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add bit-matrix, avoid flattening and/or after bit-blasting, split pdd_grobner into solver/simplifier, add xlin, add smtfd option for incremental mode logic

Browse source 
Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
This commit is contained in:
Nikolaj Bjorner 2020-01-01 20:14:12 -08:00
parent 09dbacdf50
commit 1d0572354b
17 changed files with 991 additions and 386 deletions
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2
scripts/mk_project.py
View file

@ -33,7 +33,7 @@ def init_project_def():
add_lib('tactic', ['ast', 'model'])
add_lib('substitution', ['ast', 'rewriter'], 'ast/substitution')
add_lib('parser_util', ['ast'], 'parsers/util')
add_lib('grobner', ['ast', 'dd'], 'math/grobner')
add_lib('grobner', ['ast', 'dd', 'simplex'], 'math/grobner')
add_lib('euclid', ['util'], 'math/euclid')
add_lib('proofs', ['rewriter', 'util'], 'ast/proofs')
add_lib('solver', ['model', 'tactic', 'proofs'])

8
src/cmd_context/check_logic.cpp
View file

@ -197,6 +197,14 @@ struct check_logic::imp {
m_dt = true;
m_nonlinear = true; // non-linear 0-1 variables may get eliminated
}
else if (logic == "SMTFD") {
m_bvs = true;
m_uf = true;
m_arrays = true;
m_ints = false;
m_dt = false;
m_nonlinear = false;
}
else {
m_unknown_logic = true;
}

54
src/math/dd/dd_pdd.cpp
View file

@ -1007,4 +1007,58 @@ namespace dd {
std::ostream& operator<<(std::ostream& out, pdd const& b) { return b.display(out); }
void pdd_iterator::next() {
auto& m = m_pdd.m;
while (!m_nodes.empty()) {
auto& p = m_nodes.back();
if (p.first && !m.is_val(p.second)) {
p.first = false;
m_mono.vars.pop_back();
unsigned n = m.lo(p.second);
if (m.is_val(n) && m.val(n).is_zero()) {
m_nodes.pop_back();
continue;
}
while (!m.is_val(n)) {
m_nodes.push_back(std::make_pair(true, n));
m_mono.vars.push_back(m.var(n));
n = m.hi(n);
}
m_mono.coeff = m.val(n);
break;
}
else {
m_nodes.pop_back();
}
}
}
void pdd_iterator::first() {
unsigned n = m_pdd.root;
auto& m = m_pdd.m;
while (!m.is_val(n)) {
m_nodes.push_back(std::make_pair(true, n));
m_mono.vars.push_back(m.var(n));
n = m.hi(n);
}
m_mono.coeff = m.val(n);
}
pdd_iterator pdd::begin() const { return pdd_iterator(*this, true); }
pdd_iterator pdd::end() const { return pdd_iterator(*this, false); }
std::ostream& operator<<(std::ostream& out, pdd_monomial const& m) {
if (!m.coeff.is_one()) {
out << m.coeff;
if (!m.vars.empty()) out << "*";
}
bool first = true;
for (auto v : m.vars) {
if (first) first = false; else out << "*";
out << "v" << v;
}
return out;
}
}

37
src/math/dd/dd_pdd.h
View file

@ -39,12 +39,15 @@ Author:
namespace dd {
class pdd;
class pdd_manager;
class pdd_iterator;
class pdd_manager {
public:
enum semantics { free_e, mod2_e, zero_one_vars_e };
private:
friend pdd;
friend pdd_iterator;
typedef unsigned PDD;
typedef vector<std::pair<rational,unsigned_vector>> monomials_t;
@ -241,6 +244,8 @@ namespace dd {
pdd_manager(unsigned nodes, semantics s = free_e);
~pdd_manager();
semantics get_semantics() const { return m_semantics; }
void reset(unsigned_vector const& level2var);
void set_max_num_nodes(unsigned n) { m_max_num_nodes = n; }
unsigned_vector const& get_level2var() const { return m_level2var; }
@ -274,6 +279,7 @@ namespace dd {
bool lt(pdd const& a, pdd const& b);
bool different_leading_term(pdd const& a, pdd const& b);
double tree_size(pdd const& p);
unsigned num_vars() const { return m_var2pdd.size(); }
unsigned_vector const& free_vars(pdd const& p);
@ -285,6 +291,7 @@ namespace dd {
class pdd {
friend class pdd_manager;
friend class pdd_iterator;
unsigned root;
pdd_manager& m;
pdd(unsigned root, pdd_manager& m): root(root), m(m) { m.inc_ref(root); }
@ -303,6 +310,7 @@ namespace dd {
bool is_val() const { return m.is_val(root); }
bool is_zero() const { return m.is_zero(root); }
bool is_linear() const { return m.is_linear(root); }
bool is_unary() const { return !is_val() && lo().is_zero() && hi().is_val(); }
bool is_binary() const { return m.is_binary(root); }
bool is_monomial() const { return m.is_monomial(root); }
bool var_is_leaf(unsigned v) const { return m.var_is_leaf(root, v); }
@ -330,6 +338,11 @@ namespace dd {
unsigned dag_size() const { return m.dag_size(*this); }
double tree_size() const { return m.tree_size(*this); }
unsigned degree() const { return m.degree(*this); }
unsigned_vector const& free_vars() const { return m.free_vars(*this); }
pdd_iterator begin() const;
pdd_iterator end() const;
};
inline pdd operator*(rational const& r, pdd const& b) { return b * r; }
@ -353,6 +366,30 @@ namespace dd {
std::ostream& operator<<(std::ostream& out, pdd const& b);
struct pdd_monomial {
rational coeff;
unsigned_vector vars;
};
std::ostream& operator<<(std::ostream& out, pdd_monomial const& m);
class pdd_iterator {
friend class pdd;
pdd m_pdd;
svector<std::pair<bool, unsigned>> m_nodes;
pdd_monomial m_mono;
pdd_iterator(pdd const& p, bool at_start): m_pdd(p) { if (at_start) first(); }
void first();
void next();
public:
pdd_monomial const& operator*() const { return m_mono; }
pdd_monomial const* operator->() const { return &m_mono; }
pdd_iterator& operator++() { next(); return *this; }
pdd_iterator operator++(int) { auto tmp = *this; next(); return tmp; }
bool operator==(pdd_iterator const& other) const { return m_nodes == other.m_nodes; }
bool operator!=(pdd_iterator const& other) const { return m_nodes != other.m_nodes; }
};
}

