// Rigorous proof of the existence of zero for second iteration of Henon map
// Instead of IMap class we use defined above class MapIterator.
void HenonProof()
{
std::cout << "\n\n SECOND ITERATION OF HENON IMap: \n H(x,y) = (1 - a*x*x + y, b*x) where a=1.4, b=0.3 ";
IMap Henon("par:a,b;var:x,y;fun:1-a*x*x+y,b*x;");
Henon.setParameter("a",interval(1.4));
Henon.setParameter("b",interval(0.3));
MapIterator IMap(Henon, 2);
capd::newton::Krawczyk<MapIterator> henon(IMap);
std::cout << "\n\n KRAWCZYK PROOF \n\n ";
DVector x(2); // Good quess for zero for second iteration of Henon map
x[0] = -10./3; x[1]=131./9; // It is also a center of the set.
double size = 1.e-5; // Size of the set in which we will search for a zero
try{
capd::newton::KrawczykResult code = henon.proof(x, size);
std::cout << resultToText(code) ;
std::cout << "\n\n afer " << henon.numberOfIterations << " iteration of Krawczyk method"
<< "\n set X " << henon.X
<< "\n Krawczyk operator K " << henon.K << std::endl;
}
catch(std::exception& e)
{
std::cout << e.what();
}
}
bool ode_solver::inner_loop_backward(ITaylor & solver, interval prevTime, vector<pair<interval, IVector>> & bucket) {
interval const stepMade = solver.getStep();
const ITaylor::CurveType& curve = solver.getCurve();
interval domain = interval(0, 1) * stepMade;
list<interval> intvs;
if (prevTime.rightBound() < m_T.leftBound()) {
interval pre_T = interval(0, m_T.leftBound() - prevTime.rightBound());
domain.setLeftBound(m_T.leftBound() - prevTime.rightBound());
intvs = split(domain, m_config.nra_ODE_grid_size);
intvs.push_front(pre_T);
} else {
intvs = split(domain, m_config.nra_ODE_grid_size);
}
for (interval subsetOfDomain : intvs) {
interval dt = prevTime + subsetOfDomain;
IVector v = curve(subsetOfDomain);
if (!check_invariant(v, m_inv)) {
// TODO(soonhok): invariant
return true;
}
DREAL_LOG_INFO << dt << "\t" << v;
if (prevTime + subsetOfDomain.rightBound() > m_T.leftBound()) {
bucket.emplace_back(dt, v);
}
// TODO(soonhok): visualization
// if (m_config.nra_json) {
// m_trajectory.emplace_back(m_T.rightBound() - (prevTime + subsetOfDomain), v);
// }
}
return false;
}
IVector ode_solver::extract_invariants() {
map<Enode*, pair<double, double>> inv_map;
for (auto inv : m_invs) {
Enode * p = inv->getCdr()->getCdr()->getCdr()->getCdr()->getCar();
Enode * op = p->getCar();
bool pos = true;
// Handle Negation
if (op->getId() == ENODE_ID_NOT) {
p = p->getCdr()->getCar();
op = p->getCar();
pos = false;
}
switch (op->getId()) {
case ENODE_ID_GEQ:
case ENODE_ID_GT:
// Handle >= & >
pos = !pos;
case ENODE_ID_LEQ:
case ENODE_ID_LT: {
// Handle <= & <
Enode * lhs = pos ? p->getCdr()->getCar() : p->getCdr()->getCdr()->getCar();
Enode * rhs = pos ? p->getCdr()->getCdr()->getCar() : p->getCdr()->getCar();
if (lhs->isVar() && rhs->isConstant()) {
if (inv_map.find(lhs) != inv_map.end()) {
inv_map[lhs].second = rhs->getValue();
} else {
inv_map.emplace(lhs, make_pair(lhs->getLowerBound(), rhs->getValue()));
}
} else if (lhs->isConstant() && rhs->isVar()) {
if (inv_map.find(rhs) != inv_map.end()) {
inv_map[rhs].first = lhs->getValue();
} else {
inv_map.emplace(rhs, make_pair(lhs->getValue(), rhs->getUpperBound()));
}
} else {
cerr << "ode_solver::extract_invariant: error:" << p << endl;
}
}
break;
default:
cerr << "ode_solver::extract_invariant: error" << p << endl;
}
}
IVector ret (m_t_vars.size());
unsigned i = 0;
for (auto const & m_t_var : m_t_vars) {
if (inv_map.find(m_t_var) != inv_map.end()) {
auto inv = interval(inv_map[m_t_var].first, inv_map[m_t_var].second);
DREAL_LOG_INFO << "Invariant extracted from " << m_t_var << " = " << inv;
ret[i++] = inv;
} else {
auto inv = interval(m_t_var->getLowerBound(), m_t_var->getUpperBound());
DREAL_LOG_INFO << "Default Invariant set for " << m_t_var << " = " << inv;
ret[i++] = inv;
}
}
return ret;
}
ode_solver::ODE_result ode_solver::simple_ODE_backward(IVector & X_0, IVector const & X_t, interval const & T,
