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Diffstat (limited to 'src/theory/arith/nl/transcendental_solver.cpp')
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diff --git a/src/theory/arith/nl/transcendental_solver.cpp b/src/theory/arith/nl/transcendental_solver.cpp new file mode 100644 index 000000000..1377aa54a --- /dev/null +++ b/src/theory/arith/nl/transcendental_solver.cpp @@ -0,0 +1,1476 @@ +/********************* */ +/*! \file transcendental_solver.cpp + ** \verbatim + ** Top contributors (to current version): + ** Andrew Reynolds, Tim King, Mathias Preiner + ** This file is part of the CVC4 project. + ** Copyright (c) 2009-2020 by the authors listed in the file AUTHORS + ** in the top-level source directory) and their institutional affiliations. + ** All rights reserved. See the file COPYING in the top-level source + ** directory for licensing information.\endverbatim + ** + ** \brief Implementation of solver for handling transcendental functions. + **/ + +#include "theory/arith/nl/transcendental_solver.h" + +#include <cmath> +#include <set> + +#include "expr/node_algorithm.h" +#include "expr/node_builder.h" +#include "options/arith_options.h" +#include "theory/arith/arith_msum.h" +#include "theory/arith/arith_utilities.h" +#include "theory/rewriter.h" + +using namespace CVC4::kind; + +namespace CVC4 { +namespace theory { +namespace arith { +namespace nl { + +TranscendentalSolver::TranscendentalSolver(NlModel& m) : d_model(m) +{ + NodeManager* nm = NodeManager::currentNM(); + d_true = nm->mkConst(true); + d_false = nm->mkConst(false); + d_zero = nm->mkConst(Rational(0)); + d_one = nm->mkConst(Rational(1)); + d_neg_one = nm->mkConst(Rational(-1)); + d_taylor_real_fv = nm->mkBoundVar("x", nm->realType()); + d_taylor_real_fv_base = nm->mkBoundVar("a", nm->realType()); + d_taylor_real_fv_base_rem = nm->mkBoundVar("b", nm->realType()); + d_taylor_degree = options::nlExtTfTaylorDegree(); +} + +TranscendentalSolver::~TranscendentalSolver() {} + +void TranscendentalSolver::initLastCall(const std::vector<Node>& assertions, + const std::vector<Node>& false_asserts, + const std::vector<Node>& xts, + std::vector<NlLemma>& lems) +{ + d_funcCongClass.clear(); + d_funcMap.clear(); + d_tf_region.clear(); + + NodeManager* nm = NodeManager::currentNM(); + + // register the extended function terms + std::vector<Node> trNeedsMaster; + bool needPi = false; + // for computing congruence + std::map<Kind, ArgTrie> argTrie; + for (unsigned i = 0, xsize = xts.size(); i < xsize; i++) + { + Node a = xts[i]; + Kind ak = a.getKind(); + bool consider = true; + // if is an unpurified application of SINE, or it is a transcendental + // applied to a trancendental, purify. + if (isTranscendentalKind(ak)) + { + // if we've already computed master for a + if (d_trMaster.find(a) != d_trMaster.end()) + { + // a master has at least one slave + consider = (d_trSlaves.find(a) != d_trSlaves.end()); + } + else + { + if (ak == SINE) + { + // always not a master + consider = false; + } + else + { + for (const Node& ac : a) + { + if (isTranscendentalKind(ac.getKind())) + { + consider = false; + break; + } + } + } + if (!consider) + { + // wait to assign a master below + trNeedsMaster.push_back(a); + } + else + { + d_trMaster[a] = a; + d_trSlaves[a].insert(a); + } + } + } + if (ak == EXPONENTIAL || ak == SINE) + { + needPi = needPi || (ak == SINE); + // if we didn't indicate that it should be purified above + if (consider) + { + std::vector<Node> repList; + for (const Node& ac : a) + { + Node r = d_model.computeConcreteModelValue(ac); + repList.push_back(r); + } + Node aa = argTrie[ak].add(a, repList); + if (aa != a) + { + // apply congruence to pairs of terms that are disequal and congruent + Assert(aa.getNumChildren() == a.getNumChildren()); + Node mvaa = d_model.computeAbstractModelValue(a); + Node mvaaa = d_model.computeAbstractModelValue(aa); + if (mvaa != mvaaa) + { + std::vector<Node> exp; + for (unsigned j = 0, size = a.getNumChildren(); j < size; j++) + { + exp.push_back(a[j].eqNode(aa[j])); + } + Node expn = exp.size() == 1 ? exp[0] : nm->mkNode(AND, exp); + Node cong_lemma = nm->mkNode(OR, expn.negate(), a.eqNode(aa)); + lems.push_back(cong_lemma); + } + } + else + { + // new representative of congruence class + d_funcMap[ak].push_back(a); + } + // add to congruence class + d_funcCongClass[aa].push_back(a); + } + } + else if (ak == PI) + { + Assert(consider); + needPi = true; + d_funcMap[ak].push_back(a); + d_funcCongClass[a].push_back(a); + } + } + // initialize pi if necessary + if (needPi && d_pi.isNull()) + { + mkPi(); + getCurrentPiBounds(lems); + } + + if (!lems.empty()) + { + return; + } + + // process SINE phase shifting + for (const Node& a : trNeedsMaster) + { + // should not have processed this already + Assert(d_trMaster.find(a) == d_trMaster.end()); + Kind k = a.getKind(); + Assert(k == SINE || k == EXPONENTIAL); + Node