2
src/math/grobner/CMakeLists.txt
View file

@ -1,8 +1,10 @@
z3_add_component(grobner
SOURCES
grobner.cpp
pdd_simplifier.cpp
pdd_solver.cpp
COMPONENT_DEPENDENCIES
ast
dd
simplex
)

574
src/math/grobner/pdd_simplifier.cpp Normal file
View file

@ -0,0 +1,574 @@
/*++
Copyright (c) 2020 Microsoft Corporation
Abstract:
simplification routines for pdd polys
Author:
Nikolaj Bjorner (nbjorner)
Lev Nachmanson (levnach)
Notes:
Linear Elimination:
- comprises of a simplification pass that puts linear equations in to_processed
- so before simplifying with respect to the variable ordering, eliminate linear equalities.
Extended Linear Simplification (as exploited in Bosphorus AAAI 2019):
- multiply each polynomial by one variable from their orbits.
- The orbit of a varible are the variables that occur in the same monomial as it in some polynomial.
- The extended set of polynomials is fed to a linear Gauss Jordan Eliminator that extracts
additional linear equalities.
- Bosphorus uses M4RI to perform efficient GJE to scale on large bit-matrices.
Long distance vanishing polynomials (used by PolyCleaner ICCAD 2019):
- identify polynomials p, q, such that p*q = 0
- main case is half-adders and full adders (p := x + y, q := x * y) over GF2
because (x+y)*x*y = 0 over GF2
To work beyond GF2 we would need to rely on simplification with respect to asserted equalities.
The method seems rather specific to hardware multipliers so not clear it is useful to
generalize.
- find monomials that contain pairs of vanishing polynomials, transitively
withtout actually inlining.
Then color polynomial variables w by p, resp, q if they occur in polynomial equalities
w - r = 0, such that all paths in r contain a node colored by p, resp q.
polynomial variables that get colored by both p and q can be set to 0.
When some variable gets colored, other variables can be colored.
- We can walk pdd nodes by level to perform coloring in a linear sweep.
PDD nodes that are equal to 0 using some equality are marked as definitions.
First walk definitions to search for vanishing polynomial pairs.
Given two definition polynomials d1, d2, it must be the case that
level(lo(d1)) = level(lo(d1)) for the polynomial lo(d1)*lo(d2) to be vanishing.
Then starting from the lowest level examine pdd nodes.
Let the current node be called p, check if the pdd node p is used in an equation
w - r = 0. In which case, w inherits the labels from r.
Otherwise, label the node by the intersection of vanishing polynomials from lo(p) and hi(p).
Eliminating multiplier variables, but not adders [Kaufmann et al FMCAD 2019 for GF2];
- Only apply GB saturation with respect to variables that are part of multipliers.
- Perhaps this amounts to figuring out whether a variable is used in an xor or more
--*/
#pragma once
#include "math/grobner/pdd_simplifier.h"
#include "math/simplex/bit_matrix.h"
#include "util/uint_set.h"
namespace dd {
void simplifier::operator()() {
try {
while (!s.done() &&
(simplify_linear_step(true) ||
simplify_elim_pure_step() ||
simplify_cc_step() ||
simplify_leaf_step() ||
simplify_linear_step(false) ||
/*simplify_elim_dual_step() ||*/
simplify_exlin() ||
false)) {
DEBUG_CODE(s.invariant(););
TRACE("dd.solver", s.display(tout););
}
}
catch (pdd_manager::mem_out) {
// done reduce
DEBUG_CODE(s.invariant(););
}
}
struct simplifier::compare_top_var {
bool operator()(equation* a, equation* b) const {
return a->poly().var() < b->poly().var();
}
};
bool simplifier::simplify_linear_step(bool binary) {
TRACE("dd.solver", tout << "binary " << binary << "\n";);
IF_VERBOSE(2, verbose_stream() << "binary " << binary << "\n");
equation_vector linear;
for (equation* e : s.m_to_simplify) {
pdd p = e->poly();
if (binary) {
if (p.is_binary()) linear.push_back(e);
}
else if (p.is_linear()) {
linear.push_back(e);
}
}
return simplify_linear_step(linear);
}
/**
\brief simplify linear equations by using top variable as solution.
The linear equation is moved to set of solved equations.
*/
bool simplifier::simplify_linear_step(equation_vector& linear) {
if (linear.empty()) return false;
use_list_t use_list = get_use_list();
compare_top_var ctv;
std::stable_sort(linear.begin(), linear.end(), ctv);
equation_vector trivial;
unsigned j = 0;
bool has_conflict = false;
for (equation* src : linear) {
if (has_conflict) {
break;
}
unsigned v = src->poly().var();
equation_vector const& uses = use_list[v];
TRACE("dd.solver",
s.display(tout << "uses of: ", *src) << "\n";
for (equation* e : uses) {
s.display(tout, *e) << "\n";
});
bool changed_leading_term;
bool all_reduced = true;
for (equation* dst : uses) {
if (src == dst || s.is_trivial(*dst)) {
continue;
}
pdd q = dst->poly();
if (!src->poly().is_binary() && !q.is_linear()) {
all_reduced = false;
continue;
}
remove_from_use(dst, use_list, v);
s.simplify_using(*dst, *src, changed_leading_term);
