Use zero slope tangent planes for transcendental functions (#2803)

3 files changed, 26 insertions, 22 deletions

diff --git a/src/theory/arith/nonlinear_extension.cpp b/src/theory/arith/nonlinear_extension.cpp
index 929c7808d..c43b9458b 100644
--- a/src/theory/arith/nonlinear_extension.cpp
+++ b/src/theory/arith/nonlinear_extension.cpp
@@ -2621,8 +2621,10 @@ void NonlinearExtension::mkPi(){
     d_pi_neg = Rewriter::rewrite(NodeManager::currentNM()->mkNode(
         MULT, d_pi, NodeManager::currentNM()->mkConst(Rational(-1))));
     //initialize bounds
-    d_pi_bound[0] = NodeManager::currentNM()->mkConst( Rational(333)/Rational(106) );
-    d_pi_bound[1] = NodeManager::currentNM()->mkConst( Rational(355)/Rational(113) );
+    d_pi_bound[0] =
+        NodeManager::currentNM()->mkConst(Rational(103993) / Rational(33102));
+    d_pi_bound[1] =
+        NodeManager::currentNM()->mkConst(Rational(104348) / Rational(33215));
   }
 }
 
@@ -4358,31 +4360,21 @@ bool NonlinearExtension::checkTfTangentPlanesFun(Node tf,
   {
     // compute tangent plane
     // Figure 3: T( x )
-    Node tplane;
-    Node poly_approx_deriv = getDerivative(poly_approx, d_taylor_real_fv);
-    Assert(!poly_approx_deriv.isNull());
-    poly_approx_deriv = Rewriter::rewrite(poly_approx_deriv);
-    Trace("nl-ext-tftp-debug2") << "...derivative of " << poly_approx << " is "
-                                << poly_approx_deriv << std::endl;
-    std::vector<Node> taylor_subs;
-    taylor_subs.push_back(c);
-    Assert(taylor_vars.size() == taylor_subs.size());
-    Node poly_approx_c_deriv = poly_approx_deriv.substitute(taylor_vars.begin(),
-                                                            taylor_vars.end(),
-                                                            taylor_subs.begin(),
-                                                            taylor_subs.end());
-    tplane = nm->mkNode(
-        PLUS,
-        poly_approx_c,
-        nm->mkNode(MULT, poly_approx_c_deriv, nm->mkNode(MINUS, tf[0], c)));
+    // 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++)
     {
-      if (!bounds[i].isNull())
+      // 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], bounds[i]);
+        Node ant = nm->mkNode(i == 0 ? GEQ : LEQ, tf[0], use_bound);
         antec.push_back(ant);
       }
     }
diff --git a/test/regress/CMakeLists.txt b/test/regress/CMakeLists.txt
index 64d8e3598..9b1ce17e4 100644
--- a/test/regress/CMakeLists.txt
+++ b/test/regress/CMakeLists.txt
@@ -518,6 +518,7 @@ set(regress_0_tests
   regress0/nl/nta/cos-sig-value.smt2
   regress0/nl/nta/exp-n0.5-lb.smt2
   regress0/nl/nta/exp-n0.5-ub.smt2
+  regress0/nl/nta/exp-neg2-unsat-unsound.smt2
   regress0/nl/nta/exp1-ub.smt2
   regress0/nl/nta/real-pi.smt2
   regress0/nl/nta/sin-sym.smt2
@@ -1174,7 +1175,6 @@ set(regress_1_tests
   regress1/lemmas/clocksynchro_5clocks.main_invar.base.smt
   regress1/lemmas/pursuit-safety-8.smt
   regress1/lemmas/simple_startup_9nodes.abstract.base.smt
-  regress1/nl/NAVIGATION2.smt2
   regress1/nl/approx-sqrt-unsat.smt2
   regress1/nl/approx-sqrt.smt2
   regress1/nl/arctan2-expdef.smt2
@@ -2075,6 +2075,8 @@ set(regression_disabled_tests
   regress1/ho/hoa0102.smt2
   # issue1048-arrays-int-real.smt2 -- different errors on debug and production.
   regress1/issue1048-arrays-int-real.smt2
+  # times out after update to tangent planes
+  regress1/nl/NAVIGATION2.smt2
   # ajreynol: disabled these since they give different error messages on
   # production and debug:
   regress1/quantifiers/macro-subtype-param.smt2
diff --git a/test/regress/regress0/nl/nta/exp-neg2-unsat-unsound.smt2 b/test/regress/regress0/nl/nta/exp-neg2-unsat-unsound.smt2
new file mode 100644
index 000000000..69c36179a
--- /dev/null
+++ b/test/regress/regress0/nl/nta/exp-neg2-unsat-unsound.smt2
@@ -0,0 +1,10 @@
+; COMMAND-LINE: --nl-ext-tf-tplanes --no-check-models
+; EXPECT: sat
+(set-logic QF_NRAT)
+(declare-fun x () Real)
+(assert (or 
+(and (<= (exp x) 0.01) (= x (- 2.0)))
+(and (> (exp x) 0.2) (= x (- 1.0)))
+)
+)
+(check-sat)