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z3-z3-4.12.6.src.qe.mbp.mbp_arrays.cpp Maven / Gradle / Ivy
/*++
Copyright (c) 2015 Microsoft Corporation
Module Name:
mbp_arrays.cpp
Abstract:
Model based projection for arrays
Author:
Nikolaj Bjorner (nbjorner) 2015-06-13
Revision History:
--*/
#include "util/lbool.h"
#include "ast/rewriter/rewriter_def.h"
#include "ast/expr_functors.h"
#include "ast/for_each_expr.h"
#include "ast/rewriter/expr_safe_replace.h"
#include "ast/rewriter/th_rewriter.h"
#include "ast/ast_util.h"
#include "ast/ast_pp.h"
#include "ast/array_peq.h"
#include "model/model_evaluator.h"
#include "qe/mbp/mbp_arrays.h"
#include "qe/mbp/mbp_term_graph.h"
namespace mbp {
static bool is_eq(expr_ref_vector const& xs, expr_ref_vector const& ys) {
for (unsigned i = 0; i < xs.size(); ++i) if (xs[i] != ys[i]) return false;
return true;
}
static expr_ref mk_eq(expr_ref_vector const& xs, expr_ref_vector const& ys) {
ast_manager& m = xs.get_manager();
expr_ref_vector eqs(m);
for (unsigned i = 0; i < xs.size(); ++i) eqs.push_back(m.mk_eq(xs[i], ys[i]));
return mk_and(eqs);
}
/**
* 1. Extract all equalities (store (store .. (store x i1 v1) i2 v2) .. ) = ...
* where x appears and the equalities do not evaluate to false in the current model.
* 2. Track them as partial equivalence relations.
* 3. Sort them according to nesting depth.
* 4. Solve for x by potentially introducing fresh variables.
* Thus, (store x i v) = y, then
* x = (store y i w), (select y i) = v, where w is fresh and evaluates to (select x i).
* Generally, equalities are tracked as x =_idxs y, where x, y are equal up to idxs.
*/
class array_project_eqs_util {
ast_manager& m;
array_util m_arr_u;
model_ref M;
model_evaluator* m_mev;
app_ref m_v; // array var to eliminate
ast_mark m_has_stores_v; // has stores for m_v
expr_ref m_subst_term_v; // subst term for m_v
expr_safe_replace m_true_sub_v; // subst for true equalities
expr_safe_replace m_false_sub_v; // subst for false equalities
expr_ref_vector m_aux_lits_v;
expr_ref_vector m_idx_lits_v;
app_ref_vector m_aux_vars;
void reset_v () {
m_v = nullptr;
m_has_stores_v.reset ();
m_subst_term_v = nullptr;
m_true_sub_v.reset ();
m_false_sub_v.reset ();
m_aux_lits_v.reset ();
m_idx_lits_v.reset ();
}
void reset () {
M = nullptr;
m_mev = nullptr;
reset_v ();
m_aux_vars.reset ();
}
/**
* find all array equalities on m_v or containing stores on/of m_v
*
* also mark terms containing stores on/of m_v
*/
void find_arr_eqs (expr_ref const& fml, app_ref_vector& eqs) {
if (!is_app (fml)) return;
ast_mark done;
ptr_vector todo;
todo.push_back (to_app (fml));
while (!todo.empty ()) {
app* a = todo.back ();
if (done.is_marked (a)) {
todo.pop_back ();
continue;
}
bool all_done = true;
bool args_have_stores = false;
for (expr * arg : *a) {
if (!is_app (arg)) continue;
if (!done.is_marked (arg)) {
all_done = false;
todo.push_back (to_app (arg));
}
else if (!args_have_stores && m_has_stores_v.is_marked (arg)) {
args_have_stores = true;
}
}
if (!all_done) continue;
todo.pop_back ();
// mark if a has stores
if ((!m_arr_u.is_select (a) && args_have_stores) ||
(m_arr_u.is_store (a) && (a->get_arg (0) == m_v))) {
m_has_stores_v.mark (a, true);
TRACE ("qe",
tout << "has stores:\n";
tout << mk_pp (a, m) << "\n";
);
}
// check if a is a relevant array equality
expr * a0 = nullptr, *a1 = nullptr;
if (m.is_eq (a, a0, a1)) {
if (a0 == m_v || a1 == m_v ||
(m_arr_u.is_array (a0) && m_has_stores_v.is_marked (a))) {
eqs.push_back (a);
}
}
// else, we can check for disequalities and handle them using extensionality,
// but it's not necessary
done.mark (a, true);
}
}
/**
* factor out select terms on m_v using fresh consts
*/
void factor_selects (app_ref& fml) {
expr_map sel_cache (m);
ast_mark done;
ptr_vector todo;
expr_ref_vector pinned (m); // to ensure a reference
todo.push_back (fml);
while (!todo.empty ()) {
app* a = todo.back ();
if (done.is_marked (a)) {
todo.pop_back ();
continue;
}
expr_ref_vector args (m);
bool all_done = true;
for (expr * arg : *a) {
if (!is_app (arg)) {
args.push_back(arg);
}
else if (!done.is_marked (arg)) {
all_done = false;
todo.push_back (to_app (arg));
}
else if (all_done) { // all done so far..
