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quant_elim.cpp
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/*++
Copyright (c) 2015 Microsoft Corporation
--*/
#include "ast/ast.h"
#include "smt/params/smt_params.h"
#include "qe/qe.h"
#include "ast/ast_pp.h"
#include "util/lbool.h"
#include <sstream>
#include "ast/reg_decl_plugins.h"
#if 0
static void test_qe(ast_manager& m, lbool expected_outcome, expr* fml, char const* option) {
// enable_trace("bit2int");
//enable_trace("gomory_cut");
enable_trace("final_check_arith");
enable_trace("arith_final_check");
//enable_trace("arith_branching");
enable_trace("theory_arith_int");
enable_trace("presburger");
enable_trace("quant_elim");
// enable_trace("arith_simplifier_plugin");
// enable_trace("non_linear");
// enable_trace("gomory_cut_detail");
// enable_trace("arith");
// enable_trace("bv");
// enable_trace("after_search");
// enable_trace("bv_bit_prop");
smt_params params;
// params.m_quant_elim = true;
std::cout << mk_pp(fml, m) << "\n";
qe::expr_quant_elim qe(m, params);
expr_ref result(m);
qe(m.mk_true(), fml, result);
std::cout << " -> " << mk_pp(result, m) << " " << expected_outcome << "\n";
if (expected_outcome == l_true && !m.is_true(result)) {
std::cout << "ERROR: expected true, instead got " << mk_pp(result, m) << "\n";
//exit(-1);
}
if (expected_outcome == l_false && !m.is_false(result)) {
std::cout << "ERROR: expected false, instead got " << mk_pp(result, m) << "\n";
//exit(-1);
}
}
#endif
static void test_formula(lbool expected_outcome, char const* fml) {
ast_manager m;
reg_decl_plugins(m);
// No-op requires SMTLIB2
#if 0
scoped_ptr<smtlib::parser> parser = smtlib::parser::create(m);
parser->initialize_smtlib();
std::ostringstream buffer;
buffer << "(benchmark presburger :status unknown :logic AUFLIA :extrapreds ((p1) (p2) (p3)) "
<< ":extrafuns ((a Int) (b Int))\n"
<< ":extrapreds ((p) (q) (r))\n"
<< ":datatypes ((list (nil) (cons (hd Int) (tl list))))\n"
<< ":datatypes ((cell (cnil) (ccons (car cell) (cdr cell))))\n"
<< ":extrasorts (U)\n"
<< ":extrafuns ((f U U))\n"
<< ":formula " << fml << ")";
parser->parse_string(buffer.str().c_str());
smtlib::benchmark* b = parser->get_benchmark();
smtlib::theory::expr_iterator it = b->begin_formulas();
smtlib::theory::expr_iterator end = b->end_formulas();
for (; it != end; ++it) {
test_qe(m, expected_outcome, *it, 0);
}
#endif
}
void tst_quant_elim() {
disable_debug("heap");
test_formula(l_undef, "(exists ((p1 Bool) (q1 Bool) (r1 Bool))\
(and (or (not p1) (not q1) r1)\
(or (and (not p) (not q) (not p1) q1)\
(and (not p) q p1 (not q1))\
(and p (not q) p1 q1)\
(and p q p1 q1))\
(or (and (not r) (not r1))\
(and (= p p1) (= q q1) r r1)\
(and (not (and (= p p1) (= q q1))) (not (= r r1))))))");
test_formula(l_false,"(forall (x Int) (y Int) (or (= x 0) (< (* 5 y) (* 6 x)) (> (* 5 y) (* 6 x))))");
test_formula(l_false, "(forall (a Int) (b Int) (exists (x Int) (and (< a (* 20 x)) (< (* 20 x) b))))");
test_formula(l_undef, "(exists (u U) (= (f u) u))");
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 3 u) (* 7 v))))))))");