if (s.is_trivial(*dst)) {
trivial.push_back(dst);
}
else if (s.is_conflict(dst)) {
s.pop_equation(dst);
s.set_conflict(dst);
has_conflict = true;
}
else if (changed_leading_term) {
s.pop_equation(dst);
s.push_equation(solver::to_simplify, dst);
}
// v has been eliminated.
SASSERT(!dst->poly().free_vars().contains(v));
add_to_use(dst, use_list);
}
if (all_reduced) {
linear[j++] = src;
}
}
if (!has_conflict) {
linear.shrink(j);
for (equation* src : linear) {
s.pop_equation(src);
s.push_equation(solver::solved, src);
}
}
for (equation* e : trivial) {
s.del_equation(e);
}
DEBUG_CODE(s.invariant(););
return j > 0 || has_conflict;
}
/**
\brief simplify using congruences
replace pair px + q and ry + q by
px + q, px - ry
since px = ry
*/
bool simplifier::simplify_cc_step() {
TRACE("dd.solver", tout << "cc\n";);
IF_VERBOSE(2, verbose_stream() << "cc\n");
u_map<equation*> los;
bool reduced = false;
unsigned j = 0;
for (equation* eq1 : s.m_to_simplify) {
SASSERT(eq1->state() == solver::to_simplify);
pdd p = eq1->poly();
auto* e = los.insert_if_not_there2(p.lo().index(), eq1);
equation* eq2 = e->get_data().m_value;
pdd q = eq2->poly();
if (eq2 != eq1 && (p.hi().is_val() || q.hi().is_val()) && !p.lo().is_val()) {
*eq1 = p - eq2->poly();
*eq1 = s.m_dep_manager.mk_join(eq1->dep(), eq2->dep());
reduced = true;
if (s.is_trivial(*eq1)) {
s.retire(eq1);
continue;
}
else if (s.check_conflict(*eq1)) {
continue;
}
}
s.m_to_simplify[j] = eq1;
eq1->set_index(j++);
}
s.m_to_simplify.shrink(j);
return reduced;
}
/**
\brief remove ax+b from p if x occurs as a leaf in p and a is a constant.
*/
bool simplifier::simplify_leaf_step() {
TRACE("dd.solver", tout << "leaf\n";);
IF_VERBOSE(2, verbose_stream() << "leaf\n");
use_list_t use_list = get_use_list();
equation_vector leaves;
for (unsigned i = 0; i < s.m_to_simplify.size(); ++i) {
equation* e = s.m_to_simplify[i];
pdd p = e->poly();
if (!p.hi().is_val()) {
continue;
}
leaves.reset();
for (equation* e2 : use_list[p.var()]) {
if (e != e2 && e2->poly().var_is_leaf(p.var())) {
leaves.push_back(e2);
}
}
for (equation* e2 : leaves) {
bool changed_leading_term;
remove_from_use(e2, use_list);
s.simplify_using(*e2, *e, changed_leading_term);
add_to_use(e2, use_list);
if (s.is_trivial(*e2)) {
s.pop_equation(e2);
s.retire(e2);
}
else if (e2->poly().is_val()) {
s.pop_equation(e2);
s.set_conflict(*e2);
return true;
}
else if (changed_leading_term) {
s.pop_equation(e2);
s.push_equation(solver::to_simplify, e2);
}
}
}
return false;
}
/**
\brief treat equations as processed if top variable occurs only once.
*/
bool simplifier::simplify_elim_pure_step() {
TRACE("dd.solver", tout << "pure\n";);
IF_VERBOSE(2, verbose_stream() << "pure\n");
use_list_t use_list = get_use_list();
unsigned j = 0;
for (equation* e : s.m_to_simplify) {
pdd p = e->poly();
if (!p.is_val() && p.hi().is_val() && use_list[p.var()].size() == 1) {
s.push_equation(solver::solved, e);
}
else {
s.m_to_simplify[j] = e;
e->set_index(j++);
}
}
if (j != s.m_to_simplify.size()) {
s.m_to_simplify.shrink(j);
return true;
}
return false;
}
/**
\brief
reduce equations where top variable occurs only twice and linear in one of the occurrences.
*/
bool simplifier::simplify_elim_dual_step() {
use_list_t use_list = get_use_list();
unsigned j = 0;
bool reduced = false;
for (unsigned i = 0; i < s.m_to_simplify.size(); ++i) {
equation* e = s.m_to_simplify[i];
pdd p = e->poly();
// check that e is linear in top variable.
if (e->state() != solver::to_simplify) {
reduced = true;
}
else if (!s.done() && !s.is_trivial(*e) && p.hi().is_val() && use_list[p.var()].size() == 2) {
for (equation* e2 : use_list[p.var()]) {
if (e2 == e) continue;
bool changed_leading_term;
remove_from_use(e2, use_list);
s.simplify_using(*e2, *e, changed_leading_term);
if (s.is_conflict(e2)) {
s.pop_equation(e2);
s.set_conflict(e2);
}
// when e2 is trivial, leading term is changed
SASSERT(!s.is_trivial(*e2) || changed_leading_term);
if (changed_leading_term) {
s.pop_equation(e2);
s.push_equation(solver::to_simplify, e2);
}
add_to_use(e2, use_list);
break;
}
reduced = true;
s.push_equation(solver::solved, e);
}
else {
s.m_to_simplify[j] = e;
e->set_index(j++);
}
}
if (reduced) {
// clean up elements in s.m_to_simplify
// they may have moved.
s.m_to_simplify.shrink(j);
j = 0;
for (equation* e : s.m_to_simplify) {
if (s.is_trivial(*e)) {
s.retire(e);
}
else if (e->state() == solver::to_simplify) {
s.m_to_simplify[j] = e;
e->set_index(j++);
}
}
s.m_to_simplify.shrink(j);
return true;
}
else {
return false;
}
}
void simplifier::add_to_use(equation* e, use_list_t& use_list) {
unsigned_vector const& fv = e->poly().free_vars();
for (unsigned v : fv) {
use_list.reserve(v + 1);
use_list[v].push_back(e);
}
}
void simplifier::remove_from_use(equation* e, use_list_t& use_list) {
unsigned_vector const& fv = e->poly().free_vars();