expr* arg_new = nullptr; proof* pr;
sel_cache.get (arg, arg_new, pr);
if (!arg_new) {
arg_new = arg;
}
args.push_back (arg_new);
}
}
if (!all_done) continue;
todo.pop_back ();
expr_ref a_new (m.mk_app (a->get_decl (), args.size (), args.data ()), m);
// if a_new is select on m_v, introduce new constant
if (m_arr_u.is_select (a) &&
(args.get (0) == m_v || m_has_stores_v.is_marked (args.get (0)))) {
sort* val_sort = get_array_range (m_v->get_sort());
app_ref val_const (m.mk_fresh_const ("sel", val_sort), m);
m_aux_vars.push_back (val_const);
// extend M to include val_const
expr_ref val = (*m_mev)(a_new);
M->register_decl (val_const->get_decl (), val);
// add equality
m_aux_lits_v.push_back (m.mk_eq (val_const, a_new));
// replace select by const
a_new = val_const;
}
if (a != a_new) {
sel_cache.insert (a, a_new, nullptr);
pinned.push_back (a_new);
}
done.mark (a, true);
}
expr* res = nullptr; proof* pr;
sel_cache.get (fml, res, pr);
if (res) {
fml = to_app (res);
}
}
/**
* convert partial equality expression p_exp to an equality by
* recursively adding stores on diff indices
*
* add stores on lhs or rhs depending on whether stores_on_rhs is false/true
*/
void convert_peq_to_eq (expr* p_exp, app_ref& eq, bool stores_on_rhs = true) {
peq p (to_app (p_exp), m);
app_ref_vector diff_val_consts (m);
eq = p.mk_eq (diff_val_consts, stores_on_rhs);
m_aux_vars.append (diff_val_consts);
// extend M to include diff_val_consts
vector I;
expr_ref arr = p.lhs ();
p.get_diff_indices (I);
expr_ref val (m);
unsigned num_diff = diff_val_consts.size ();
SASSERT (num_diff == I.size ());
for (unsigned i = 0; i < num_diff; i++) {
// mk val term
ptr_vector sel_args;
sel_args.push_back (arr);
sel_args.append(I[i].size(), I[i].data());
expr_ref val_term (m_arr_u.mk_select (sel_args), m);
// evaluate and assign to ith diff_val_const
val = (*m_mev)(val_term);
M->register_decl (diff_val_consts.get (i)->get_decl (), val);
}
}
void find_subst_term (app* eq) {
SASSERT(m.is_eq(eq));
vector empty;
app_ref p_exp = mk_peq (eq->get_arg (0), eq->get_arg (1), empty, m);
bool subst_eq_found = false;
while (true) {
TRACE ("qe", tout << "processing peq:\n" << p_exp << "\n";);
peq p (p_exp, m);
expr_ref lhs = p.lhs(), rhs = p.rhs();
if (!m_has_stores_v.is_marked (lhs)) {
std::swap (lhs, rhs);
}
if (m_has_stores_v.is_marked (lhs) && m_arr_u.is_store(lhs)) {
/** project using the equivalence:
*
* (store(arr0,idx,x) ==I arr1) <->
*
* (idx \in I => (arr0 ==I arr1)) /\
* (idx \not\in I => (arr0 ==I+idx arr1) /\ (arr1[idx] == x)))
*/
vector I;
expr_ref_vector idxs (m);
p.get_diff_indices (I);
app* a_lhs = to_app (lhs);
expr* arr0 = a_lhs->get_arg (0);
idxs.append(a_lhs->get_num_args() - 2, a_lhs->get_args() + 1);
expr* x = a_lhs->get_arg (2);
expr* arr1 = rhs;
// check if (idx \in I) in M
bool idx_in_I = false;
expr_ref_vector idx_diseq (m);
if (!I.empty ()) {
expr_ref_vector vals = (*m_mev)(idxs);
for (unsigned i = 0; i < I.size () && !idx_in_I; i++) {
if (is_eq(idxs, I.get(i))) {
idx_in_I = true;
}
else {
expr_ref idx_eq = mk_eq(idxs, I[i]);
expr_ref_vector vals1 = (*m_mev)(I[i]);
if (is_eq(vals, vals1)) {
idx_in_I = true;
m_idx_lits_v.push_back (idx_eq);
}
else {
idx_diseq.push_back (m.mk_not (idx_eq));
}
}
}
}
if (idx_in_I) {
TRACE ("qe",
tout << "store index in diff indices:\n";
tout << mk_pp (m_idx_lits_v.back (), m) << "\n";
);
// arr0 ==I arr1
p_exp = mk_peq (arr0, arr1, I, m);
TRACE ("qe",
tout << "new peq:\n";
tout << mk_pp (p_exp, m) << "\n";
);
}
else {
m_idx_lits_v.append (idx_diseq);
// arr0 ==I+idx arr1
I.push_back (idxs);
p_exp = mk_peq (arr0, arr1, I, m);
TRACE ("qe", tout << "new peq:\n" << p_exp << "\n"; );
// arr1[idx] == x
ptr_vector sel_args;
sel_args.push_back (arr1);
sel_args.append(idxs.size(), idxs.data());
expr_ref arr1_idx (m_arr_u.mk_select (sel_args), m);
expr_ref eq (m.mk_eq (arr1_idx, x), m);
m_aux_lits_v.push_back (eq);
TRACE ("qe",
tout << "new eq:\n";
tout << mk_pp (eq, m) << "\n";
);
}
}
else if (lhs == rhs) { // trivial peq (a ==I a)
break;
}
else if (lhs == m_v || rhs == m_v) {
subst_eq_found = true;
TRACE ("qe",
tout << "subst eq found!\n";
);
break;
}
else {
UNREACHABLE ();
}
}
// factor out select terms on m_v from p_exp using fresh constants
if (subst_eq_found) {
factor_selects (p_exp);
TRACE ("qe",
tout << "after factoring selects:\n";
tout << mk_pp (p_exp, m) << "\n";
for (unsigned i = m_aux_lits_v.size () - m_aux_vars.size (); i < m_aux_lits_v.size (); i++) {
tout << mk_pp (m_aux_lits_v.get (i), m) << "\n";
}
);
// find subst_term
bool stores_on_rhs = true;
app* a = to_app (p_exp);
if (a->get_arg (1) == m_v) {
stores_on_rhs = false;
}
app_ref eq (m);
convert_peq_to_eq (p_exp, eq, stores_on_rhs);
m_subst_term_v = eq->get_arg (1);
TRACE ("qe",
tout << "subst term found:\n";
tout << mk_pp (m_subst_term_v, m) << "\n";
);
}
}
/**
* compute nesting depths of stores on m_v in true_eqs, as follows:
* 0 if m_v appears on both sides of equality
* 1 if equality is (m_v=t)
* 2 if equality is (store(m_v,i,v)=t)
* ...