y = + nm->mkSkolem("y", nm->realType(), "phase shifted trigonometric arg"); + Node new_a = nm->mkNode(k, y); + d_trSlaves[new_a].insert(new_a); + d_trSlaves[new_a].insert(a); + d_trMaster[a] = new_a; + d_trMaster[new_a] = new_a; + Node lem; + if (k == SINE) + { + Trace("nl-ext-tf") << "Basis sine : " << new_a << " for " << a + << std::endl; + Assert(!d_pi.isNull()); + Node shift = nm->mkSkolem("s", nm->integerType(), "number of shifts"); + // TODO : do not introduce shift here, instead needs model-based + // refinement for constant shifts (cvc4-projects #1284) + lem = nm->mkNode( + AND, + mkValidPhase(y, d_pi), + nm->mkNode( + ITE, + mkValidPhase(a[0], d_pi), + a[0].eqNode(y), + a[0].eqNode(nm->mkNode( + PLUS, + y, + nm->mkNode(MULT, nm->mkConst(Rational(2)), shift, d_pi)))), + new_a.eqNode(a)); + } + else + { + // do both equalities to ensure that new_a becomes a preregistered term + lem = nm->mkNode(AND, a.eqNode(new_a), a[0].eqNode(y)); + } + // note we must do preprocess on this lemma + Trace("nl-ext-lemma") << "NonlinearExtension::Lemma : purify : " << lem + << std::endl; + NlLemma nlem(lem); + nlem.d_preprocess = true; + lems.push_back(nlem); + } + + if (Trace.isOn("nl-ext-mv")) + { + Trace("nl-ext-mv") << "Arguments of trancendental functions : " + << std::endl; + for (std::pair<const Kind, std::vector<Node> >& tfl : d_funcMap) + { + Kind k = tfl.first; + if (k == SINE || k == EXPONENTIAL) + { + for (const Node& tf : tfl.second) + { + Node v = tf[0]; + d_model.computeConcreteModelValue(v); + d_model.computeAbstractModelValue(v); + d_model.printModelValue("nl-ext-mv", v); + } + } + } + } +} + +bool TranscendentalSolver::preprocessAssertionsCheckModel( + std::vector<Node>& assertions) +{ + std::vector<Node> pvars; + std::vector<Node> psubs; + for (const std::pair<const Node, Node>& tb : d_trMaster) + { + pvars.push_back(tb.first); + psubs.push_back(tb.second); + } + + // initialize representation of assertions + std::vector<Node> passertions; + for (const Node& a : assertions) + + { + Node pa = a; + if (!pvars.empty()) + { + pa = arithSubstitute(pa, pvars, psubs); + pa = Rewriter::rewrite(pa); + } + if (!pa.isConst() || !pa.getConst<bool>()) + { + Trace("nl-ext-cm-assert") << "- assert : " << pa << std::endl; + passertions.push_back(pa); + } + } + // get model bounds for all transcendental functions + Trace("nl-ext-cm") << "----- Get bounds for transcendental functions..." + << std::endl; + for (std::pair<const Kind, std::vector<Node> >& tfs : d_funcMap) + { + Kind k = tfs.first; + for (const Node& tf : tfs.second) + { + Trace("nl-ext-cm") << "- Term: " << tf << std::endl; + bool success = true; + // tf is Figure 3 : tf( x ) + Node bl; + Node bu; + if (k == PI) + { + bl = d_pi_bound[0]; + bu = d_pi_bound[1]; + } + else + { + std::pair<Node, Node> bounds = getTfModelBounds(tf, d_taylor_degree); + bl = bounds.first; + bu = bounds.second; + if (bl != bu) + { + d_model.setUsedApproximate(); + } + } + if (!bl.isNull() && !bu.isNull()) + { + // for each function in the congruence classe + for (const Node& ctf : d_funcCongClass[tf]) + { + // each term in congruence classes should be master terms + Assert(d_trSlaves.find(ctf) != d_trSlaves.end()); + // we set the bounds for each slave of tf + for (const Node& stf : d_trSlaves[ctf]) + { + Trace("nl-ext-cm") << "...bound for " << stf << " : [" << bl << ", " + << bu << "]" << std::endl; + success = d_model.addCheckModelBound(stf, bl, bu); + } + } + } + else + { + Trace("nl-ext-cm") << "...no bound for " << tf << std::endl; + } + if (!success) + { + // a bound was conflicting + Trace("nl-ext-cm") << "...failed to set bound for " << tf << std::endl; + Trace("nl-ext-cm") << "-----" << std::endl; + return false; + } + } + } + // replace the assertions + assertions = passertions; + return true; +} + +void TranscendentalSolver::incrementTaylorDegree() { d_taylor_degree++; } +unsigned TranscendentalSolver::getTaylorDegree() const +{ + return d_taylor_degree; +} + +void TranscendentalSolver::processSideEffect(const NlLemma& se) +{ + for (const std::tuple<Node, unsigned, Node>& sp : se.d_secantPoint) + { + Node tf = std::get<0>(sp); + unsigned d = std::get<1>(sp); + Node c = std::get<2>(sp); + d_secant_points[tf][d].push_back(c); + } +} + +void TranscendentalSolver::mkPi() +{ + NodeManager* nm = NodeManager::currentNM(); + if (d_pi.isNull()) + { + d_pi = nm->mkNullaryOperator(nm->realType(), PI); + d_pi_2 = Rewriter::rewrite( + nm->mkNode(MULT, d_pi, nm->mkConst(Rational(1) / Rational(2)))); + d_pi_neg_2 = Rewriter::rewrite( + nm->mkNode(MULT, d_pi, nm->mkConst(Rational(-1) / Rational(2)))); + d_pi_neg = + Rewriter::rewrite(nm->mkNode(MULT, d_pi, nm->mkConst(Rational(-1)))); + // initialize bounds + d_pi_bound[0] = nm->mkConst(Rational(103993) / Rational(33102)); + d_pi_bound[1] = nm->mkConst(Rational(104348) / Rational(33215)); + } +} + +void