IVector const & inv, vector<IFunction> & funcs) {
// X_0 = X_0 \cup (X_t - + (d/dt Inv) * T)
for (int i = 0; i < X_0.dimension(); i++) {
interval & x_0 = X_0[i];
interval const & x_t = X_t[i];
IFunction & dxdt = funcs[i];
for (Enode * par : m_pars) {
double lb = get_lb(par);
double ub = get_ub(par);
string name = par->getCar()->getName();
dxdt.setParameter(name, interval(lb, ub));
}
try {
interval const new_x_0 = x_t - dxdt(inv) * T;
if (!intersection(new_x_0, x_0, x_0)) {
DREAL_LOG_INFO << "Simple_ODE: no intersection for X_0";
return ODE_result::UNSAT;
}
} catch (exception& e) {
DREAL_LOG_INFO << "Exception in Simple_ODE: " << e.what();
}
}
// update
IVector_to_varlist(X_0, m_0_vars);
return ODE_result::SAT;
}
IVector ode_solver::varlist_to_IVector(vector<Enode*> const & vars) {
IVector intvs (vars.size());
/* Assign current interval values */
for (unsigned i = 0; i < vars.size(); i++) {
Enode* const & var = vars[i];
interval & intv = intvs[i];
double lb = get_lb(var);
double ub = get_ub(var);
intv = interval(lb, ub);
DREAL_LOG_INFO << "The interval on " << var->getCar()->getName() << ": " << intv;
}
return intvs;
}
// For Henon map we iterate Krawczyk and Newton Interval Operators.
// In this example Newton Method requires much smaller initial set then Krawczyk to work.
void HenonMap()
{
std::cout << "\n\n HENON map: H(x,y) = (1 - a*x*x + y, b*x) where a=1.4, b=0.3 ";
// We define henon map and set its parameters
IMap Henon("par:a,b;var:x,y;fun:1-a*x*x+y,b*x;");
Henon.setParameter("a", interval(1.4));
Henon.setParameter("b",interval(0.3));
IVector x0(2), K(2), N(2);
// We define an initial set for iterations of taking Krawczyk Operator
K[0] = interval(-100,110); K[1]=interval(-100,110);
std::cout << "\n\n Interval KRAWCZYK OPERATOR "
<< "\n for set X = " << K;
for(int i=1; i<=5; ++i)
{
x0=midVector(K);
K = capd::newton::KrawczykOperator(x0, K, Henon);
std::cout << "\n iteration "<< i << " = " << K;
}
// We define an initial set for iterations of taking Newton Operator
N[0]=interval(-.1,.1); N[1] = interval(0.9,1.1);
std::cout << "\n\n Interval NEWTON OPERATOR "
<< "\n for set X = " << N;
for(int i=1; i<=5; ++i)
{
x0=midVector(N);
N = capd::newton::NewtonOperator(x0, N, Henon);
std::cout << "\n iteration "<< i << " = " << N;
}
}
ode_solver::ode_solver(SMTConfig& c,
Egraph & e,
Enode * l_int,
vector<Enode*> invs,
unordered_map<Enode*, int>& enode_to_rp_id) :
m_config(c),
m_egraph(e),
m_int(l_int),
m_invs(invs),
m_enode_to_rp_id(enode_to_rp_id),
m_stepControl(c.nra_ODE_step),
m_time(nullptr) {
// Pick the right flow_map (var |-> ODE) using current mode
m_mode = l_int->getCdr()->getCar()->getValue();
map<string, Enode *> & flow_map = m_egraph.flow_maps[string("flow_") + to_string(m_mode)];
m_time = l_int->getCdr()->getCdr()->getCdr()->getCar();
string time_str = m_time->getCar()->getName(); // i.e. "time_1"
m_step = stoi(time_str.substr(time_str.find_last_of("_") + 1)); // i.e. 1
Enode * var_list = l_int->getCdr()->getCdr()->getCdr()->getCdr();
// Collect _0, _t variables from variable list in integral literal
while (!var_list->isEnil()) {
string name = var_list->getCar()->getCar()->getName();
size_t second_ = name.find_last_of("_");
size_t first_ = name.find_last_of("_", second_ - 1);
string name_prefix, name_postfix;
if (first_ == string::npos) {
name_prefix = name.substr(0, second_);
name_postfix = name.substr(second_);
} else {
name_prefix = name.substr(0, first_);
name_postfix = name.substr(first_);
}
if (flow_map.find(name_prefix) == flow_map.end()) {
cerr << name_prefix << " is not found in flow_map." << endl;
assert(flow_map.find(name_prefix) != flow_map.end());
}
Enode * rhs = flow_map[name_prefix];
stringstream ss;
rhs->print_infix(ss, true, name_postfix);
if (rhs->isConstant() && rhs->getValue() == 0.0) {
// If RHS of ODE == 0.0, we treat it as a parameter in CAPD
m_pars.push_back(var_list->getCar());
m_par_list.push_back(name);
} else {
// Otherwise, we treat it as an ODE variable.