*/
unsigned get_nesting_depth(app* eq) {
expr* lhs = nullptr, *rhs = nullptr;
VERIFY(m.is_eq(eq, lhs, rhs));
bool lhs_has_v = (lhs == m_v || m_has_stores_v.is_marked (lhs));
bool rhs_has_v = (rhs == m_v || m_has_stores_v.is_marked (rhs));
app* store = nullptr;
SASSERT (lhs_has_v || rhs_has_v);
if (!lhs_has_v && is_app(rhs)) {
store = to_app (rhs);
}
else if (!rhs_has_v && is_app(lhs)) {
store = to_app (lhs);
}
else {
// v appears on both sides -- trivial equality
// put it in the beginning to simplify it away
return 0;
}
unsigned nd = 0; // nesting depth
for (nd = 1; m_arr_u.is_store (store); nd++, store = to_app (store->get_arg (0))) {
/* empty */ ;
}
if (store != m_v) {
TRACE("qe", tout << "not a store " << mk_pp(eq, m) << " " << lhs_has_v << " " << rhs_has_v << " " << mk_pp(m_v, m) << "\n";);
return UINT_MAX;
}
return nd;
}
struct compare_nd {
bool operator()(std::pair const& x, std::pair const& y) const {
return x < y;
}
};
/**
* try to substitute for m_v, using array equalities
*
* compute substitution term and aux lits
*/
bool project (expr_ref const& fml) {
app_ref_vector eqs (m);
svector > true_eqs;
find_arr_eqs (fml, eqs);
TRACE ("qe", tout << "array equalities:\n" << eqs << "\n";);
// evaluate eqs in M
for (app * eq : eqs) {
TRACE ("qe", tout << "array equality:\n" << mk_pp (eq, m) << "\n"; );
if (m_mev->is_false(eq)) {
m_false_sub_v.insert (eq, m.mk_false());
}
else {
true_eqs.push_back (std::make_pair(get_nesting_depth(eq), eq));
}
}
std::sort(true_eqs.begin(), true_eqs.end(), compare_nd());
DEBUG_CODE(for (unsigned i = 0; i + 1 < true_eqs.size(); ++i) SASSERT(true_eqs[i].first <= true_eqs[i+1].first););
// search for subst term
for (unsigned i = 0; !m_subst_term_v && i < true_eqs.size(); i++) {
app* eq = true_eqs[i].second;
m_true_sub_v.insert (eq, m.mk_true ());
// try to find subst term
find_subst_term (eq);
}
return true;
}
void mk_result (expr_ref& fml) {
th_rewriter rw(m);
rw (fml);
// add in aux_lits and idx_lits
expr_ref_vector lits (m);
// TODO: eliminate possible duplicates, especially in idx_lits
// theory rewriting is a possibility, but not sure if it
// introduces unwanted terms such as ite's
lits.append (m_idx_lits_v);
lits.append (m_aux_lits_v);
lits.push_back (fml);
fml = mk_and(lits);
if (m_subst_term_v) {
m_true_sub_v.insert (m_v, m_subst_term_v);
m_true_sub_v (fml);
}
else {
m_true_sub_v (fml);
m_false_sub_v (fml);
}
rw(fml);
SASSERT (!m.is_false (fml));
}
public:
array_project_eqs_util (ast_manager& m):
m (m),
m_arr_u (m),
m_v (m),
m_subst_term_v (m),
m_true_sub_v (m),
m_false_sub_v (m),
m_aux_lits_v (m),
m_idx_lits_v (m),
m_aux_vars (m)
{}
void operator () (model& mdl, app_ref_vector& arr_vars, expr_ref& fml, app_ref_vector& aux_vars) {
reset ();
model_evaluator mev(mdl);
mev.set_model_completion(true);
M = &mdl;
m_mev = &mev;
unsigned j = 0;
for (unsigned i = 0; i < arr_vars.size (); i++) {
reset_v ();
m_v = arr_vars.get (i);
if (!m_arr_u.is_array (m_v)) {
TRACE ("qe",
tout << "not an array variable: " << m_v << "\n";
);
aux_vars.push_back (m_v);
continue;
}
TRACE ("qe", tout << "projecting equalities on variable: " << m_v << "\n"; );
if (project (fml)) {
mk_result (fml);
contains_app contains_v (m, m_v);
if (!m_subst_term_v || contains_v (m_subst_term_v)) {
arr_vars[j++] = m_v;
}
TRACE ("qe", tout << "after projection: \n" << fml << "\n";);
}
else {
IF_VERBOSE(2, verbose_stream() << "can't project:" << m_v << "\n";);
TRACE ("qe", tout << "Failed to project: " << m_v << "\n";);
arr_vars[j++] = m_v;
}
}
arr_vars.shrink(j);
aux_vars.append (m_aux_vars);
}
};
/**
* Eliminate (select (store ..) ..) redexes by evaluating indices under model M.
* This does not eliminate variables, but introduces additional constraints on index equalities.