for (unsigned v : fv) {
use_list.reserve(v + 1);
use_list[v].erase(e);
}
}
void simplifier::remove_from_use(equation* e, use_list_t& use_list, unsigned except_v) {
unsigned_vector const& fv = e->poly().free_vars();
for (unsigned v : fv) {
if (v != except_v) {
use_list.reserve(v + 1);
use_list[v].erase(e);
}
}
}
simplifier::use_list_t simplifier::get_use_list() {
use_list_t use_list;
for (equation * e : s.m_to_simplify) {
add_to_use(e, use_list);
}
for (equation * e : s.m_processed) {
add_to_use(e, use_list);
}
return use_list;
}
/**
\brief use Gauss elimination to extract linear equalities.
So far just for GF(2) semantics.
*/
bool simplifier::simplify_exlin() {
if (s.m.get_semantics() != pdd_manager::mod2_e ||
!s.m_config.m_enable_exlin) {
return false;
}
vector<pdd> eqs, simp_eqs;
for (auto* e : s.m_to_simplify) if (!e->dep()) eqs.push_back(e->poly());
for (auto* e : s.m_processed) if (!e->dep()) eqs.push_back(e->poly());
exlin_augment(eqs);
simplify_exlin(eqs, simp_eqs);
for (pdd const& p : simp_eqs) {
s.add(p);
}
return !simp_eqs.empty() && simplify_linear_step(false);
}
/**
augment set of equations by multiplying with selected variables.
Uses orbits to prune which variables are multiplied.
TBD: could also prune added polynomials based on a maximal degree.
*/
void simplifier::exlin_augment(vector<pdd>& eqs) {
unsigned nv = s.m.num_vars();
vector<uint_set> orbits(nv);
random_gen rand(s.m_config.m_random_seed);
unsigned modest_num_eqs = std::min(eqs.size(), 500u);
unsigned max_xlin_eqs = modest_num_eqs*modest_num_eqs + eqs.size();
for (pdd p : eqs) {
auto const& fv = p.free_vars();
for (unsigned i = fv.size(); i-- > 0; ) {
for (unsigned j = i; j-- > 0; ) {
orbits[fv[i]].insert(fv[j]);
orbits[fv[j]].insert(fv[i]);
}
}
}
TRACE("dd.solver", tout << "augment " << nv << "\n";
for (auto const& o : orbits) tout << o.num_elems() << "\n";
);
vector<pdd> n_eqs;
unsigned start = rand();
for (unsigned _v = 0; _v < nv; ++_v) {
unsigned v = (_v + start) % nv;
auto const& orbitv = orbits[v];
if (orbitv.empty()) continue;
pdd pv = s.m.mk_var(v);
for (pdd p : eqs) {
for (unsigned w : p.free_vars()) {
if (orbitv.contains(w)) {
n_eqs.push_back(pv * p);
break;
}
}
if (n_eqs.size() > max_xlin_eqs) {
goto end_of_new_eqs;
}
}
}
start = rand();
for (unsigned _v = 0; _v < nv; ++_v) {
unsigned v = (_v + start) % nv;
auto const& orbitv = orbits[v];
if (orbitv.empty()) continue;
pdd pv = s.m.mk_var(v);
for (unsigned w : orbitv) {
if (v > w) continue;
pdd pw = s.m.mk_var(w);
for (pdd p : eqs) {
if (n_eqs.size() > max_xlin_eqs) {
goto end_of_new_eqs;
}
for (unsigned u : p.free_vars()) {
if (orbits[w].contains(u) || orbits[v].contains(u)) {
n_eqs.push_back(pw * pv * p);
break;
}
}
}
}
}
end_of_new_eqs:
s.m_config.m_random_seed = rand();
eqs.append(n_eqs);
TRACE("dd.solver", for (pdd const& p : eqs) tout << p << "\n";);
}
void simplifier::simplify_exlin(vector<pdd> const& eqs, vector<pdd>& simp_eqs) {
// create variables for each non-constant monomial.
u_map<unsigned> mon2idx;
vector<pdd> idx2mon;
// insert monomials of degree > 1
for (pdd const& p : eqs) {
for (auto const& m : p) {
if (m.vars.size() <= 1) continue;
pdd r = s.m.mk_val(m.coeff);
for (unsigned i = m.vars.size(); i-- > 0; )
r *= s.m.mk_var(m.vars[i]);
if (!mon2idx.contains(r.index())) {
mon2idx.insert(r.index(), idx2mon.size());
idx2mon.push_back(r);
}
}
}
// insert variables last.
unsigned nv = s.m.num_vars();
for (unsigned v = 0; v < nv; ++v) {
pdd r = s.m.mk_var(v);
mon2idx.insert(r.index(), idx2mon.size());
idx2mon.push_back(r);
}
bit_matrix bm;
unsigned const_idx = idx2mon.size();
bm.reset(const_idx + 1);
// populate rows
for (pdd const& p : eqs) {
if (p.is_zero()) {
continue;
}
auto row = bm.add_row();
for (auto const& m : p) {
SASSERT(m.coeff.is_one());
if (m.vars.empty()) {
row.set(const_idx);
continue;
}
pdd r = s.m.one();
for (unsigned i = m.vars.size(); i-- > 0; )
r *= s.m.mk_var(m.vars[i]);
unsigned v = mon2idx[r.index()];
row.set(v);
}
}
TRACE("dd.solver", tout << bm << "\n";);
bm.solve();
TRACE("dd.solver", tout << bm << "\n";);
for (auto const& r : bm) {
bool is_linear = true;
for (unsigned c : r) {
SASSERT(r[c]);
if (c == const_idx) {
break;
}
pdd const& p = idx2mon[c];
if (!p.is_unary()) {
is_linear = false;
break;
}
}
if (is_linear) {
pdd p = s.m.zero();
for (unsigned c : r) {
if (c == const_idx) {
p += s.m.one();
}
else {
p += idx2mon[c];
}
}
if (!p.is_zero()) {
TRACE("dd.solver", tout << "new linear: " << p << "\n";);
simp_eqs.push_back(p);
}
}
// could also consider singleton monomials as Bosphorus does
// Singleton monomials are of the form v*w*u*v == 0
// Generally want to deal with negations too
// v*(w+1)*u will have shared pdd under w,
// e.g, test every variable in p whether it has hi() == lo().
// maybe easier to read out of a pdd than the expanded form.
}
}
}