test_formula(l_true, "(forall (x Int) (y Int) (implies (= (* 6 x) (* 5 y)) (exists (d Int) (= y (* 3 d)))))");
test_formula(l_undef, "(exists (x Int) (= (- a (mod x 4)) 0))");
// return;
// test_formula(l_true, "(exists (x Int) (y Int) (= 1 (+ (* 5 x) (* 3 y))))");
test_formula(l_undef, "(exists (a Bool) (b Bool) (or (and p1 a) (and p2 (not b))))");
test_formula(l_false,
"(forall (x Int) (q1 Int) (q2 Int) (r1 Int) (r2 Int) "
" (implies "
" (and (< x 4699) "
" (= (* 2622 x) (+ (* 65536 q1) r1)) "
" (<= 0 q1) "
" (<= 0 r1) "
" (< r1 65536) "
" (= x (+ (* 100 q2) r2)) "
" (<= 0 q2) "
" (<= 0 r2) "
" (< r2 100)) "
" (= q1 q2)))");
test_formula(l_undef,
"(forall (l list) (or (= l nil) (exists (x Int) (ll list) (= l (cons x ll)))))");
test_formula(l_false, "(exists (x Real) (forall (y Real) (>= x y)))");
test_formula(l_false, "(exists (x Real) (forall (y Real) (> x y)))");
test_formula(l_false, "(exists (x Real) (forall (y Real) (< x y)))");
test_formula(l_false, "(exists (x Real) (forall (y Real) (<= x y)))");
test_formula(l_true, "(exists (x Real) (exists (y Real) (< x y)))");
test_formula(l_true, "(exists (x Real) (exists (y Real) (<= x y)))");
test_formula(l_true, "(exists (x Real) (exists (y Real) (>= x y)))");
test_formula(l_true, "(exists (x Real) (exists (y Real) (> x y)))");
test_formula(l_true, "(forall (x Real) (exists (y Real) (< x y)))");
test_formula(l_true, "(forall (x Real) (exists (y Real) (<= x y)))");
test_formula(l_true, "(forall (x Real) (exists (y Real) (>= x y)))");
test_formula(l_true, "(forall (x Real) (exists (y Real) (> x y)))");
test_formula(l_false, "(forall (x Real) (forall (y Real) (< x y)))");
test_formula(l_false, "(forall (x Real) (forall (y Real) (<= x y)))");
test_formula(l_false, "(forall (x Real) (forall (y Real) (>= x y)))");
test_formula(l_false, "(forall (x Real) (forall (y Real) (> x y)))");
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 3 u) (* 5 v))))))))");
test_formula(l_false, "(forall (d Int) (implies (>= d 0) (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= d (+ (* 3 x) (* 5 y)))))))");
test_formula(l_true, "(forall (y Int) (implies (exists (d Int) (= y (* 6 d))) (exists (d Int) (= y (* 2 d)))))");
test_formula(l_true, "(forall (y Int) (implies (exists (d Int) (= y (* 65 d))) (exists (d Int) (= y (* 5 d)))))");
test_formula(l_true,
"(exists (z Int) (forall (w Int) (exists (x Int) (y Int) "
" (or (and (< (+ (* 3 x) w) 2) (< 1 (- (+ (* 2 x) z) w))) "
" (and (< z (* 2 y)) (> z y))))))");
test_formula(l_true, "(exists (x Int) (y Int) (and (> x 0) (>= y 0) (= 1 (- (* 3 x) (* 5 y)))))");
test_formula(l_true,
"(exists (a Int) (b Int) "
" (and (not (= a 1)) (= a b) (or (= a (* 2 b)) (= (* 2 b) (+ 1 (* 3 a))))))");
test_formula(l_true,
"(forall (x Int) (iff (and (not (= 0 (mod x 2))) (= 0 (mod (- x 1) 3))) "
" (or (= 0 (mod (- x 1) 12)) (= 0 (mod (- x 7) 12)))))");
test_formula(l_false, "(exists (x Int) (and (< (* 3 x) 2) (< 1 (* 2 x))))");
test_formula(l_true, "(forall (x Int) (y Int) (or (= 0 (mod x 5)) (not (= (* 6 x) (* 5 y)))))");
test_formula(l_false, "(forall (x Int) (exists (y Int) (= x (* 2 y))))");
test_formula(l_false,
"(forall (x Int) "
" (implies (not (= 0 (mod x 2))) "