*/
class array_select_reducer {
ast_manager& m;
array_util m_arr_u;
obj_map m_cache;
expr_ref_vector m_pinned; // to ensure a reference
expr_ref_vector m_idx_lits;
model_ref M;
model_evaluator* m_mev;
th_rewriter m_rw;
ast_mark m_arr_test;
ast_mark m_has_stores;
bool m_reduce_all_selects;
void reset () {
m_cache.reset ();
m_pinned.reset ();
m_idx_lits.reset ();
M = nullptr;
m_mev = nullptr;
m_arr_test.reset ();
m_has_stores.reset ();
m_reduce_all_selects = false;
}
bool is_equals (expr *e1, expr *e2) {
return e1 == e2 || (*m_mev)(e1) == (*m_mev)(e2);
}
bool is_equals (unsigned arity, expr * const* xs, expr * const * ys) {
for (unsigned i = 0; i < arity; ++i) {
if (!is_equals(xs[i], ys[i])) return false;
}
return true;
}
expr_ref mk_eq(unsigned arity, expr * const* xs, expr * const * ys) {
expr_ref_vector r(m);
for (unsigned i = 0; i < arity; ++i) {
r.push_back(m.mk_eq(xs[i], ys[i]));
}
return mk_and(r);
}
void add_idx_cond (expr_ref& cond) {
m_rw (cond);
if (!m.is_true (cond)) m_idx_lits.push_back (cond);
}
bool has_stores (expr* e) {
if (m_reduce_all_selects) return true;
return m_has_stores.is_marked (e);
}
void mark_stores (app* a, bool args_have_stores) {
if (m_reduce_all_selects) return;
if (args_have_stores ||
(m_arr_u.is_store (a) && m_arr_test.is_marked (a->get_arg (0)))) {
m_has_stores.mark (a, true);
}
}
bool reduce (expr_ref& e) {
if (!is_app (e)) return true;
expr *r = nullptr;
if (m_cache.find (e, r)) {
e = r;
return true;
}
ptr_vector todo;
todo.push_back (to_app (e));
expr_ref_vector args (m);
while (!todo.empty ()) {
app *a = todo.back ();
unsigned sz = todo.size ();
bool dirty = false;
bool args_have_stores = false;
args.reset();
for (expr * arg : *a) {
expr *narg = nullptr;
if (!is_app (arg)) {
args.push_back (arg);
}
else if (m_cache.find (arg, narg)) {
args.push_back (narg);
dirty |= (arg != narg);
if (!args_have_stores && has_stores (narg)) {
args_have_stores = true;
}
}
else {
todo.push_back (to_app (arg));
}
}
if (todo.size () > sz) continue;
todo.pop_back ();
if (dirty) {
r = m.mk_app (a->get_decl (), args.size (), args.data ());
m_pinned.push_back (r);
}
else {
r = a;
}
if (m_arr_u.is_select (r) && has_stores (to_app (r)->get_arg (0))) {
r = reduce_core (to_app(r));
}
else {
mark_stores (to_app (r), args_have_stores);
}
m_cache.insert (a, r);
}
SASSERT (r);
e = r;
return true;
}
/**
* \brief reduce (select (store (store x i1 v1) i2 v2) idx) under model M
* such that the result is v2 if idx = i2 under M, it is v1 if idx = i1, idx != i2 under M,
* and it is (select x idx) if idx != i1, idx !+ i2 under M.
*/
expr* reduce_core (app *a) {
if (!m_arr_u.is_store (a->get_arg (0))) return a;
expr* array = a->get_arg(0);
unsigned arity = get_array_arity(array->get_sort());
expr* const* js = a->get_args() + 1;
while (m_arr_u.is_store (array)) {
a = to_app (array);
expr* const* idxs = a->get_args() + 1;
expr_ref cond = mk_eq(arity, idxs, js);
if (is_equals (arity, idxs, js)) {
add_idx_cond (cond);
return a->get_arg (a->get_num_args() - 1);
}
else {
cond = m.mk_not (cond);
add_idx_cond (cond);
array = a->get_arg (0);
}
}
ptr_vector args;
args.push_back(array);
args.append(arity, js);
expr* r = m_arr_u.mk_select (args);
m_pinned.push_back (r);
return r;
}
void mk_result (expr_ref& fml) {
// conjoin idx lits
expr_ref_vector lits (m);
lits.append (m_idx_lits);
lits.push_back (fml);
fml = mk_and(lits);
// simplify all trivial expressions introduced
m_rw (fml);
TRACE ("qe", tout << "after reducing selects:\n" << fml << "\n";);
}
public:
array_select_reducer (ast_manager& m):
m (m),
m_arr_u (m),
m_pinned (m),
m_idx_lits (m),
m_rw (m),
m_reduce_all_selects (false)
{}
void operator () (model& mdl, app_ref_vector const& arr_vars, expr_ref& fml, bool reduce_all_selects = false) {
if (!reduce_all_selects && arr_vars.empty ()) return;
reset ();
model_evaluator mev(mdl);
mev.set_model_completion(true);
M = &mdl;
m_mev = &mev;
m_reduce_all_selects = reduce_all_selects;
// mark vars to eliminate
for (app* v : arr_vars) {
m_arr_test.mark (v, true);
}
// assume all arr_vars are of array sort
// and assume no store equalities on arr_vars
if (reduce (fml)) {
mk_result (fml);
}
else {
IF_VERBOSE(2, verbose_stream() << "can't project arrays:" << "\n";);
TRACE ("qe", tout << "Failed to project arrays\n";);
}
}
};
/**
* Perform Ackerman reduction on arrays.
* for occurrences (select a i1), (select a i2), ... assuming these are all occurrences.
* - collect i1, i2, i3, into equivalence classes according to M
* - for distinct index classes accumulate constraint i1 < i2 < i3 .. (for arithmetic)
* and generally distinct(i1, i2, i3) for arbitrary index sorts.
* - for equal indices accumulate constraint i1 = i2, i3 = i5, ..
* - introduce variables v1, v2, .., for each equivalence class.
* - replace occurrences of select by v1, v2, ...