55
src/math/grobner/pdd_simplifier.h Normal file
View file

@ -0,0 +1,55 @@
/*++
Copyright (c) 2017 Microsoft Corporation
Abstract:
simplification routines for pdd polys
Author:
Nikolaj Bjorner (nbjorner)
Lev Nachmanson (levnach)
--*/
#pragma once
#include "math/grobner/pdd_solver.h"
namespace dd {
class simplifier {
typedef solver::equation equation;
typedef ptr_vector<equation> equation_vector;
solver& s;
public:
simplifier(solver& s): s(s) {}
~simplifier() {}
void operator()();
private:
struct compare_top_var;
bool simplify_linear_step(bool binary);
bool simplify_linear_step(equation_vector& linear);
typedef vector<equation_vector> use_list_t;
use_list_t get_use_list();
void add_to_use(equation* e, use_list_t& use_list);
void remove_from_use(equation* e, use_list_t& use_list);
void remove_from_use(equation* e, use_list_t& use_list, unsigned except_v);
bool simplify_cc_step();
bool simplify_elim_pure_step();
bool simplify_elim_dual_step();
bool simplify_leaf_step();
bool simplify_exlin();
void exlin_augment(vector<pdd>& eqs);
void simplify_exlin(vector<pdd> const& eqs, vector<pdd>& simp_eqs);
};
}