TranscendentalSolver::getCurrentPiBounds(std::vector<NlLemma>& lemmas) +{ + NodeManager* nm = NodeManager::currentNM(); + Node pi_lem = nm->mkNode(AND, + nm->mkNode(GEQ, d_pi, d_pi_bound[0]), + nm->mkNode(LEQ, d_pi, d_pi_bound[1])); + lemmas.push_back(pi_lem); +} + +std::vector<NlLemma> TranscendentalSolver::checkTranscendentalInitialRefine() +{ + NodeManager* nm = NodeManager::currentNM(); + std::vector<NlLemma> lemmas; + Trace("nl-ext") + << "Get initial refinement lemmas for transcendental functions..." + << std::endl; + for (std::pair<const Kind, std::vector<Node> >& tfl : d_funcMap) + { + Kind k = tfl.first; + for (const Node& t : tfl.second) + { + // initial refinements + if (d_tf_initial_refine.find(t) == d_tf_initial_refine.end()) + { + d_tf_initial_refine[t] = true; + Node lem; + if (k == SINE) + { + Node symn = nm->mkNode(SINE, nm->mkNode(MULT, d_neg_one, t[0])); + symn = Rewriter::rewrite(symn); + // Can assume it is its own master since phase is split over 0, + // hence -pi <= t[0] <= pi implies -pi <= -t[0] <= pi. + d_trMaster[symn] = symn; + d_trSlaves[symn].insert(symn); + Assert(d_trSlaves.find(t) != d_trSlaves.end()); + std::vector<Node> children; + + lem = nm->mkNode(AND, + // bounds + nm->mkNode(AND, + nm->mkNode(LEQ, t, d_one), + nm->mkNode(GEQ, t, d_neg_one)), + // symmetry + nm->mkNode(PLUS, t, symn).eqNode(d_zero), + // sign + nm->mkNode(EQUAL, + nm->mkNode(LT, t[0], d_zero), + nm->mkNode(LT, t, d_zero)), + // zero val + nm->mkNode(EQUAL, + nm->mkNode(GT, t[0], d_zero), + nm->mkNode(GT, t, d_zero))); + lem = nm->mkNode( + AND, + lem, + // zero tangent + nm->mkNode(AND, + nm->mkNode(IMPLIES, + nm->mkNode(GT, t[0], d_zero), + nm->mkNode(LT, t, t[0])), + nm->mkNode(IMPLIES, + nm->mkNode(LT, t[0], d_zero), + nm->mkNode(GT, t, t[0]))), + // pi tangent + nm->mkNode( + AND, + nm->mkNode(IMPLIES, + nm->mkNode(LT, t[0], d_pi), + nm->mkNode(LT, t, nm->mkNode(MINUS, d_pi, t[0]))), + nm->mkNode( + IMPLIES, + nm->mkNode(GT, t[0], d_pi_neg), + nm->mkNode(GT, t, nm->mkNode(MINUS, d_pi_neg, t[0]))))); + } + else if (k == EXPONENTIAL) + { + // ( exp(x) > 0 ) ^ ( x=0 <=> exp( x ) = 1 ) ^ ( x < 0 <=> exp( x ) < + // 1 ) ^ ( x <= 0 V exp( x ) > x + 1 ) + lem = nm->mkNode( + AND, + nm->mkNode(GT, t, d_zero), + nm->mkNode(EQUAL, t[0].eqNode(d_zero), t.eqNode(d_one)), + nm->mkNode(EQUAL, + nm->mkNode(LT, t[0], d_zero), + nm->mkNode(LT, t, d_one)), + nm->mkNode(OR, + nm->mkNode(LEQ, t[0], d_zero), + nm->mkNode(GT, t, nm->mkNode(PLUS, t[0], d_one)))); + } + if (!lem.isNull()) + { + lemmas.push_back(lem); + } + } + } + } + + return lemmas; +} + +std::vector<NlLemma> TranscendentalSolver::checkTranscendentalMonotonic() +{ + std::vector<NlLemma> lemmas; + Trace("nl-ext") << "Get monotonicity lemmas for transcendental functions..." + << std::endl; + + // sort arguments of all transcendentals + std::map<Kind, std::vector<Node> > sorted_tf_args; + std::map<Kind, std::map<Node, Node> > tf_arg_to_term; + + for (std::pair<const Kind, std::vector<Node> >& tfl : d_funcMap) + { + Kind k = tfl.first; + if (k == EXPONENTIAL || k == SINE) + { + for (const Node& tf : tfl.second) + { + Node a = tf[0]; + Node mvaa = d_model.computeAbstractModelValue(a); + if (mvaa.isConst()) + { + Trace("nl-ext-tf-mono-debug") << "...tf term : " << a << std::endl; + sorted_tf_args[k].push_back(a); + tf_arg_to_term[k][a] = tf; + } + } + } + } + + SortNlModel smv; + smv.d_nlm = &d_model; + // sort by concrete values + smv.d_isConcrete = true; + smv.d_reverse_order = true; + for (std::pair<const Kind, std::vector<Node> >& tfl : d_funcMap) + { + Kind k = tfl.first; + if (!sorted_tf_args[k].empty()) + { + std::sort(sorted_tf_args[k].begin(), sorted_tf_args[k].end(), smv); + Trace("nl-ext-tf-mono") << "Sorted transcendental function list for " << k + << " : " << std::endl; + for (unsigned i = 0; i < sorted_tf_args[k].size(); i++) + { + Node targ = sorted_tf_args[k][i]; + Node mvatarg = d_model.computeAbstractModelValue(targ); + Trace("nl-ext-tf-mono") + << " " << targ << " -> " << mvatarg << std::endl; + Node t = tf_arg_to_term[k][targ]; + Node mvat = d_model.computeAbstractModelValue(t); + Trace("nl-ext-tf-mono") << " f-val : " << mvat << std::endl; + } + std::vector<Node> mpoints; + std::vector<Node> mpoints_vals; + if (k == SINE) + { + mpoints.push_back(d_pi); + mpoints.push_back(d_pi_2); + mpoints.push_back(d_zero); + mpoints.push_back(d_pi_neg_2); + mpoints.push_back(d_pi_neg); + } + else if (k == EXPONENTIAL) + { + mpoints.push_back(Node::null()); + } + if (!mpoints.empty()) + { + // get model values for points + for (unsigned i = 0; i < mpoints.size(); i++) + { + Node mpv; + if (!mpoints[i].isNull()) + { + mpv = d_model.computeAbstractModelValue(mpoints[i]); + Assert(mpv.isConst()); + } + mpoints_vals.push_back(mpv); + } + + unsigned mdir_index = 0; + int monotonic_dir = -1; + Node mono_bounds[2]; + Node targ, targval, t, tval; + for (unsigned i = 0, size = sorted_tf_args[k].size(); i < size; i++) + { + Node sarg = sorted_tf_args[k][i]; + Node sargval = d_model.computeAbstractModelValue(sarg); + Assert(sargval.isConst()); + Node s = tf_arg_to_term[k][sarg]; + Node sval = d_model.computeAbstractModelValue(s); + Assert(sval.isConst()); + + // increment to the proper monotonicity region + bool increment = true; + while (increment && mdir_index < mpoints.size()) + { + increment = false; + if (mpoints[mdir_index].isNull()) + { + increment = true; + } + else + { + Node pval = mpoints_vals[mdir_index]; + Assert(pval.isConst()); + if (sargval.getConst<Rational>() < pval.getConst<Rational>()) + { + increment = true; + Trace("nl-ext-tf-mono") << "...increment at " << sarg + << " since model value is less than " + << mpoints[mdir_index] << std::endl; + } + } + if (increment) + { + tval = Node::null(); + mono_bounds[1] = mpoints[mdir_index]; + mdir_index++; + monotonic_dir = regionToMonotonicityDir(k, mdir_index); + if (mdir_index < mpoints.size()) + { + mono_bounds[0] = mpoints[mdir_index]; + } + else + { + mono_bounds[0] = Node::null(); + } + } + } + // store the concavity region + d_tf_region[s] = mdir_index; + Trace("nl-ext-concavity") << "Transcendental function " << s + << " is in region #" << mdir_index; + Trace("nl-ext-concavity") + << ", arg model value = " << sargval << std::endl; + + if (!tval.isNull()) + { + NodeManager* nm = NodeManager::currentNM(); + Node mono_lem; + if (monotonic_dir == 1 + && sval.getConst<Rational>() > tval.getConst<Rational>()) + { + mono_lem = nm->mkNode( + IMPLIES, nm->mkNode(GEQ, targ, sarg), nm->mkNode(GEQ, t, s)); + } + else if (monotonic_dir == -1 + && sval.getConst<Rational>() < tval.getConst<Rational>()) + { + mono_lem = nm->mkNode( + IMPLIES, nm->mkNode(LEQ, targ, sarg), nm->mkNode(LEQ, t, s)); + } + if (!mono_lem.isNull()) + { + if (!mono_bounds[0].isNull()) + { + Assert(!mono_bounds[1].isNull()); + mono_lem = nm->mkNode( + IMPLIES, + nm->mkNode(AND, + mkBounded(mono_bounds[0], targ, mono_bounds[1]), + mkBounded(mono_bounds[0], sarg, mono_bounds[1])), + mono_lem); + } + Trace("nl-ext-tf-mono") + << "Monotonicity lemma : " << mono_lem << std::endl; + lemmas.push_back(mono_lem); + } + } + // store the previous values + targ = sarg; + targval = sargval; + t = s; + tval = sval; + } + } + } + } + return lemmas; +} + +std::vector<NlLemma> TranscendentalSolver::checkTranscendentalTangentPlanes() +{ + std::vector<NlLemma> lemmas; + Trace("nl-ext") << "Get tangent plane lemmas for transcendental functions..." + << std::endl; + // this implements Figure 3 of "Satisfiaility Modulo Transcendental Functions + // via Incremental Linearization" by Cimatti et al + for (std::pair<const Kind, std::vector<Node> >& tfs : d_funcMap) + { + Kind k = tfs.first; + if (k == PI) + { + // We do not use Taylor approximation for PI currently. + // This is because the convergence is extremely slow, and hence an + // initial approximation is superior. + continue; + } + Trace("nl-ext-tftp-debug2") << "Taylor variables: " << std::endl; + Trace("nl-ext-tftp-debug2") + << " taylor_real_fv : " << d_taylor_real_fv << std::endl; + Trace("nl-ext-tftp-debug2") + << " taylor_real_fv_base : " << d_taylor_real_fv_base << std::endl; + Trace("nl-ext-tftp-debug2") + << " taylor_real_fv_base_rem : " << d_taylor_real_fv_base_rem + << std::endl; + Trace("nl-ext-tftp-debug2") << std::endl; + + // we substitute into the Taylor sum P_{n,f(0)}( x ) + + for (const Node& tf : tfs.second) + { + // tf is Figure 3 : tf( x ) + Trace("nl-ext-tftp") << "Compute tangent planes " << tf << std::endl; + // go until max degree is reached, or we don't meet bound criteria + for (unsigned d = 1; d <= d_taylor_degree; d++) + { + Trace("nl-ext-tftp") << "- run at degree " << d << "..." << std::endl; + unsigned prev = lemmas.size(); + if (checkTfTangentPlanesFun(tf, d, lemmas)) + { + Trace("nl-ext-tftp") + << "...fail, #lemmas = " << (lemmas.size() - prev) << std::endl; + break; + } + else + { + Trace("nl-ext-tftp") << "...success" << std::endl; + } + } + } + } + + return lemmas; +} + +bool TranscendentalSolver::checkTfTangentPlanesFun(Node tf, + unsigned d, + std::vector<NlLemma>& lemmas) +{ + NodeManager* nm = NodeManager::currentNM(); + Kind k = tf.getKind(); + // this should only be run on master applications + Assert(d_trSlaves.find(tf) != d_trSlaves.end()); + + // Figure 3 : c + Node c = d_model.computeAbstractModelValue(tf[0]); + int csign = c.getConst<Rational>().sgn(); + if (csign == 0) + { + // no secant/tangent plane is necessary + return true; + } + Assert(csign == 1 || csign == -1); + + // Figure 3: P_l, P_u + // mapped to for signs of c + std::map<int, Node> poly_approx_bounds[2]; + std::vector<Node> pbounds; + getPolynomialApproximationBoundForArg(k, c, d, pbounds); + poly_approx_bounds[0][1] = pbounds[0]; + poly_approx_bounds[0][-1] = pbounds[1]; + poly_approx_bounds[1][1] = pbounds[2]; + poly_approx_bounds[1][-1] = pbounds[3]; + + // Figure 