* - update M to evaluate v1, v2, the same as (select a i1) (select a i2)
*/
class array_project_selects_util {
typedef obj_map*> sel_map;
struct idx_val {
expr_ref_vector idx;
expr_ref_vector val;
vector rval;
idx_val(expr_ref_vector && idx, expr_ref_vector && val, vector && rval):
idx(std::move(idx)), val(std::move(val)), rval(std::move(rval)) {}
};
ast_manager& m;
array_util m_arr_u;
arith_util m_ari_u;
bv_util m_bv_u;
sel_map m_sel_terms;
// representative indices for eliminating selects
vector m_idxs;
app_ref_vector m_sel_consts;
expr_ref_vector m_idx_lits;
model_ref M;
model_evaluator* m_mev;
expr_safe_replace m_sub;
ast_mark m_arr_test;
void reset () {
m_sel_terms.reset ();
m_idxs.reset();
m_sel_consts.reset ();
m_idx_lits.reset ();
M = nullptr;
m_mev = nullptr;
m_sub.reset ();
m_arr_test.reset ();
}
/**
* collect sel terms on array vars as given by m_arr_test
*/
void collect_selects (expr* fml) {
if (!is_app (fml)) return;
ast_mark done;
ptr_vector todo;
todo.push_back (to_app (fml));
for (unsigned i = 0; i < todo.size(); ++i) {
app* a = todo[i];
if (done.is_marked (a)) continue;
done.mark (a, true);
for (expr* arg : *a) {
if (!done.is_marked (arg) && is_app (arg)) {
todo.push_back (to_app (arg));
}
}
if (m_arr_u.is_select (a)) {
expr* arr = a->get_arg (0);
if (m_arr_test.is_marked (arr)) {
ptr_vector* lst = m_sel_terms.find (to_app (arr));;
lst->push_back (a);
}
}
}
}
vector to_num(expr_ref_vector const& vals) {
vector rs;
rational r;
for (expr* v : vals) {
if (m_bv_u.is_bv(v)) {
VERIFY (m_bv_u.is_numeral(v, r));
}
else if (m_ari_u.is_real(v) || m_ari_u.is_int(v)) {
VERIFY (m_ari_u.is_numeral(v, r));
}
else {
r.reset();
}
rs.push_back(std::move(r));
}
return rs;
}
struct compare_idx {
array_project_selects_util& u;
compare_idx(array_project_selects_util& u):u(u) {}
bool operator()(idx_val const& x, idx_val const& y) {
SASSERT(x.rval.size() == y.rval.size());
for (unsigned j = 0; j < x.rval.size(); ++j) {
rational const& xv = x.rval[j];
rational const& yv = y.rval[j];
if (xv < yv) return true;
if (xv > yv) return false;
}
return false;
}
};
expr* mk_lt(expr* x, expr* y) {
if (m_bv_u.is_bv(x)) {
return m.mk_not(m_bv_u.mk_ule(y, x));
}
else {
return m_ari_u.mk_lt(x, y);
}
}
expr_ref mk_lex_lt(expr_ref_vector const& xs, expr_ref_vector const& ys) {
SASSERT(xs.size() == ys.size() && !xs.empty());
expr_ref result(mk_lt(xs.back(), ys.back()), m);
for (unsigned i = xs.size()-1; i-- > 0; ) {
result = m.mk_or(mk_lt(xs[i], ys[i]),
m.mk_and(m.mk_eq(xs[i], ys[i]), result));
}
return result;
}
/**
* model based ackermannization for sel terms of some array
*
* update sub with val consts for sel terms
*/
void ackermann (ptr_vector const& sel_terms) {
if (sel_terms.empty ()) return;
expr* v = sel_terms.get (0)->get_arg (0); // array variable
sort* v_sort = v->get_sort ();
sort* val_sort = get_array_range (v_sort);
unsigned arity = get_array_arity(v_sort);
bool is_numeric = true;
for (unsigned i = 0; i < arity && is_numeric; ++i) {
sort* srt = get_array_domain(v_sort, i);
if (!m_ari_u.is_real(srt) && !m_ari_u.is_int(srt) && !m_bv_u.is_bv_sort(srt)) {
TRACE("qe", tout << "non-numeric index sort for Ackerman" << mk_pp(srt, m) << "\n";);
is_numeric = false;
}
}
unsigned start = m_idxs.size (); // append at the end
for (app * a : sel_terms) {
expr_ref_vector idxs(m, arity, a->get_args() + 1);
expr_ref_vector vals = (*m_mev)(idxs);
bool is_new = true;
for (unsigned j = start; j < m_idxs.size (); j++) {
if (!is_eq(m_idxs[j].val, vals)) continue;
// idx belongs to the jth equivalence class;
// substitute sel term with ith sel const
expr* c = m_sel_consts.get (j);
m_sub.insert (a, c);
// add equality (idx == repr)
m_idx_lits.push_back (mk_eq (idxs, m_idxs[j].idx));
is_new = false;
break;
}
if (is_new) {
// new repr, val, and sel const
m_idxs.push_back(idx_val(std::move(idxs), std::move(vals), to_num(vals)));
app_ref c (m.mk_fresh_const ("sel", val_sort), m);
m_sel_consts.push_back (c);
// substitute sel term with new const
m_sub.insert (a, c);
// extend M to include c
expr_ref val = (*m_mev)(a);
M->register_decl (c->get_decl (), val);
}
}
if (start + 1 == m_idxs.size()) {
// nothing to differentiate.