m_0_vars.push_back(var_list->getCar());
m_t_vars.push_back(var_list->getCdr()->getCar());
m_var_list.push_back(name);
m_ode_list.push_back(ss.str());
}
var_list = var_list->getCdr()->getCdr();
}
// join var_list to make diff_var, ode_list to diff_fun_forward
string diff_var = "";
if (!m_var_list.empty()) {
diff_var = "var:" + join(m_var_list, ", ") + ";";
}
string diff_fun_forward = "";
string diff_fun_backward = "";
if (!m_ode_list.empty()) {
diff_fun_forward = "fun:" + join(m_ode_list, ", ") + ";";
diff_fun_backward = "fun: -" + join(m_ode_list, ", -") + ";";
}
// construct diff_sys_forward (string to CAPD)
string diff_par;
if (m_par_list.size() > 0) {
diff_par = "par:" + join(m_par_list, ", ") + ";";
m_diff_sys_forward = diff_par;
m_diff_sys_backward = diff_par;
}
m_diff_sys_forward += diff_var + diff_fun_forward;
m_diff_sys_backward += diff_var + diff_fun_backward;
DREAL_LOG_INFO << "diff_par : " << diff_par;
DREAL_LOG_INFO << "diff_var : " << diff_var;
DREAL_LOG_INFO << "diff_fun_forward : " << diff_fun_forward;
DREAL_LOG_INFO << "diff_fun_backward : " << diff_fun_backward;
DREAL_LOG_INFO << "diff_sys_forward : " << m_diff_sys_forward;
DREAL_LOG_INFO << "diff_sys_backward : " << m_diff_sys_backward;
for (auto ode_str : m_ode_list) {
string const func_str = diff_par + diff_var + "fun:" + ode_str + ";";
m_funcs.push_back(IFunction(func_str));
};
m_inv = extract_invariants();
}
bool ode_solver::contain_NaN(C0Rect2Set const & s) {
return contain_NaN(IVector(s));
}
bool ode_solver::check_invariant(C0Rect2Set & s, IVector const & inv) {
IVector v(s);
bool r = check_invariant(v, inv);
s = C0Rect2Set(v);
return r;
}
ode_solver::ODE_result ode_solver::compute_backward(vector<pair<interval, IVector>> & bucket) {
ODE_result ret = ODE_result::SAT;
auto start = high_resolution_clock::now();
bool invariantViolated = false;
try {
// Set up VectorField
IMap vectorField(m_diff_sys_backward);
for (Enode * par : m_pars) {
double lb = get_lb(par);
double ub = get_ub(par);
string name = par->getCar()->getName();
vectorField.setParameter(name, interval(lb, ub));
}
ITaylor solver(vectorField, m_config.nra_ODE_taylor_order, .001);
ITimeMap timeMap(solver);
C0Rect2Set s(m_X_t);
timeMap.stopAfterStep(true);
timeMap.turnOnStepControl();
// TODO(soonhok): visualization
// if (m_config.nra_json) {
// m_trajectory.clear();
// m_trajectory.emplace_back(m_T.rightBound() - timeMap.getCurrentTime(), IVector(s));
// }
interval prevTime(0.);
do {
// Handle Timeout
if (m_config.nra_ODE_timeout > 0.0) {
auto end = high_resolution_clock::now();
if (duration_cast<milliseconds>(end - start).count() >= m_config.nra_ODE_timeout) {
return ODE_result::TIMEOUT;
}
}
// Check Invariant
invariantViolated = !check_invariant(s, m_inv);
if (invariantViolated) {
// TODO(soonhok): invariant
if (timeMap.getCurrentTime().rightBound() < m_T.leftBound()) {
ret = ODE_result::UNSAT;
} else {
ret = ODE_result::SAT;
}
break;
}
// Control TimeStep
timeMap.turnOnStepControl();
if (m_stepControl > 0 && solver.getStep() < m_stepControl) {
timeMap.turnOffStepControl();
solver.setStep(m_stepControl);
timeMap.setStep(m_stepControl);
}
// Move s toward m_T.rightBound()
timeMap(m_T.rightBound(), s);
if (contain_NaN(s)) { return ODE_result::SAT; }
if (m_T.leftBound() <= timeMap.getCurrentTime().rightBound()) {
invariantViolated = inner_loop_backward(solver, prevTime, bucket);
if (invariantViolated) {
// TODO(soonhok): invariant
ret = ODE_result::SAT;
break;
}
}
prevTime = timeMap.getCurrentTime();
} while (!invariantViolated && !timeMap.completed());
} catch (exception& e) {
ret = ODE_result::EXCEPTION;
}
if (m_config.nra_json) {
prune_trajectory(m_T, m_X_0);
}
return ret;
}
void ode_solver::update(rp_box b) {
m_b = b;
m_X_0 = varlist_to_IVector(m_0_vars);
m_X_t = varlist_to_IVector(m_t_vars);
m_T = interval(get_lb(m_time), get_ub(m_time));
}