3 : v + Node v = d_model.computeAbstractModelValue(tf); + + // check value of tf + Trace("nl-ext-tftp-debug") << "Process tangent plane refinement for " << tf + << ", degree " << d << "..." << std::endl; + Trace("nl-ext-tftp-debug") << " value in model : " << v << std::endl; + Trace("nl-ext-tftp-debug") << " arg value in model : " << c << std::endl; + + std::vector<Node> taylor_vars; + taylor_vars.push_back(d_taylor_real_fv); + + // compute the concavity + int region = -1; + std::unordered_map<Node, int, NodeHashFunction>::iterator itr = + d_tf_region.find(tf); + if (itr != d_tf_region.end()) + { + region = itr->second; + Trace("nl-ext-tftp-debug") << " region is : " << region << std::endl; + } + // Figure 3 : conc + int concavity = regionToConcavity(k, itr->second); + Trace("nl-ext-tftp-debug") << " concavity is : " << concavity << std::endl; + if (concavity == 0) + { + // no secant/tangent plane is necessary + return true; + } + // bounds for which we are this concavity + // Figure 3: < l, u > + Node bounds[2]; + if (k == SINE) + { + bounds[0] = regionToLowerBound(k, region); + Assert(!bounds[0].isNull()); + bounds[1] = regionToUpperBound(k, region); + Assert(!bounds[1].isNull()); + } + + // Figure 3: P + Node poly_approx; + + // compute whether this is a tangent refinement or a secant refinement + bool is_tangent = false; + bool is_secant = false; + std::pair<Node, Node> mvb = getTfModelBounds(tf, d); + for (unsigned r = 0; r < 2; r++) + { + Node pab = poly_approx_bounds[r][csign]; + Node v_pab = r == 0 ? mvb.first : mvb.second; + if (!v_pab.isNull()) + { + Trace("nl-ext-tftp-debug2") + << "...model value of " << pab << " is " << v_pab << std::endl; + + Assert(v_pab.isConst()); + Node comp = nm->mkNode(r == 0 ? LT : GT, v, v_pab); + Trace("nl-ext-tftp-debug2") << "...compare : " << comp << std::endl; + Node compr = Rewriter::rewrite(comp); + Trace("nl-ext-tftp-debug2") << "...got : " << compr << std::endl; + if (compr == d_true) + { + // beyond the bounds + if (r == 0) + { + poly_approx = poly_approx_bounds[r][csign]; + is_tangent = concavity == 1; + is_secant = concavity == -1; + } + else + { + poly_approx = poly_approx_bounds[r][csign]; + is_tangent = concavity == -1; + is_secant = concavity == 1; + } + if (Trace.isOn("nl-ext-tftp")) + { + Trace("nl-ext-tftp") << "*** Outside boundary point ("; + Trace("nl-ext-tftp") << (r == 0 ? "low" : "high") << ") "; + printRationalApprox("nl-ext-tftp", v_pab); + Trace("nl-ext-tftp") << ", will refine..." << std::endl; + Trace("nl-ext-tftp") + << " poly_approx = " << poly_approx << std::endl; + Trace("nl-ext-tftp") + << " is_tangent = " << is_tangent << std::endl; + Trace("nl-ext-tftp") << " is_secant = " << is_secant << std::endl; + } + break; + } + else + { + Trace("nl-ext-tftp") + << " ...within " << (r == 0 ? "low" : "high") << " bound : "; + printRationalApprox("nl-ext-tftp", v_pab); + Trace("nl-ext-tftp") << std::endl; + } + } + } + + // Figure 3: P( c ) + Node poly_approx_c; + if (is_tangent || is_secant) + { + Assert(!poly_approx.isNull()); + std::vector<Node> taylor_subs; + taylor_subs.push_back(c); + Assert(taylor_vars.size() == taylor_subs.size()); + poly_approx_c = poly_approx.substitute(taylor_vars.begin(), + taylor_vars.end(), + taylor_subs.begin(), + taylor_subs.end()); + Trace("nl-ext-tftp-debug2") + << "...poly approximation at c is " << poly_approx_c << std::endl; + } + else + { + // we may want to continue getting better bounds + return false; + } + + if (is_tangent) + { + // compute tangent plane + // Figure 3: T( x ) + // We use zero slope tangent planes, since the concavity of the Taylor + // approximation cannot be easily established. + Node tplane = poly_approx_c; + + Node lem = nm->mkNode(concavity == 1 ? GEQ : LEQ, tf, tplane); + std::vector<Node> antec; + int mdir = regionToMonotonicityDir(k, region); + for (unsigned i = 0; i < 2; i++) + { + // Tangent plane is valid in the interval [c,u) if the slope of the + // function matches its concavity, and is valid in (l, c] otherwise. + Node use_bound = (mdir == concavity) == (i == 0) ? c : bounds[i]; + if (!use_bound.isNull()) + { + Node ant = nm->mkNode(i == 0 ? GEQ : LEQ, tf[0], use_bound); + antec.push_back(ant); + } + } + if (!antec.empty()) + { + Node antec_n = antec.size() == 1 ? antec[0] : nm->mkNode(AND, antec); + lem = nm->mkNode(IMPLIES, antec_n, lem); + } + Trace("nl-ext-tftp-debug2") + << "*** Tangent plane lemma (pre-rewrite): " << lem << std::endl; + lem = Rewriter::rewrite(lem); + Trace("nl-ext-tftp-lemma") + << "*** Tangent plane lemma : " << lem << std::endl; + Assert(d_model.computeAbstractModelValue(lem) == d_false); + // Figure 3 : line 9 + lemmas.push_back(lem); + } + else if (is_secant) + { + // bounds are the minimum and maximum previous secant points + // should not repeat secant points: secant lemmas should suffice to + // rule