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improve quantifier elimination for arithmetic

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This update changes the handling of mod and adds support for nested div terms.

Simple use cases that are handled using small results are given below.

```
(declare-const x Int)
(declare-const y Int)
(declare-const z Int)
(assert (exists ((x Int)) (and (<= y (* 10 x)) (<= (* 10 x) z))))
(apply qe2)
(reset)

(declare-const y Int)
(assert (exists ((x Int)) (and (> x 0) (= (div x 41) y))))
(apply qe2)
(reset)

(declare-const y Int)
(assert (exists ((x Int)) (= (mod x 41) y)))
(apply qe2)
(reset)
```

The main idea is to introduce definition rows for mod/div terms.
Elimination of variables under mod/div is defined by rewriting the variable to multiples of the mod/divisior and remainder.

The functionality is disabled in this push.
This commit is contained in:
Nikolaj Bjorner 2022-08-12 10:20:43 -04:00
parent 786280c646
commit 03385bf78d
3 changed files with 1795 additions and 1432 deletions
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2864
src/math/simplex/model_based_opt.cpp
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38
src/math/simplex/model_based_opt.h
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@ -30,7 +30,9 @@ namespace opt {
t_eq,
t_lt,
t_le,
t_mod
t_divides,
t_mod,
t_div
};
@ -57,6 +59,7 @@ namespace opt {
ineq_type m_type; // inequality type
rational m_value; // value of m_vars + m_coeff under interpretation of m_var2value.
bool m_alive; // rows can be marked dead if they have been processed.
unsigned m_id; // variable defined by row (used for mod_t and div_t)
void reset() { m_vars.reset(); m_coeff.reset(); m_value.reset(); }
row& normalize();
@ -85,9 +88,9 @@ namespace opt {
static const unsigned m_objective_id = 0;
vector<unsigned_vector> m_var2row_ids;
vector<rational> m_var2value;
bool_vector m_var2is_int;
bool_vector m_var2is_int;
vector<var> m_new_vars;
unsigned_vector m_lub, m_glb, m_mod;
unsigned_vector m_lub, m_glb, m_divides, m_mod, m_div;
unsigned_vector m_above, m_below;
unsigned_vector m_retired_rows;
@ -122,14 +125,18 @@ namespace opt {
void sub(unsigned dst, rational const& c);
void del_var(unsigned dst, unsigned x);
void set_row(unsigned row_id, vector<var> const& coeffs, rational const& c, rational const& m, ineq_type rel);
void set_row(unsigned row_id, vector<var> const& coeffs, rational const& c, rational const& m, ineq_type rel);
void add_lower_bound(unsigned x, rational const& lo);
void add_upper_bound(unsigned x, rational const& hi);
void add_constraint(vector<var> const& coeffs, rational const& c, rational const& m, ineq_type r);
void replace_var(unsigned row_id, unsigned x, rational const& A, unsigned y, rational const& B);
void replace_var(unsigned row_id, unsigned x, rational const& A, unsigned y, rational const& B, unsigned z);
void replace_var(unsigned row_id, unsigned x, rational const& C);
void normalize(unsigned row_id);
@ -138,7 +145,7 @@ namespace opt {
unsigned new_row();
unsigned copy_row(unsigned row_id);
unsigned copy_row(unsigned row_id, unsigned excl = UINT_MAX);
rational n_sign(rational const& b) const;
@ -150,8 +157,12 @@ namespace opt {
def solve_for(unsigned row_id, unsigned x, bool compute_def);
def solve_mod(unsigned x, unsigned_vector const& mod_rows, bool compute_def);
def solve_divides(unsigned x, unsigned_vector const& divide_rows, bool compute_def);
def solve_mod(unsigned x, unsigned_vector const& divide_rows, bool compute_def);
def solve_div(unsigned x, unsigned_vector const& divide_rows, bool compute_def);
bool is_int(unsigned x) const { return m_var2is_int[x]; }
void retire_row(unsigned row_id);
@ -173,6 +184,17 @@ namespace opt {
// add a divisibility constraint. The row should divide m.
void add_divides(vector<var> const& coeffs, rational const& c, rational const& m);
// add sub-expression for modulus
// v = add_mod(coeffs, m) means
// v = coeffs mod m
unsigned add_mod(vector<var> const& coeffs, rational const& c, rational const& m);
// add sub-expression for div
// v = add_div(coeffs, m) means
// v = coeffs div m
unsigned add_div(vector<var> const& coeffs, rational const& c, rational const& m);
// Set the objective function (linear).
void set_objective(vector<var> const& coeffs, rational const& c);