360
src/math/grobner/pdd_solver.cpp
View file

@ -12,8 +12,10 @@
--*/
#include "math/grobner/pdd_solver.h"
#include "math/grobner/pdd_simplifier.h"
#include "util/uint_set.h"
namespace dd {
/***
@ -71,47 +73,6 @@ namespace dd {
Justification:
- elements in S have no variables watched
- elements in A are always reduced modulo all variables above the current x_i.
TBD:
Linear Elimination:
- comprises of a simplification pass that puts linear equations in to_processed
- so before simplifying with respect to the variable ordering, eliminate linear equalities.
Extended Linear Simplification (as exploited in Bosphorus AAAI 2019):
- multiply each polynomial by one variable from their orbits.
- The orbit of a varible are the variables that occur in the same monomial as it in some polynomial.
- The extended set of polynomials is fed to a linear Gauss Jordan Eliminator that extracts
additional linear equalities.
- Bosphorus uses M4RI to perform efficient GJE to scale on large bit-matrices.
Long distance vanishing polynomials (used by PolyCleaner ICCAD 2019):
- identify polynomials p, q, such that p*q = 0
- main case is half-adders and full adders (p := x + y, q := x * y) over GF2
because (x+y)*x*y = 0 over GF2
To work beyond GF2 we would need to rely on simplification with respect to asserted equalities.
The method seems rather specific to hardware multipliers so not clear it is useful to
generalize.
- find monomials that contain pairs of vanishing polynomials, transitively
withtout actually inlining.
Then color polynomial variables w by p, resp, q if they occur in polynomial equalities
w - r = 0, such that all paths in r contain a node colored by p, resp q.
polynomial variables that get colored by both p and q can be set to 0.
When some variable gets colored, other variables can be colored.
- We can walk pdd nodes by level to perform coloring in a linear sweep.
PDD nodes that are equal to 0 using some equality are marked as definitions.
First walk definitions to search for vanishing polynomial pairs.
Given two definition polynomials d1, d2, it must be the case that
level(lo(d1)) = level(lo(d1)) for the polynomial lo(d1)*lo(d2) to be vanishing.
Then starting from the lowest level examine pdd nodes.
Let the current node be called p, check if the pdd node p is used in an equation
w - r = 0. In which case, w inherits the labels from r.
Otherwise, label the node by the intersection of vanishing polynomials from lo(p) and hi(p).
Eliminating multiplier variables, but not adders [Kaufmann et al FMCAD 2019 for GF2];
- Only apply GB saturation with respect to variables that are part of multipliers.
- Perhaps this amounts to figuring out whether a variable is used in an xor or more
*/
@ -165,323 +126,10 @@ namespace dd {
}
void solver::simplify() {
try {
while (!done() &&
(simplify_linear_step(true) ||
simplify_elim_pure_step() ||
simplify_cc_step() ||
simplify_leaf_step() ||
simplify_linear_step(false) ||
/*simplify_elim_dual_step() ||*/
false)) {
DEBUG_CODE(invariant(););
TRACE("dd.solver", display(tout););
}
}
catch (pdd_manager::mem_out) {
// done reduce
DEBUG_CODE(invariant(););
}
simplifier s(*this);
s();
}
struct solver::compare_top_var {
bool operator()(equation* a, equation* b) const {
return a->poly().var() < b->poly().var();
}
};
bool solver::simplify_linear_step(bool binary) {
TRACE("dd.solver", tout << "binary " << binary << "\n";);
IF_VERBOSE(2, verbose_stream() << "binary " << binary << "\n");
equation_vector linear;
for (equation* e : m_to_simplify) {
pdd p = e->poly();
if (binary) {
if (p.is_binary()) linear.push_back(e);
}
else if (p.is_linear()) {
linear.push_back(e);
}
}
return simplify_linear_step(linear);
}
/**
\brief simplify linear equations by using top variable as solution.
The linear equation is moved to set of solved equations.
*/
bool solver::simplify_linear_step(equation_vector& linear) {
if (linear.empty()) return false;
use_list_t use_list = get_use_list();
compare_top_var ctv;
std::stable_sort(linear.begin(), linear.end(), ctv);
equation_vector trivial;
unsigned j = 0;
bool has_conflict = false;
for (equation* src : linear) {
if (has_conflict) {
break;
}
unsigned v = src->poly().var();
equation_vector const& uses = use_list[v];
TRACE("dd.solver",
display(tout << "uses of: ", *src) << "\n";
for (equation* e : uses) {
display(tout, *e) << "\n";
});
bool changed_leading_term;
bool all_reduced = true;
for (equation* dst : uses) {
if (src == dst || is_trivial(*dst)) {
continue;
}
pdd q = dst->poly();
if (!src->poly().is_binary() && !q.is_linear()) {
all_reduced = false;
continue;
}
remove_from_use(dst, use_list, v);
simplify_using(*dst, *src, changed_leading_term);
if (is_trivial(*dst)) {
trivial.push_back(dst);
}
else if (is_conflict(dst)) {
pop_equation(dst);
set_conflict(dst);
has_conflict = true;
}
else if (changed_leading_term) {
pop_equation(dst);
push_equation(to_simplify, dst);
}
// v has been eliminated.
SASSERT(!m.free_vars(dst->poly()).contains(v));
add_to_use(dst, use_list);
}
if (all_reduced) {
linear[j++] = src;
}
}
if (!has_conflict) {
linear.shrink(j);
for (equation* src : linear) {
pop_equation(src);
push_equation(solved, src);
}
}
for (equation* e : trivial) {
del_equation(e);
}
DEBUG_CODE(invariant(););
return j > 0 || has_conflict;
}
/**
\brief simplify using congruences
replace pair px + q and ry + q by
px + q, px - ry
since px = ry
*/
bool solver::simplify_cc_step() {
TRACE("dd.solver", tout << "cc\n";);
IF_VERBOSE(2, verbose_stream() << "cc\n");
u_map<equation*> los;
bool reduced = false;
unsigned j = 0;
for (equation* eq1 : m_to_simplify) {
SASSERT(eq1->state() == to_simplify);
pdd p = eq1->poly();
auto* e = los.insert_if_not_there2(p.lo().index(), eq1);
equation* eq2 = e->get_data().m_value;
pdd q = eq2->poly();
if (eq2 != eq1 && (p.hi().is_val() || q.hi().is_val()) && !p.lo().is_val()) {
*eq1 = p - eq2->poly();
*eq1 = m_dep_manager.mk_join(eq1->dep(), eq2->dep());
reduced = true;
if (is_trivial(*eq1)) {
retire(eq1);
continue;
}
else if (check_conflict(*eq1)) {
continue;
}
}
m_to_simplify[j] = eq1;
eq1->set_index(j++);
}
m_to_simplify.shrink(j);
return reduced;
}
/**
\brief remove ax+b from p if x occurs as a leaf in p and a is a constant.
*/
bool solver::simplify_leaf_step() {
TRACE("dd.solver", tout << "leaf\n";);
IF_VERBOSE(2, verbose_stream() << "leaf\n");
use_list_t use_list = get_use_list();
equation_vector leaves;
for (unsigned i = 0; i < m_to_simplify.size(); ++i) {
equation* e = m_to_simplify[i];
pdd p = e->poly();
if (!p.hi().is_val()) {
continue;
}
leaves.reset();
for (equation* e2 : use_list[p.var()]) {
if (e != e2 && e2->poly().var_is_leaf(p.var())) {
leaves.push_back(e2);
}
}
for (equation* e2 : leaves) {
bool changed_leading_term;
remove_from_use(e2, use_list);
simplify_using(*e2, *e, changed_leading_term);
add_to_use(e2, use_list);
if (is_trivial(*e2)) {
pop_equation(e2);
retire(e2);
}
else if (e2->poly().is_val()) {
pop_equation(e2);
set_conflict(*e2);
return true;
}
else if (changed_leading_term) {
pop_equation(e2);
push_equation(to_simplify, e2);
}
}
}
return false;
}
/**
\brief treat equations as processed if top variable occurs only once.
*/
bool solver::simplify_elim_pure_step() {
TRACE("dd.solver", tout << "pure\n";);
IF_VERBOSE(2, verbose_stream() << "pure\n");
use_list_t use_list = get_use_list();
unsigned j = 0;
for (equation* e : m_to_simplify) {
pdd p = e->poly();
if (!p.is_val() && p.hi().is_val() && use_list[p.var()].size() == 1) {
push_equation(solved, e);
}
else {
m_to_simplify[j] = e;
e->set_index(j++);
}
}
if (j != m_to_simplify.size()) {
m_to_simplify.shrink(j);
return true;
}
return false;
}
/**
\brief
reduce equations where top variable occurs only twice and linear in one of the occurrences.
*/
bool solver::simplify_elim_dual_step() {
use_list_t use_list = get_use_list();
unsigned j = 0;
bool reduced = false;
for (unsigned i = 0; i < m_to_simplify.size(); ++i) {
equation* e = m_to_simplify[i];
pdd p = e->poly();
// check that e is linear in top variable.
if (e->state() != to_simplify) {
reduced = true;
}
else if (!done() && !is_trivial(*e) && p.hi().is_val() && use_list[p.var()].size() == 2) {
for (equation* e2 : use_list[p.var()]) {
if (e2 == e) continue;
bool changed_leading_term;
remove_from_use(e2, use_list);
simplify_using(*e2, *e, changed_leading_term);
if (is_conflict(e2)) {
pop_equation(e2);
set_conflict(e2);
}
// when e2 is trivial, leading term is changed
SASSERT(!is_trivial(*e2) || changed_leading_term);
if (changed_leading_term) {
pop_equation(e2);
push_equation(to_simplify, e2);
}
add_to_use(e2, use_list);
break;
}
reduced = true;
push_equation(solved, e);
}
else {
m_to_simplify[j] = e;
e->set_index(j++);
}
}
if (reduced) {
// clean up elements in m_to_simplify
// they may have moved.
m_to_simplify.shrink(j);
j = 0;
for (equation* e : m_to_simplify) {
if (is_trivial(*e)) {
retire(e);
}
else if (e->state() == to_simplify) {
m_to_simplify[j] = e;
e->set_index(j++);
}
}
m_to_simplify.shrink(j);
return true;
}
else {
return false;
}
}
void solver::add_to_use(equation* e, use_list_t& use_list) {
unsigned_vector const& fv = m.free_vars(e->poly());
for (unsigned v : fv) {
use_list.reserve(v + 1);
use_list[v].push_back(e);
}
}
void solver::remove_from_use(equation* e, use_list_t& use_list) {
unsigned_vector const& fv = m.free_vars(e->poly());
for (unsigned v : fv) {
use_list.reserve(v + 1);
use_list[v].erase(e);
}
}
void solver::remove_from_use(equation* e, use_list_t& use_list, unsigned except_v) {
unsigned_vector const& fv = m.free_vars(e->poly());
for (unsigned v : fv) {
if (v != except_v) {
use_list.reserve(v + 1);
use_list[v].erase(e);
}
}
}
solver::use_list_t solver::get_use_list() {
use_list_t use_list;
for (equation * e : m_to_simplify) {
add_to_use(e, use_list);
}
for (equation * e : m_processed) {
add_to_use(e, use_list);
}
return use_list;
}
void solver::superpose(equation const & eq) {
for (equation* target : m_processed) {