}
else if (is_numeric) {
// sort reprs by their value and add a chain of strict inequalities
compare_idx cmp(*this);
std::sort(m_idxs.begin() + start, m_idxs.end(), cmp);
for (unsigned i = start; i + 1 < m_idxs.size(); ++i) {
m_idx_lits.push_back (mk_lex_lt(m_idxs[i].idx, m_idxs[i+1].idx));
}
}
else if (arity == 1) {
// create distinct constraint.
expr_ref_vector xs(m);
for (unsigned i = start; i < m_idxs.size(); ++i) {
xs.append(m_idxs[i].idx);
}
m_idx_lits.push_back(m.mk_distinct(xs.size(), xs.data()));
}
else {
datatype::util dt(m);
sort_ref_vector srts(m);
ptr_vector acc;
unsigned i = 0;
for (expr * x : m_idxs[0].idx) {
std::stringstream name;
name << "get" << (i++);
acc.push_back(mk_accessor_decl(m, symbol(name.str()), type_ref(x->get_sort())));
}
constructor_decl* constrs[1] = { mk_constructor_decl(symbol("tuple"), symbol("is-tuple"), acc.size(), acc.data()) };
datatype::def* dts = mk_datatype_decl(dt, symbol("tuple"), 0, nullptr, 1, constrs);
VERIFY(dt.plugin().mk_datatypes(1, &dts, 0, nullptr, srts));
del_datatype_decl(dts);
sort* tuple = srts.get(0);
ptr_vector const & decls = *dt.get_datatype_constructors(tuple);
expr_ref_vector xs(m);
for (unsigned i = start; i < m_idxs.size(); ++i) {
xs.push_back(m.mk_app(decls[0], m_idxs[i].idx.size(), m_idxs[i].idx.data()));
}
m_idx_lits.push_back(m.mk_distinct(xs.size(), xs.data()));
}
}
void mk_result (expr_ref& fml) {
// conjoin idx lits
m_idx_lits.push_back(fml);
fml = mk_and (m_idx_lits);
// substitute for sel terms
m_sub (fml);
TRACE ("qe", tout << "after projection of selects:\n" << fml << "\n";);
}
/**
* project selects
* populates idx lits and obtains substitution for sel terms
*/
bool project (expr* fml) {
// collect sel terms -- populate the map m_sel_terms
collect_selects (fml);
// model based ackermannization
for (auto & kv : m_sel_terms) {
TRACE ("qe",tout << "ackermann for var: " << mk_pp (kv.m_key, m) << "\n";);
ackermann (*(kv.m_value));
}
TRACE ("qe", tout << "idx lits:\n" << m_idx_lits; );
return true;
}
public:
array_project_selects_util (ast_manager& m):
m (m),
m_arr_u (m),
m_ari_u (m),
m_bv_u (m),
m_sel_consts (m),
m_idx_lits (m),
m_sub (m)
{}
void operator () (model& mdl, app_ref_vector& arr_vars, expr_ref& fml, app_ref_vector& aux_vars) {
if (arr_vars.empty()) return;
reset ();
model_evaluator mev(mdl);
mev.set_model_completion(true);
M = &mdl;
m_mev = &mev;
// mark vars to eliminate
// alloc empty map from array var to sel terms over it
for (app* v : arr_vars) {
m_arr_test.mark(v, true);
m_sel_terms.insert(v, alloc (ptr_vector));
}
// assume all arr_vars are of array sort
// and they only appear in select terms
if (project (fml)) {
mk_result (fml);
aux_vars.append (m_sel_consts);
arr_vars.reset ();
}
else {
IF_VERBOSE(2, verbose_stream() << "can't project arrays:" << "\n";);
TRACE ("qe", tout << "Failed to project arrays\n";);
}
// dealloc
for (auto & kv : m_sel_terms) dealloc(kv.m_value);
m_sel_terms.reset ();
}
};
struct array_project_plugin::imp {
struct indices {
expr_ref_vector m_values;
expr* const* m_vars;
indices(ast_manager& m, model& model, unsigned n, expr* const* vars):
m_values(m), m_vars(vars) {
expr_ref val(m);
for (unsigned i = 0; i < n; ++i) {
m_values.push_back(model(vars[i]));
}
}
};
ast_manager& m;
array_util a;
scoped_ptr m_var;
imp(ast_manager& m): m(m), a(m), m_stores(m) {}
~imp() {}
bool solve(model& model, app_ref_vector& vars, expr_ref_vector& lits) {
return false;
}
void remove_true(expr_ref_vector& lits) {
for (unsigned i = 0; i < lits.size(); ++i) {
if (m.is_true(lits[i].get())) {
project_plugin::erase(lits, i);
}
}
}
bool contains_x(expr* e) {
return (*m_var)(e);
}
void mk_eq(indices const& x, indices const& y, expr_ref_vector& lits) {
SASSERT(x.m_values.size() == y.m_values.size());
unsigned n = x.m_values.size();
for (unsigned j = 0; j < n; ++j) {
lits.push_back(m.mk_eq(x.m_vars[j], y.m_vars[j]));
}
}
bool solve_eq(model& model, app_ref_vector& vars, expr_ref_vector& lits) {
// find an equality to solve for.
expr* s, *t;
for (unsigned i = 0; i < lits.size(); ++i) {
if (m.is_eq(lits[i].get(), s, t)) {
vector idxs;
expr_ref save(m), back(m);
save = lits[i].get();
back = lits.back();
lits[i] = back;
lits.pop_back();
unsigned sz = lits.size();
if (contains_x(s) && !contains_x(t) && is_app(s)) {
if (solve(model, to_app(s), t, idxs, vars, lits)) {
return true;
}
}
else if (contains_x(t) && !contains_x(s) && is_app(t)) {
if (solve(model, to_app(t), s, idxs, vars, lits)) {
return true;
}
}
// put back the equality literal.
lits.resize(sz);
lits.push_back(back);
lits[i] = save;
}
// TBD: not distinct?