" (or (= 0 (mod (- x 1) 4)) "
" (= 0 (mod (- x 1) 8)) "
" (= 0 (mod (- x 3) 8)) "
" (= 0 (mod (- x 1) 6)) "
" (= 0 (mod (- x 1) 14)) "
" (= 0 (mod (- x 9) 14)) "
" (= 0 (mod (- x 11) 14)) "
" (= 0 (mod (- x 5) 24)) "
" (= 0 (mod (- x 11) 24))))) ");
test_formula(l_true,
"(forall (x Int) (iff (and (not (= 0 (mod x 2))) (= 0 (mod (- x 1) 3))) "
" (or (= 0 (mod (- x 1) 12)) (= 0 (mod (- x 7) 12)))))");
test_formula(l_false,
"(forall (d Int) (c Int) (b Int) "
" (and (= c 0) (= d (* b c)) (= d 0)))");
//return;
test_formula(l_undef, "(exists (k!12 Int) (k!11 Int) (and (= (ite (= k!11 0) 0 k!11) k!11) (not (= (ite (= k!12 (+ 1)) 1 0) 0))))");
//return;
test_formula(l_false,
"(forall (a Int) (b Int) (x Int) (y Int) (z Int) "
" (implies (and (= (+ a 2) b) (= x (+ 1 (- b a))) (= y (- b 2)) (= z 3)) false))");
test_formula(l_false,
"(exists (a Int) (b Int) "
" (and (> a 1) (> b 1) (= a b) (or (= a (* 2 b)) (= (* 2 b) (+ 1 (* 3 a))))))");
test_formula(l_true, "(forall (d Int) (implies true (exists (x Int) (y Int) (and true true (= d (+ (* 3 x) (* 5 y)))))))");
// This one takes forever without bit-vectors
test_formula(l_true, "(forall (d Int) (implies (>= d 8) (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= d (+ (* 3 x) (* 5 y)))))))");
test_formula(l_true, "(forall (d Int) (implies (>= d 0) (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= d (- (* 3 x) (* 5 y)))))))");
test_formula(l_false, "(exists (x Int) (y Int) (z Int) (= 1 (- (* 4 x) (* 6 y))))");
//return;
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 3 u) (* 8 v))))))))");
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 3 u) (* 8 v))))))))");
#if 0
// too slow.
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 7 u) (* 8 v))))))))");
#endif
test_formula(l_true, "(forall (x Int) (exists (y Int) (and (<= (* 2 y) x) (< x (* 2 (+ y 1))))))");
test_formula(l_false, "(exists (x Int) (y Int) (and (> y 0) (> y (* 2 x)) (< y (+ x 2)) (= 0 (mod y 2))))");
test_formula(l_false, "(exists (x Int) (and (< (* 3 x) 3) (< 1 (* 2 x))))");
test_formula(l_true, "(exists (x Int) (and (< (* 3 x) 4) (< 1 (* 2 x))))");
test_formula(l_false, "(exists (x Int) (and (< (+ (* 3 x) 1) 10) (> (- (* 7 x) 6) 7) (= 0 (mod x 3))))");
test_formula(l_false, "(exists (x Int) (y Int) (and (< (- 1 (* 5 y)) x) (< (+ 1 y) (* 13 x)) (< (+ x 2) 0) (> y 0)))");
test_formula(l_false, "(exists (x Int) (y Int) (and (< (- 1 (* 5 y)) x) (< (+ 1 y) (* 13 x)) (< x -2)))");
test_formula(l_true, "(exists (w Int) (z Int) (y Int) (x Int) (and (< (- 1 (* 5 y)) (+ x (* 2 z))) (< (+ 1 y w (* -4 z)) (* 13 x)) (< x -2) (> z 0)))");
test_formula(l_true,
"(forall (w Int) "
" (exists (z Int) (y Int) (x Int) "
" (and (< (- 1 (* 5 y)) (+ x (* 2 z))) "
" (< (- (+ 1 y) (* 4 z)) (* 13 x)) "
" (< x -2) (> z 0) (< x 10)))) ");
test_formula(l_false,
"(forall (d Int) (c Int) (b Int) "
" (and (= c 0) (= d (* b c)) (= d 4)))");
test_formula(l_undef,
"(exists (d Int) (c Int) (b Int) "
" (and (= c 0) (= d (* b c)) (= d 0)))");
test_formula(l_undef,
"(exists (d Int) (c Int) (b Int) "
" (and (= c 0) (= d (* b c)) (= d 4)))");
// Tests from Harrison's HOL-light version of Cooper.