21
src/math/grobner/pdd_solver.h
View file

@ -29,6 +29,7 @@
namespace dd {
class solver {
friend class simplifier;
public:
struct stats {
unsigned m_simplified;
@ -44,10 +45,14 @@ public:
unsigned m_eqs_threshold;
unsigned m_expr_size_limit;
unsigned m_max_steps;
unsigned m_random_seed;
bool m_enable_exlin;
config() :
m_eqs_threshold(UINT_MAX),
m_expr_size_limit(UINT_MAX),
m_max_steps(UINT_MAX)
m_max_steps(UINT_MAX),
m_random_seed(0),
m_enable_exlin(false)
{}
};
@ -160,20 +165,6 @@ private:
void push_equation(eq_state st, equation& eq);
void push_equation(eq_state st, equation* eq) { push_equation(st, *eq); }
struct compare_top_var;
bool simplify_linear_step(bool binary);
bool simplify_linear_step(equation_vector& linear);
typedef vector<equation_vector> use_list_t;
use_list_t get_use_list();
void add_to_use(equation* e, use_list_t& use_list);
void remove_from_use(equation* e, use_list_t& use_list);
void remove_from_use(equation* e, use_list_t& use_list, unsigned except_v);
bool simplify_cc_step();
bool simplify_elim_pure_step();
bool simplify_elim_dual_step();
bool simplify_leaf_step();
void invariant() const;
struct scoped_process {
solver& g;

1
src/math/simplex/CMakeLists.txt
View file

@ -2,6 +2,7 @@ z3_add_component(simplex
SOURCES
simplex.cpp
model_based_opt.cpp
bit_matrix.cpp
COMPONENT_DEPENDENCIES
util
)

92
src/math/simplex/bit_matrix.cpp Normal file
View file

@ -0,0 +1,92 @@
/*++
Copyright (c) 2020 Microsoft Corporation
Module Name:
bit_matrix.cpp
Author:
Nikolaj Bjorner (nbjorner) 2020-01-1
Notes:
--*/
#include "math/simplex/bit_matrix.h"
bit_matrix::col_iterator bit_matrix::row::begin() const {
return bit_matrix::col_iterator(*this, true);
}
bit_matrix::col_iterator bit_matrix::row::end() const {
return bit_matrix::col_iterator(*this, false);
}
void bit_matrix::col_iterator::next() {
++m_column;
while (m_column < r.m.m_num_columns && !r[m_column]) {
while ((m_column % 64) == 0 && m_column + 64 < r.m.m_num_columns && !r.r[m_column >> 6]) {
m_column += 64;
}
++m_column;
}
}
bool bit_matrix::row::operator[](unsigned i) const {
SASSERT((i >> 6) < m.m_num_chunks);
return (r[i >> 6] & (1ull << static_cast<uint64_t>(i & 63))) != 0;
}
std::ostream& bit_matrix::row::display(std::ostream& out) const {
for (unsigned i = 0; i < m.m_num_columns; ++i) {
out << ((*this)[i]?"1":"0");
}
return out << "\n";
}
void bit_matrix::reset(unsigned num_columns) {
m_region.reset();
m_num_columns = num_columns;
m_num_chunks = (num_columns + 63)/64;
}
bit_matrix::row bit_matrix::add_row() {
uint64_t* r = new (m_region) uint64_t[m_num_chunks];
m_rows.push_back(r);
memset(r, 0, sizeof(uint64_t)*m_num_chunks);
return row(*this, r);
}
bit_matrix::row& bit_matrix::row::operator+=(row const& other) {
for (unsigned i = 0; i < m.m_num_chunks; ++i) {
r[i] ^= other.r[i];
}
return *this;
}
void bit_matrix::solve() {
basic_solve();
}
void bit_matrix::basic_solve() {
for (row& r : *this) {
auto ci = r.begin();
if (ci != r.end()) {
unsigned c = *ci;
for (row& r2 : *this) {
if (r2 != r && r2[c]) r2 += r;
}
}
}
}
std::ostream& bit_matrix::display(std::ostream& out) {
for (row& r : *this) {
out << r;
}
return out;
}

112
src/math/simplex/bit_matrix.h Normal file
View file

@ -0,0 +1,112 @@
/*++
Copyright (c) 2020 Microsoft Corporation
Module Name:
bit_matrix.h
Abstract:
Dense bit-matrix utilities.
Author:
Nikolaj Bjorner (nbjorner) 2020-01-1
Notes:
Exposes Gauss-Jordan simplification.
Uses extremely basic implementation, can be tuned by 4R method.
--*/
#ifndef BIT_MATRIX_H_
#define BIT_MATRIX_H_
#include "util/region.h"
#include "util/vector.h"
class bit_matrix {
region m_region;
unsigned m_num_columns;
unsigned m_num_chunks;
ptr_vector<uint64_t> m_rows;
public:
class col_iterator;
class row_iterator;
class row {
friend row_iterator;
friend bit_matrix;
bit_matrix& m;
uint64_t* r;
row(bit_matrix& m, uint64_t* r):m(m), r(r) {}
public:
col_iterator begin() const;
col_iterator end() const;
bool operator[](unsigned i) const;
void set(unsigned i, bool b) { if (b) set(i); else unset(i); }
void set(unsigned i) { SASSERT((i >> 6) < m.m_num_chunks); r[i >> 6] |= (1ull << (i & 63)); }
void unset(unsigned i) { SASSERT((i >> 6) < m.m_num_chunks); r[i >> 6] &= ~(1ull << (i & 63)); }
row& operator+=(row const& other);
// using pointer equality:
bool operator==(row const& other) const { return r == other.r; }
bool operator!=(row const& other) const { return r != other.r; }
std::ostream& display(std::ostream& out) const;
};
class col_iterator {
friend row;
friend bit_matrix;
row r;
unsigned m_column;
void next();
col_iterator(row const& r, bool at_first): r(r), m_column(at_first?0:r.m.m_num_columns) { if (at_first && !r[m_column]) next(); }
public:
unsigned const& operator*() const { return m_column; }
unsigned const* operator->() const { return &m_column; }
col_iterator& operator++() { next(); return *this; }
col_iterator operator++(int) { auto tmp = *this; next(); return tmp; }
bool operator==(col_iterator const& other) const { return m_column == other.m_column; }
bool operator!=(col_iterator const& other) const { return m_column != other.m_column; }
};
class row_iterator {
friend class bit_matrix;
row m_row;
unsigned m_index;
row_iterator(bit_matrix& m, bool at_first): m_row(m, at_first ? m.m_rows[0] : nullptr), m_index(at_first ? 0 : m.m_rows.size()) {}
void next() { m_index++; if (m_index < m_row.m.m_rows.size()) m_row.r = m_row.m.m_rows[m_index]; }
public:
row const& operator*() const { return m_row; }
row const* operator->() const { return &m_row; }
row& operator*() { return m_row; }
row* operator->() { return &m_row; }
row_iterator& operator++() { next(); return *this; }
row_iterator operator++(int) { auto tmp = *this; next(); return tmp; }
bool operator==(row_iterator const& other) const { return m_index == other.m_index; }
bool operator!=(row_iterator const& other) const { return m_index != other.m_index; }
};
bit_matrix() {}
~bit_matrix() {}
void reset(unsigned num_columns);
row_iterator begin() { return row_iterator(*this, true); }
row_iterator end() { return row_iterator(*this, false); }
row add_row();
void solve();
std::ostream& display(std::ostream& out);
private:
void basic_solve();
};
inline std::ostream& operator<<(std::ostream& out, bit_matrix& m) { return m.display(out); }
inline std::ostream& operator<<(std::ostream& out, bit_matrix::row const& r) { return r.display(out); }
#endif