}
return false;
}
bool solve(model& model, app* s, expr* t, vector& idxs, app_ref_vector& vars, expr_ref_vector& lits) {
SASSERT(contains_x(s));
SASSERT(!contains_x(t));
if (s == m_var->x()) {
expr_ref result(t, m);
expr_ref_vector args(m);
sort* range = get_array_range(s->get_sort());
for (unsigned i = 0; i < idxs.size(); ++i) {
app_ref var(m), sel(m);
expr_ref val(m);
var = m.mk_fresh_const("value", range);
vars.push_back(var);
args.reset();
args.push_back (s);
args.append(idxs[i].m_values.size(), idxs[i].m_vars);
sel = a.mk_select (args);
val = model(sel);
model.register_decl (var->get_decl (), val);
args[0] = result;
args.push_back(var);
result = a.mk_store(args.size(), args.data());
}
expr_safe_replace sub(m);
sub.insert(s, result);
for (unsigned i = 0; i < lits.size(); ++i) {
sub(lits[i].get(), result);
lits[i] = result;
}
return true;
}
if (a.is_store(s)) {
unsigned n = s->get_num_args()-2;
indices idx(m, model, n, s->get_args()+1);
for (unsigned i = 1; i < n; ++i) {
if (contains_x(s->get_arg(i))) {
return false;
}
}
unsigned i;
expr_ref_vector args(m);
switch (contains(idx, idxs, i)) {
case l_true:
mk_eq(idx, idxs[i], lits);
return solve(model, to_app(s->get_arg(0)), t, idxs, vars, lits);
case l_false:
for (unsigned i = 0; i < idxs.size(); ++i) {
expr_ref_vector eqs(m);
mk_eq(idx, idxs[i], eqs);
lits.push_back(m.mk_not(mk_and(eqs))); // TBD: extract single index of difference based on model.
}
args.push_back(t);
args.append(n, s->get_args()+1);
lits.push_back(m.mk_eq(a.mk_select(args), s->get_arg(n+1)));
idxs.push_back(idx);
return solve(model, to_app(s->get_arg(0)), t, idxs, vars, lits);
case l_undef:
return false;
}
}
return false;
}
lbool contains(indices const& idx, vector const& idxs, unsigned& j) {
for (unsigned i = 0; i < idxs.size(); ++i) {
switch (compare(idx, idxs[i])) {
case l_true:
j = i;
return l_true;
case l_false:
break;
case l_undef:
return l_undef;
}
}
return l_false;
}
lbool compare(indices const& idx1, indices const& idx2) {
unsigned n = idx1.m_values.size();
for (unsigned i = 0; i < n; ++i) {
switch (compare(idx1.m_values[i], idx2.m_values[i])) {
case l_true:
break;
case l_false:
return l_false;
case l_undef:
return l_undef;
}
}
return l_true;
}
lbool compare(expr* val1, expr* val2) {
if (m.are_equal (val1, val2)) return l_true;
if (m.are_distinct (val1, val2)) return l_false;
if (is_uninterp(val1) ||
is_uninterp(val2)) {
// TBD chase definition of nested array.
return l_undef;
}
return l_undef;
}
void saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) {
term_graph tg(m);
tg.set_vars(shared, false);
tg.add_model_based_terms(model, lits);
// need tg to take term and map it to optional rep over the
// shared vocabulary if it exists.
// . collect shared store expressions, index sorts
// . collect shared index expressions
// . assert extensionality (add shared index expressions)
// . assert store axioms for collected expressions
collect_store_expressions(tg, lits);
collect_index_expressions(tg, lits);
TRACE("qe",
tout << "indices\n";
for (auto& kv : m_indices) {
tout << sort_ref(kv.m_key, m) << " |-> " << *kv.m_value << "\n";
}
tout << "stores " << m_stores << "\n";
tout << "arrays\n";
for (auto& kv : m_arrays) {
tout << sort_ref(kv.m_key, m) << " |-> " << *kv.m_value << "\n";
});
assert_extensionality(model, tg, lits);
assert_store_select(model, tg, lits);
TRACE("qe", tout << lits << "\n";);
for (auto& kv : m_indices) {
dealloc(kv.m_value);
}
for (auto& kv : m_arrays) {
dealloc(kv.m_value);
}
m_stores.reset();
m_indices.reset();
m_arrays.reset();
TRACE("qe", tout << "done: " << lits << "\n";);
}
app_ref_vector m_stores;
obj_map m_indices;
obj_map m_arrays;
void add_index_sort(expr* n) {
sort* s = n->get_sort();
if (!m_indices.contains(s)) {
m_indices.insert(s, alloc(app_ref_vector, m));
}
}
void add_array(app* n) {
sort* s = n->get_sort();
app_ref_vector* vs = nullptr;
if (!m_arrays.find(s, vs)) {
vs = alloc(app_ref_vector, m);
m_arrays.insert(s, vs);
}
vs->push_back(n);
}
app_ref_vector* is_index(expr* n) {
app_ref_vector* result = nullptr;
m_indices.find(n->get_sort(), result);
return result;
}
struct for_each_store_proc {
imp& m_imp;
term_graph& tg;
for_each_store_proc(imp& i, term_graph& tg) : m_imp(i), tg(tg) {}
void operator()(app* n) {
if (m_imp.a.is_array(n) && tg.rep_of(n)) {
m_imp.add_array(n);
}
if (m_imp.a.is_store(n) &&
(tg.rep_of(n->get_arg(0)) ||
tg.rep_of(n->get_arg(n->get_num_args() - 1)))) {
m_imp.m_stores.push_back(n);
for (unsigned i = 1; i + 1 < n->get_num_args(); ++i) {
m_imp.add_index_sort(n->get_arg(i));
}
}
}
void operator()(expr* e) {}
};
struct for_each_index_proc {