test_formula(l_true, "(forall (x Int) (y Int) (not (= (+ 1 (* 2 x)) (* 2 y))))");
test_formula(l_false, "(exists (x Int) (y Int) (= 1 (- (* 4 x) (* 6 y))))");
// "(forall (x Int) (implies (< b x) (<= a x)))"
// "(forall (x Int) (implies (< b x) (< a x)))"
test_formula(l_false, "(forall (d Int) (implies (>= d 0) (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= d (+ (* 3 x) (* 5 y)))))))");
test_formula(l_true, "(forall (d Int) (implies true (exists (x Int) (y Int) (and true true (= d (+ (* 3 x) (* 5 y)))))))");
// This one takes forever without bit-vectors
test_formula(l_true, "(forall (d Int) (implies (>= d 8) (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= d (+ (* 3 x) (* 5 y)))))))");
test_formula(l_true, "(forall (d Int) (implies (>= d 0) (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= d (- (* 3 x) (* 5 y)))))))");
test_formula(l_true, "(exists (x Int) (y Int) (and (> x 0) (>= y 0) (= 1 (- (* 3 x) (* 5 y)))))");
test_formula(l_false, "(exists (x Int) (y Int) (z Int) (= 1 (- (* 4 x) (* 6 y))))");
// "(forall (x Int) (implies (< b (* 3 x)) (a < (* 3 x))))"
test_formula(l_false, "(forall (x Int) (y Int) (implies (<= x y) (< (+ 1 (* 2 x)) (* 2 y))))");
test_formula(l_true, "(forall (x Int) (y Int) (z Int) (implies (= (+ 1 (* 2 x)) (* 2 y)) (> (+ x y z) 129)))");
// Formula examples from Cooper's paper.
test_formula(l_true, "(forall (a Int) (exists (b Int) (or (< a (+ (* 4 b) (* 3 a))) (and (not (< a b)) (> a (+ b 1))))))");
test_formula(l_false, "(exists (y Int) (forall (x Int) (and (> (+ x (* 5 y)) 1) (> (- (* 13 x) y) 1) (< (+ x 2) 0))))");
// Harrison's formulas:
test_formula(l_false, "(forall (x Int) (y Int) (implies (and (>= x 0) (>= y 0)) (or (< (- (* 12 x) (* 8 y)) 0) (> (- (* 12 x) (* 8 y)) 2))))");
// test_formula(l_true, "(exists (x Int) (y Int) (= 1 (+ (* 5 x) (* 3 y))))");
test_formula(l_false, "(exists (x Int) (y Int) (= 1 (+ (* 5 x) (* 10 y))))");
test_formula(l_true, "(exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= 1 (- (* 5 x) (* 6 y)))))");
test_formula(l_true, "(exists (x Int) (y Int) (z Int) (w Int) (= 1 (+ (* 2 w) (* 3 x) (* 4 y) (* 5 z))))");
test_formula(l_true, "(exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= 1 (- (* 5 x) (* 3 y)))))");
test_formula(l_true, "(exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= 1 (- (* 3 x) (* 5 y)))))");
test_formula(l_false,"(exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= 1 (- (* 6 x) (* 3 y)))))");
test_formula(l_true, "(forall (x Int) (y Int) (or (= 0 (mod x 5)) (= 0 (mod y 6)) (not (= (* 6 x) (* 5 y)))))");
test_formula(l_false,"(forall (x Int) (y Int) (or (not (= (* 6 x) (* 5 y)))))");
// Positive variant of the Bezout theorem (see the exercise). *)
test_formula(l_true, "(forall (z Int) (implies (> z 7) (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= (+ (* 3 x) (* 5 y)) z)))))");
test_formula(l_false,"(forall (z Int) (implies (> z 2) (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= (+ (* 3 x) (* 5 y)) z)))))");
test_formula(l_true,
"(forall (z Int) (implies (<= z 7) "
" (iff (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= z (+ (* 3 x) (* 5 y))))) "
" (not (exists (x Int) (y Int) (and (>= x 0) (>= y 0) (= (- 7 z) (+ (* 3 x) (* 5 y))))))))) ");
// Basic result about congruences.