out previous assignment + Assert(std::find( + d_secant_points[tf][d].begin(), d_secant_points[tf][d].end(), c) + == d_secant_points[tf][d].end()); + // Insert into the (temporary) vector. We do not update this vector + // until we are sure this secant plane lemma has been processed. We do + // this by mapping the lemma to a side effect below. + std::vector<Node> spoints = d_secant_points[tf][d]; + spoints.push_back(c); + + // sort + SortNlModel smv; + smv.d_nlm = &d_model; + smv.d_isConcrete = true; + std::sort(spoints.begin(), spoints.end(), smv); + // get the resulting index of c + unsigned index = + std::find(spoints.begin(), spoints.end(), c) - spoints.begin(); + // bounds are the next closest upper/lower bound values + if (index > 0) + { + bounds[0] = spoints[index - 1]; + } + else + { + // otherwise, we use the lower boundary point for this concavity + // region + if (k == SINE) + { + Assert(!bounds[0].isNull()); + } + else if (k == EXPONENTIAL) + { + // pick c-1 + bounds[0] = Rewriter::rewrite(nm->mkNode(MINUS, c, d_one)); + } + } + if (index < spoints.size() - 1) + { + bounds[1] = spoints[index + 1]; + } + else + { + // otherwise, we use the upper boundary point for this concavity + // region + if (k == SINE) + { + Assert(!bounds[1].isNull()); + } + else if (k == EXPONENTIAL) + { + // pick c+1 + bounds[1] = Rewriter::rewrite(nm->mkNode(PLUS, c, d_one)); + } + } + Trace("nl-ext-tftp-debug2") << "...secant bounds are : " << bounds[0] + << " ... " << bounds[1] << std::endl; + + // the secant plane may be conjunction of 1-2 guarded inequalities + std::vector<Node> lemmaConj; + for (unsigned s = 0; s < 2; s++) + { + // compute secant plane + Assert(!poly_approx.isNull()); + Assert(!bounds[s].isNull()); + // take the model value of l or u (since may contain PI) + Node b = d_model.computeAbstractModelValue(bounds[s]); + Trace("nl-ext-tftp-debug2") << "...model value of bound " << bounds[s] + << " is " << b << std::endl; + Assert(b.isConst()); + if (c != b) + { + // Figure 3 : P(l), P(u), for s = 0,1 + Node poly_approx_b; + std::vector<Node> taylor_subs; + taylor_subs.push_back(b); + Assert(taylor_vars.size() == taylor_subs.size()); + poly_approx_b = poly_approx.substitute(taylor_vars.begin(), + taylor_vars.end(), + taylor_subs.begin(), + taylor_subs.end()); + // Figure 3: S_l( x ), S_u( x ) for s = 0,1 + Node splane; + Node rcoeff_n = Rewriter::rewrite(nm->mkNode(MINUS, b, c)); + Assert(rcoeff_n.isConst()); + Rational rcoeff = rcoeff_n.getConst<Rational>(); + Assert(rcoeff.sgn() != 0); + poly_approx_b = Rewriter::rewrite(poly_approx_b); + poly_approx_c = Rewriter::rewrite(poly_approx_c); + splane = nm->mkNode( + PLUS, + poly_approx_b, + nm->mkNode(MULT, + nm->mkNode(MINUS, poly_approx_b, poly_approx_c), + nm->mkConst(Rational(1) / rcoeff), + nm->mkNode(MINUS, tf[0], b))); + + Node lem = nm->mkNode(concavity == 1 ? LEQ : GEQ, tf, splane); + // With respect to Figure 3, this is slightly different. + // In particular, we chose b to be the model value of bounds[s], + // which is a constant although bounds[s] may not be (e.g. if it + // contains PI). + // To ensure that c...b does not cross an inflection point, + // we guard with the symbolic version of bounds[s]. + // This leads to lemmas e.g. of this form: + // ( c <= x <= PI/2 ) => ( sin(x) < ( P( b ) - P( c ) )*( x - + // b ) + P( b ) ) + // where b = (PI/2)^M, the current value of PI/2 in the model. + // This is sound since we are guarded by the symbolic + // representation of PI/2. + Node antec_n = + nm->mkNode(AND, + nm->mkNode(GEQ, tf[0], s == 0 ? bounds[s] : c), + nm->mkNode(LEQ, tf[0], s == 0 ? c : bounds[s])); + lem = nm->mkNode(IMPLIES, antec_n, lem); + Trace("nl-ext-tftp-debug2") + << "*** Secant plane lemma (pre-rewrite) : " << lem << std::endl; + lem = Rewriter::rewrite(lem); + Trace("nl-ext-tftp-lemma") + << "*** Secant plane lemma : " << lem << std::endl; + lemmaConj.push_back(lem); + Assert(d_model.computeAbstractModelValue(lem) == d_false); + } + } + // Figure 3 : line 22 + Assert(!lemmaConj.empty()); + Node lem = + lemmaConj.size() == 1 ? lemmaConj[0] : nm->mkNode(AND, lemmaConj); + NlLemma nlem(lem); + // The side effect says that if lem is added, then we should add the + // secant point c for (tf,d). + nlem.d_secantPoint.push_back(std::make_tuple(tf, d, c)); + lemmas.push_back(nlem); + } + return true; +} + +int TranscendentalSolver::regionToMonotonicityDir(Kind k, int region) +{ + if (k == EXPONENTIAL) + { + if (region == 1) + { + return 1; + } + } + else if (k == SINE) + { + if (region == 1 || region == 4) + { + return -1; + } + else if (region == 2 || region == 3) + { + return 1; + } + } + return 0; +} + +int TranscendentalSolver::regionToConcavity(Kind k, int region) +{ + if (k == EXPONENTIAL) + { + if (region == 1) + { + return 1; + } + } + else if (k == SINE) + { + if (region == 1 || region == 2) + { + return -1; + } + else if (region == 3 || region == 4) + { + return 1; + } + } + return 0; +} + +Node TranscendentalSolver::regionToLowerBound(Kind k, int region) +{ + if (k == SINE) + { + if (region == 1) + { + return d_pi_2; + } + else if (region == 2) + { + return d_zero; + } + else if (region == 3) + { + return d_pi_neg_2; + } + else if (region == 4) + { + return d_pi_neg; + } + } + return Node::null(); +} + +Node TranscendentalSolver::regionToUpperBound(Kind k, int region) +{ + if (k == SINE) + { + if (region == 1) + { + return d_pi; + } + else if (region == 2) + { + return d_pi_2; + } + else if (region == 3) + { + return d_zero; + } + else if (region == 4) + { + return d_pi_neg_2; + } + } + return Node::null(); +} + +Node TranscendentalSolver::getDerivative(Node n, Node x) +{ + NodeManager* nm = NodeManager::currentNM(); + Assert(x.isVar()); + // only handle the cases of the taylor expansion of d + if (n.getKind() == EXPONENTIAL) + { + if (n[0] == x) + { + return n; + } + } + else if (n.getKind() == SINE) + { + if (n[0] == x) + { + Node na = nm->mkNode(MINUS, d_pi_2, n[0]); + Node ret = nm->mkNode(SINE, na); + ret = Rewriter::rewrite(ret); + return ret; + } + } + else if (n.getKind() == PLUS) + { + std::vector<Node> dchildren; + for (unsigned i = 0; i < n.getNumChildren(); i++) + { + // PLUS is flattened in rewriter, recursion depth is bounded by 1 + Node dc = getDerivative(n[i], x); + if (dc.isNull()) + { + return dc; + } + else + { + dchildren.push_back(dc); + } + } + return nm->mkNode(PLUS, dchildren); + } + else if (n.getKind() == MULT) + { + Assert(n[0].isConst()); + Node dc = getDerivative(n[1], x); + if (!dc.isNull()) + { + return nm->mkNode(MULT, n[0], dc); + } + } + else if (n.getKind() == NONLINEAR_MULT) + { + unsigned xcount = 0; + std::vector<Node> children; + unsigned xindex = 0; + for (unsigned i = 0, size = n.getNumChildren(); i < size; i++) + { + if (n[i] == x) + { + xcount++; + xindex = i; + } + children.push_back(n[i]); + } + if (xcount == 0) + { + return d_zero; + } + else + { + children[xindex] = nm->mkConst(Rational(xcount)); + } + return nm->mkNode(MULT, children); + } + else if (n.isVar()) + { + return n == x ? d_one : d_zero; + } + else if (n.isConst()) + { + return d_zero; + } + Trace("nl-ext-debug") << "No derivative computed for " << n; + Trace("nl-ext-debug") << " for d/d{" << x << "}" << std::endl; + return Node::null(); +} + +std::pair<Node, Node> TranscendentalSolver::getTaylor(Node fa, unsigned n) +{ + NodeManager* nm = NodeManager::currentNM(); + Assert(n > 0); + Node fac; // what term we cache for fa + if (fa[0] == d_zero) + { + // optimization : simpler to compute (x-fa[0])^n if we are centered around 0 + fac = fa; + } + else + { + // otherwise we use a standard factor a in (x-a)^n + fac = nm->mkNode(fa.getKind(), d_taylor_real_fv_base); + } + Node taylor_rem; + Node taylor_sum; + // check if we have already computed this Taylor series + std::unordered_map<unsigned, Node>::iterator itt = d_taylor_sum[fac].find(n); + if (itt == d_taylor_sum[fac].end()) + { + Node i_exp_base; + if (fa[0] == d_zero) + { + i_exp_base = d_taylor_real_fv; + } + else + { + i_exp_base = Rewriter::rewrite( + nm->mkNode(MINUS, d_taylor_real_fv, d_taylor_real_fv_base)); + } + Node i_derv = fac; + Node i_fact = d_one; + Node i_exp = d_one; + int i_derv_status = 0; + unsigned counter = 0; + std::vector<Node> sum; + do + { + counter++; + if (fa.getKind() == EXPONENTIAL) + { + // unchanged + } + else if (fa.getKind() == SINE) + { + if (i_derv_status % 2 == 1) + { + Node arg = nm->mkNode(PLUS, d_pi_2, d_taylor_real_fv_base); + i_derv = nm->mkNode(SINE, arg); + } + else + { + i_derv = fa; + } + if (i_derv_status >= 2) + { + i_derv = nm->mkNode(MINUS, d_zero, i_derv); + } + i_derv = Rewriter::rewrite(i_derv); + i_derv_status = i_derv_status == 3 ? 0 : i_derv_status + 1; + } + if (counter == (n + 1)) + { + TNode x = d_taylor_real_fv_base; + i_derv = i_derv.substitute(x, d_taylor_real_fv_base_rem); + } + Node curr = nm->mkNode(MULT, nm->mkNode(DIVISION, i_derv, i_fact), i_exp); + if (counter == (n + 1)) + { + taylor_rem = curr; + } + else + { + sum.push_back(curr); + i_fact = Rewriter::rewrite( + nm->mkNode(MULT, nm->mkConst(Rational(counter)), i_fact)); + i_exp = Rewriter::rewrite(nm->mkNode(MULT, i_exp_base, i_exp)); + } + } while (counter <= n); + taylor_sum = sum.size() == 1 ? sum[0] : nm->mkNode(PLUS, sum); + + if (fac[0] != d_taylor_real_fv_base) + { + TNode x = d_taylor_real_fv_base; + taylor_sum = taylor_sum.substitute(x, fac[0]); + } + + // cache + d_taylor_sum[fac][n] = taylor_sum; + d_taylor_rem[fac][n] = taylor_rem; + } + else + { + taylor_sum = itt->second; + Assert(d_taylor_rem[fac].find(n) != d_taylor_rem[fac].end()); + taylor_rem = d_taylor_rem[fac][n]; + } + + // must substitute for the argument if we were using a different lookup + if (fa[0] != fac[0]) + { + TNode x = d_taylor_real_fv_base; + taylor_sum = taylor_sum.substitute(x, fa[0]); + } + return std::pair<Node, Node>(taylor_sum, taylor_rem); +} + +void TranscendentalSolver::getPolynomialApproximationBounds( + Kind k, unsigned d, std::vector<Node>& pbounds) +{ + if (d_poly_bounds[k][d].empty()) + { + NodeManager* nm = NodeManager::currentNM(); + Node tft = nm->mkNode(k, d_zero); + // n is the Taylor degree we are currently considering + unsigned n = 2 * d; + // n must be even + std::pair<Node, Node> taylor = getTaylor(tft, n); + Trace("nl-ext-tftp-debug2") + << "Taylor for " << k << " is : " << taylor.first << std::endl; + Node taylor_sum = Rewriter::rewrite(taylor.first); + Trace("nl-ext-tftp-debug2") + << "Taylor for " << k << " is (post-rewrite) : " << taylor_sum + << std::endl; + Assert(taylor.second.getKind() == MULT); + Assert(taylor.second.getNumChildren() == 2); + Assert(taylor.second[0].getKind() == DIVISION); + Trace("nl-ext-tftp-debug2") + << "Taylor remainder for " << k << " is " << taylor.second << std::endl; + // ru is x^{n+1}/(n+1)! + Node ru = nm->mkNode(DIVISION, taylor.second[1], taylor.second[0][1]); + ru = Rewriter::rewrite(ru); + Trace("nl-ext-tftp-debug2") + << "Taylor remainder factor is (post-rewrite) : " << ru << std::endl; + if (k == EXPONENTIAL) + { + pbounds.push_back(taylor_sum); + pbounds.push_back(taylor_sum); + pbounds.push_back(Rewriter::rewrite( + nm->mkNode(MULT, taylor_sum, nm->mkNode(PLUS, d_one, ru)))); + pbounds.push_back(Rewriter::rewrite(nm->mkNode(PLUS, taylor_sum, ru))); + } + else + { + Assert(k == SINE); + Node l = Rewriter::rewrite(nm->mkNode(MINUS, taylor_sum, ru)); + Node u = Rewriter::rewrite(nm->mkNode(PLUS, taylor_sum, ru)); + pbounds.push_back(l); + pbounds.push_back(l); + pbounds.push_back(u); + pbounds.push_back(u); + } + Trace("nl-ext-tf-tplanes") + << "Polynomial approximation for " << k << " is: " << std::endl; + Trace("nl-ext-tf-tplanes") << " Lower (pos): " << pbounds[0] << std::endl; + Trace("nl-ext-tf-tplanes") << " Upper (pos): " << pbounds[2] << std::endl; + Trace("nl-ext-tf-tplanes") << " Lower (neg): " << pbounds[1] << std::endl; + Trace("nl-ext-tf-tplanes") << " Upper (neg): " << pbounds[3] << std::endl; + d_poly_bounds[k][d].insert( + d_poly_bounds[k][d].end(), pbounds.begin(), pbounds.end()); + } + else + { + pbounds.insert( + pbounds.end(), d_poly_bounds[k][d].begin(), d_poly_bounds[k][d].end()); + } +} + +void TranscendentalSolver::getPolynomialApproximationBoundForArg( + Kind k, Node c, unsigned d, std::vector<Node>& pbounds) +{ + getPolynomialApproximationBounds(k, d, pbounds); + Assert(c.isConst()); + if (k == EXPONENTIAL && c.getConst<Rational>().sgn() == 1) + { + NodeManager* nm = NodeManager::currentNM(); + Node tft = nm->mkNode(k, d_zero); + bool success = false; + unsigned ds = d; + TNode ttrf = d_taylor_real_fv; + TNode tc = c; + do + { + success = true; + unsigned n = 2 * ds; + std::pair<Node, Node> taylor = getTaylor(tft, n); + // check that 1-c^{n+1}/(n+1)! > 0 + Node ru = nm->mkNode(DIVISION, taylor.second[1], taylor.second[0][1]); + Node rus = ru.substitute(ttrf, tc); + rus = Rewriter::rewrite(rus); + Assert(rus.isConst()); + if (rus.getConst<Rational>() > d_one.getConst<Rational>()) + { + success = false; + ds = ds + 1; + } + } while (!success); + if (ds > d) + { + Trace("nl-ext-exp-taylor") + << "*** Increase Taylor bound to " << ds << " > " << d << " for (" + << k << " " << c << ")" << std::endl; + // must use sound upper bound + std::vector<Node> pboundss; + getPolynomialApproximationBounds(k, ds, pboundss); + pbounds[2] = pboundss[2]; + } + } +} + +std::pair<Node, Node> TranscendentalSolver::getTfModelBounds(Node tf, + unsigned d) +{ + // compute the model value of the argument + Node c = d_model.computeAbstractModelValue(tf[0]); + Assert(c.isConst()); + int csign = c.getConst<Rational>().sgn(); + Kind k = tf.getKind(); + if (csign == 0) + { + // at zero, its trivial + if (k == SINE) + { + return std::pair<Node, Node>(d_zero, d_zero); + } + Assert(k == EXPONENTIAL); + return std::pair<Node, Node>(d_one, d_one); + } + bool isNeg = csign == -1; + + std::vector<Node> pbounds; + getPolynomialApproximationBoundForArg(k, c, d, pbounds); + + std::vector<Node> bounds; + TNode tfv = d_taylor_real_fv; + TNode tfs = tf[0]; + for (unsigned d2 = 0; d2 < 2; d2++) + { + int index = d2 == 0 ? (isNeg ? 1 : 0) : (isNeg ? 3 : 2); + Node pab = pbounds[index]; + if (!pab.isNull()) + { + // { x -> tf[0] } + pab = pab.substitute(tfv, tfs); + pab = Rewriter::rewrite(pab); + Node v_pab = d_model.computeAbstractModelValue(pab); + bounds.push_back(v_pab); + } + else + { + bounds.push_back(Node::null()); + } + } + return std::pair<Node, Node>(bounds[0], bounds[1]); +} + +Node TranscendentalSolver::mkValidPhase(Node a, Node pi) +{ + return mkBounded( + NodeManager::currentNM()->mkNode(MULT, mkRationalNode(-1), pi), a, pi); +} + +} // namespace nl +} // namespace arith +} // namespace theory +} // namespace CVC4 |