30
src/sat/sat_solver/inc_sat_solver.cpp
View file

@ -539,25 +539,29 @@ public:
if (!m_bb_rewriter) {
m_bb_rewriter = alloc(bit_blaster_rewriter, m, m_params);
}
params_ref simp1_p = m_params;
simp1_p.set_bool("som", true);
simp1_p.set_bool("pull_cheap_ite", true);
simp1_p.set_bool("push_ite_bv", false);
simp1_p.set_bool("local_ctx", true);
simp1_p.set_uint("local_ctx_limit", 10000000);
simp1_p.set_bool("flat", true); // required by som
simp1_p.set_bool("hoist_mul", false); // required by som
simp1_p.set_bool("elim_and", true);
simp1_p.set_bool("blast_distinct", true);
params_ref simp2_p = m_params;
simp2_p.set_bool("som", true);
simp2_p.set_bool("pull_cheap_ite", true);
simp2_p.set_bool("push_ite_bv", false);
simp2_p.set_bool("local_ctx", true);
simp2_p.set_uint("local_ctx_limit", 10000000);
simp2_p.set_bool("flat", true); // required by som
simp2_p.set_bool("hoist_mul", false); // required by som
simp2_p.set_bool("elim_and", true);
simp2_p.set_bool("blast_distinct", true);
simp2_p.set_bool("flat", false);
m_preprocess =
and_then(mk_simplify_tactic(m),
mk_propagate_values_tactic(m),
//time consuming if done in inner loop: mk_solve_eqs_tactic(m, simp2_p),
//time consuming if done in inner loop: mk_solve_eqs_tactic(m, simp1_p),
mk_card2bv_tactic(m, m_params), // updates model converter
using_params(mk_simplify_tactic(m), simp2_p),
using_params(mk_simplify_tactic(m), simp1_p),
mk_max_bv_sharing_tactic(m),
mk_bit_blaster_tactic(m, m_bb_rewriter.get()),
using_params(mk_simplify_tactic(m), simp2_p)
mk_bit_blaster_tactic(m, m_bb_rewriter.get())
/*TBD remove and check what simplifier does with expansion */ , using_params(mk_simplify_tactic(m), simp2_p)
);
while (m_bb_rewriter->get_num_scopes() < m_num_scopes) {
m_bb_rewriter->push();

4
src/solver/smt_logics.cpp
View file

@ -107,6 +107,7 @@ bool smt_logics::logic_has_bv(symbol const & s) {
s == "QF_BVFP" ||
logic_is_allcsp(s) ||
s == "QF_FD" ||
s == "SMTFD" ||
s == "HORN";
}
@ -129,6 +130,7 @@ bool smt_logics::logic_has_array(symbol const & s) {
logic_is_allcsp(s) ||
s == "QF_ABV" ||
s == "QF_AUFBV" ||
s == "SMTFD" ||
s == "HORN";
}
@ -145,7 +147,7 @@ bool smt_logics::logic_has_fpa(symbol const & s) {
}
bool smt_logics::logic_has_uf(symbol const & s) {
return s == "QF_UF" || s == "UF" || s == "QF_DT";
return s == "QF_UF" || s == "UF" || s == "QF_DT" || s == "SMTFD";
}
bool smt_logics::logic_has_horn(symbol const& s) {

3
src/tactic/portfolio/smt_strategic_solver.cpp
View file

@ -34,6 +34,7 @@ Notes:
#include "tactic/smtlogics/nra_tactic.h"
#include "tactic/portfolio/default_tactic.h"
#include "tactic/fd_solver/fd_solver.h"
#include "tactic/fd_solver/smtfd_solver.h"
#include "tactic/ufbv/ufbv_tactic.h"
#include "tactic/fpa/qffp_tactic.h"
#include "muz/fp/horn_tactic.h"
@ -107,6 +108,8 @@ static solver* mk_special_solver_for_logic(ast_manager & m, params_ref const & p
parallel_params pp(p);
if ((logic == "QF_FD" || logic == "SAT") && !m.proofs_enabled() && !pp.enable())
return mk_fd_solver(m, p);
if (logic == "SMTFD" && !m.proofs_enabled() && !pp.enable())
return mk_smtfd_solver(m, p);
return nullptr;
}

15
src/test/pdd.cpp
View file

@ -103,6 +103,20 @@ namespace dd {
std::cout << (a + b)*(c + d) << "\n";
}
void test_iterator() {
std::cout << "test iterator\n";
pdd_manager m(4);
pdd a = m.mk_var(0);
pdd b = m.mk_var(1);
pdd c = m.mk_var(2);
pdd d = m.mk_var(3);
pdd p = (a + b)*(c + 3*d) + 2;
std::cout << p << "\n";
for (auto const& m : p) {
std::cout << m << "\n";
}
}
}
void tst_pdd() {
@ -110,4 +124,5 @@ void tst_pdd() {
dd::test2();
dd::test3();
dd::test_reset();
dd::test_iterator();
}

7
src/test/pdd_solver.cpp
View file

@ -217,6 +217,13 @@ namespace dd {
g.display(std::cout);
g.simplify();
g.display(std::cout);
if (use_mod2) {
solver::config cfg;
cfg.m_enable_exlin = true;
g = cfg;
g.simplify();
g.display(std::cout << "after exlin\n");
}
g.saturate();
g.display(std::cout);
}