test_formula(l_true,
"(forall (x Int) "
" (iff (and (not (exists (m Int) (= x (* 2 m)))) (exists (m Int) (= x (+ (* 3 m) 1)))) "
" (or (exists (m Int) (= x (+ (* 12 m) 1))) (exists (m Int) (= x (+ (* 12 m) 7))))))");
// Inspired by the Collatz conjecture.
test_formula(l_false,
"(forall (a Int) (b Int) (x Int) (y Int) (z Int) "
" (implies (and (= (+ a 2) b) (= x (+ 1 (- b a))) (= y (- b 2)) (= z 3)) false))");
test_formula(l_true,
"(exists (a Int) (b Int) "
" (and (not (= a 1)) (= a b) (or (= a (* 2 b)) (= (* 2 b) (+ 1 (* 3 a))))))");
test_formula(l_false,
"(exists (a Int) (b Int) "
" (and (> a 1) (> b 1) (= a b) (or (= a (* 2 b)) (= (* 2 b) (+ 1 (* 3 a))))))");
test_formula(l_false,
"(exists (a Int) (b Int) "
" (and (> a 1) (> b 1) "
" (or (= a (* 2 b)) (= (* 2 b) (+ 1 (* 3 a)))) "
" (or (= b (* 2 a)) (= (* 2 a) (+ 1 (* 3 b))))))");
#if 0
// Bob Constable's "stamp problem".
test_formula(l_true,
"(forall (x Int) (implies (>= x 8) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 3 u) (* 5 v)))))))");
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 3 u) (* 5 v))))))))");
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 3 u) (* 7 v))))))))");
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 3 u) (* 8 v))))))))");
test_formula(l_true,
"(exists (l Int) (forall (x Int) (implies (>= x l) "
" (exists (u Int) (v Int) (and (>= u 0) (>= v 0) (= x (+ (* 7 u) (* 8 v))))))))");
#endif
// Example from reciprocal mult: (2622 * x)>>16 = x/100 within a range.
test_formula(l_true,
"(forall (x Int) (y Int) "
" (iff (exists (d Int) (= (+ x y) (* 2 d))) "
" (iff (exists (d Int) (= x (* 2 d))) (exists (d Int) (= y (* 2 d))))))");
test_formula(l_true,
"(forall (n Int) "
" (implies (and (< 0 n) (< n 2400)) "
" (or (and (<= n 2) (<= 2 (* 2 n))) "
" (and (<= n 3) (<= 3 (* 2 n))) "
" (and (<= n 5) (<= 5 (* 2 n))) "
" (and (<= n 7) (<= 7 (* 2 n))) "
" (and (<= n 13) (<= 13 (* 2 n))) "
" (and (<= n 23) (<= 23 (* 2 n))) "
" (and (<= n 43) (<= 43 (* 2 n))) "
" (and (<= n 83) (<= 83 (* 2 n))) "
" (and (<= n 163) (<= 163 (* 2 n))) "
" (and (<= n 317) (<= 317 (* 2 n))) "
" (and (<= n 631) (<= 631 (* 2 n))) "
" (and (<= n 1259) (<= 1259 (* 2 n))) "
" (and (<= n 2503) (<= 2503 (* 2 n)))))) ");
memory::finalize();
#ifdef _WINDOWS
_CrtDumpMemoryLeaks();
#endif
exit(0);
}