imp& m_imp;
term_graph& tg;
for_each_index_proc(imp& i, term_graph& tg) : m_imp(i), tg(tg) {}
void operator()(app* n) {
auto* v = m_imp.is_index(n);
if (v && tg.rep_of(n)) {
v->push_back(n);
}
}
void operator()(expr* e) {}
};
void collect_store_expressions(term_graph& tg, expr_ref_vector const& terms) {
for_each_store_proc proc(*this, tg);
for_each_expr(proc, terms);
}
void collect_index_expressions(term_graph& tg, expr_ref_vector const& terms) {
for_each_index_proc proc(*this, tg);
for_each_expr(proc, terms);
}
bool are_equal(model& mdl, expr* s, expr* t) {
return mdl.are_equal(s, t);
}
void assert_extensionality(model & mdl, term_graph& tg, expr_ref_vector& lits) {
for (auto& kv : m_arrays) {
app_ref_vector const& vs = *kv.m_value;
if (vs.size() <= 1) continue;
func_decl_ref_vector ext(m);
sort* s = kv.m_key;
unsigned arity = get_array_arity(s);
for (unsigned i = 0; i < arity; ++i) {
ext.push_back(a.mk_array_ext(s, i));
}
expr_ref_vector args(m);
args.resize(arity + 1);
for (unsigned i = 0; i < vs.size(); ++i) {
expr* s = vs[i];
for (unsigned j = i + 1; j < vs.size(); ++j) {
expr* t = vs[j];
if (are_equal(mdl, s, t)) {
lits.push_back(m.mk_eq(s, t));
}
else {
for (unsigned k = 0; k < arity; ++k) {
args[k+1] = m.mk_app(ext.get(k), s, t);
}
args[0] = t;
expr* t1 = a.mk_select(args);
args[0] = s;
expr* s1 = a.mk_select(args);
lits.push_back(m.mk_not(m.mk_eq(t1, s1)));
}
}
}
}
}
void assert_store_select(model & mdl, term_graph& tg, expr_ref_vector& lits) {
for (auto& store : m_stores) {
assert_store_select(store, mdl, tg, lits);
}
}
void assert_store_select(app* store, model & mdl, term_graph& tg, expr_ref_vector& lits) {
SASSERT(a.is_store(store));
ptr_vector indices;
for (unsigned i = 1; i + 1 < store->get_num_args(); ++i) {
SASSERT(indices.empty());
assert_store_select(indices, store, mdl, tg, lits);
}
}
void assert_store_select(ptr_vector& indices, app* store, model & mdl, term_graph& tg, expr_ref_vector& lits) {
unsigned sz = store->get_num_args();
if (indices.size() + 2 == sz) {
ptr_vector args;
args.push_back(store);
for (expr* idx : indices) args.push_back(idx);
for (unsigned i = 1; i + 1 < sz; ++i) {
expr* idx1 = store->get_arg(i);
expr* idx2 = indices[i - 1];
if (!are_equal(mdl, idx1, idx2)) {
lits.push_back(m.mk_not(m.mk_eq(idx1, idx2)));
lits.push_back(m.mk_eq(store->get_arg(sz-1), a.mk_select(args)));
return;
}
}
for (unsigned i = 1; i + 1 < sz; ++i) {
expr* idx1 = store->get_arg(i);
expr* idx2 = indices[i - 1];
lits.push_back(m.mk_eq(idx1, idx2));
}
expr* a1 = a.mk_select(args);
args[0] = store->get_arg(0);
expr* a2 = a.mk_select(args);
lits.push_back(m.mk_eq(a1, a2));
}
else {
sort* s = store->get_arg(indices.size() + 1)->get_sort();
for (app* idx : *m_indices[s]) {
indices.push_back(idx);
assert_store_select(indices, store, mdl, tg, lits);
indices.pop_back();
}
}
}
};
array_project_plugin::array_project_plugin(ast_manager& m):project_plugin(m) {
m_imp = alloc(imp, m);
}
array_project_plugin::~array_project_plugin() {
dealloc(m_imp);
}
bool array_project_plugin::operator()(model& model, app* var, app_ref_vector& vars, expr_ref_vector& lits) {
ast_manager& m = vars.get_manager();
app_ref_vector vvars(m, 1, &var);
expr_ref fml = mk_and(lits);
(*this)(model, vvars, fml, vars, false);
lits.reset();
flatten_and(fml, lits);
return true;
}
bool array_project_plugin::solve(model& model, app_ref_vector& vars, expr_ref_vector& lits) {
return m_imp->solve(model, vars, lits);
}
family_id array_project_plugin::get_family_id() {
return m_imp->a.get_family_id();
}
void array_project_plugin::operator()(model& mdl, app_ref_vector& arr_vars, expr_ref& fml, app_ref_vector& aux_vars, bool reduce_all_selects) {
// 1. project array equalities
ast_manager& m = fml.get_manager();
array_project_eqs_util pe (m);
pe (mdl, arr_vars, fml, aux_vars);
TRACE ("qe",
tout << "Projected array eqs: " << fml << "\n";
tout << "Remaining array vars: " << arr_vars << "\n";
tout << "Aux vars: " << aux_vars << "\n";
);
// 2. reduce selects
array_select_reducer rs (m);
rs (mdl, arr_vars, fml, reduce_all_selects);
TRACE ("qe", tout << "Reduced selects:\n" << fml << "\n"; );
// 3. project selects using model based ackermannization
array_project_selects_util ps (m);
ps (mdl, arr_vars, fml, aux_vars);
TRACE ("qe",
tout << "Projected array selects: " << fml << "\n";
tout << "All aux vars: " << aux_vars << "\n";
);
}
bool array_project_plugin::project(model& model, app_ref_vector& vars, expr_ref_vector& lits, vector& defs) {
return true;
}
void array_project_plugin::saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) {
m_imp->saturate(model, shared, lits);
}
};
