs),
m_cancel(false) {
}
@ -95,6 +98,14 @@ struct expr2polynomial::imp {
cooperate("expr2polynomial");
}
void throw_not_polynomial() {
throw default_exception("the given expression is not a polynomial");
}
void throw_no_int_var() {
throw default_exception("integer variables are not allowed in the given polynomial");
}
void push_frame(app * t) {
m_frame_stack.push_back(frame(t));
}
@ -127,14 +138,26 @@ struct expr2polynomial::imp {
}
void store_var_poly(expr * t) {
polynomial::var x = m_expr2var->to_var(t);
if (x == UINT_MAX) {
bool is_int = m_autil.is_int(t);
x = m_wrapper.mk_var(is_int);
m_expr2var->insert(t, x);
if (x >= m_var2expr.size())
m_var2expr.resize(x+1, 0);
m_var2expr.set(x, t);
polynomial::var x;
if (m_use_var_idxs) {
SASSERT(::is_var(t));
if (m_autil.is_int(t))
throw_no_int_var();
unsigned idx = to_var(t)->get_idx();
while (idx >= m_pm.num_vars())
m_pm.mk_var();
x = static_cast<polynomial::var>(idx);
}
else {
x = m_expr2var->to_var(t);
if (x == UINT_MAX) {
bool is_int = m_autil.is_int(t);
x = m_wrapper.mk_var(is_int);
m_expr2var->insert(t, x);
if (x >= m_var2expr.size())
m_var2expr.resize(x+1, 0);
m_var2expr.set(x, t);
}
}
polynomial::numeral one(1);
store_result(t, pm().mk_polynomial(x), one);
@ -160,7 +183,10 @@ struct expr2polynomial::imp {
rational k;
SASSERT(t->get_num_args() == 2);
if (!m_autil.is_numeral(t->get_arg(1), k) || !k.is_int() || !k.is_unsigned()) {
store_var_poly(t);
if (m_use_var_idxs)
throw_not_polynomial();
else
store_var_poly(t);
return true;
}
push_frame(t);
@ -168,6 +194,8 @@ struct expr2polynomial::imp {
}
default:
// can't handle operator
if (m_use_var_idxs)
throw_not_polynomial();
store_var_poly(t);
return true;
}
@ -190,6 +218,8 @@ struct expr2polynomial::imp {
SASSERT(is_app(t));
if (!m_autil.is_arith_expr(t)) {
if (m_use_var_idxs)
throw_not_polynomial();
store_var_poly(t);
return true;
}
@ -378,19 +408,25 @@ struct expr2polynomial::imp {
for (unsigned i = 0; i < sz; i++) {
margs.reset();
polynomial::monomial * m = pm().get_monomial(p, i);
polynomial::monomial * _m = pm().get_monomial(p, i);
polynomial::numeral const & a = pm().coeff(p, i);
if (!nm().is_one(a)) {
margs.push_back(m_autil.mk_numeral(rational(a), is_int));
}
unsigned msz = pm().size(m);
unsigned msz = pm().size(_m);
for (unsigned j = 0; j < msz; j++) {
polynomial::var x = pm().get_var(m, j);
expr * t = m_var2expr.get(x);
if (m_wrapper.is_int(x) && !is_int) {
t = m_autil.mk_to_real(t);
polynomial::var x = pm().get_var(_m, j);
expr * t;
if (m_use_var_idxs) {
t = m().mk_var(x, m_autil.mk_real());
}
unsigned d = pm().degree(m, j);
else {
t = m_var2expr.get(x);
if (m_wrapper.is_int(x) && !is_int) {
t = m_autil.mk_to_real(t);
}
}
unsigned d = pm().degree(_m, j);
if (use_power && d > 1) {
margs.push_back(m_autil.mk_power(t, m_autil.mk_numeral(rational(d), is_int)));
}
@ -426,8 +462,8 @@ struct expr2polynomial::imp {
}
};
expr2polynomial::expr2polynomial(ast_manager & am, polynomial::manager & pm, expr2var * e2v) {
m_imp = alloc(imp, *this, am, pm, e2v);
expr2polynomial::expr2polynomial(ast_manager & am, polynomial::manager & pm, expr2var * e2v, bool use_var_idxs) {
m_imp = alloc(imp, *this, am, pm, e2v, use_var_idxs);
}
expr2polynomial::~expr2polynomial() {
@ -451,10 +487,12 @@ void expr2polynomial::to_expr(polynomial::polynomial_ref const & p, bool use_pow
}
bool expr2polynomial::is_var(expr * t) const {
SASSERT(!m_imp->m_use_var_idxs);
return m_imp->m_expr2var->is_var(t);
}
expr2var const & expr2polynomial::get_mapping() const {
SASSERT(!m_imp->m_use_var_idxs);
return *(m_imp->m_expr2var);
}

25
src/ast/expr2polynomial.h
View file

@ -29,7 +29,24 @@ class expr2polynomial {
struct imp;
imp * m_imp;
public:
expr2polynomial(ast_manager & am, polynomial::manager & pm, expr2var * e2v);
expr2polynomial(ast_manager & am,
polynomial::manager & pm,
expr2var * e2v,
/*
If true, the expressions converted into
polynomials should only contain Z3 free variables.
A Z3 variable x, with idx i, is converted into
the variable i of the polynomial manager pm.
An exception is thrown if there is a mismatch between
the sorts x and the variable in the polynomial manager.
The argument e2v is ignored when use_var_idxs is true.
Moreover, only real variables are allowed.
*/
bool use_var_idxs = false
);
virtual ~expr2polynomial();
ast_manager & m() const;
@ -63,6 +80,8 @@ public:
/**
\brief Return the mapping from expressions to variables
\pre the object was created using use_var_idxs = false.
*/
expr2var const & get_mapping() const;
@ -74,10 +93,10 @@ public:
/**
\brief Return true if the variable is associated with an expression of integer sort.
*/
virtual bool is_int(polynomial::var x) const = 0;
virtual bool is_int(polynomial::var x) const { UNREACHABLE(); return false; }
protected:
virtual polynomial::var mk_var(bool is_int) = 0;
virtual polynomial::var mk_var(bool is_int) { UNREACHABLE(); return polynomial::null_var; }
};
class default_expr2polynomial : public expr2polynomial {

2
src/math/polynomial/algebraic_numbers.h
View file

@ -64,6 +64,8 @@ namespace algebraic_numbers {
static void collect_param_descrs(param_descrs & r) { get_param_descrs(r); }
void set_cancel(bool f);
void cancel() { set_cancel(true); }
void updt_params(params_ref const & p);