mirror of
https://github.com/Z3Prover/z3
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prep for bilinear adt
Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
This commit is contained in:
Nikolaj Bjorner
parent
9e1afc5916
commit
50630bf8f5
2 changed files with 128 additions and 71 deletions
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@ -1546,36 +1546,36 @@ namespace polysat {
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*
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* If y == null_var, chooses some variable y != x from p (if one exists).
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*/
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bool saturation::extract_bilinear_form(pvar x, pdd const& p, pvar& y, rational& a, rational& b, rational& c, rational& d) {
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bool saturation::extract_bilinear_form(pvar x, pdd const& p, pvar& y, bilinear& b) {
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auto& m = s.var2pdd(x);
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rational const& M = m.two_to_N();
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switch (p.degree(x)) {
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case 0:
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if (!s.try_eval(p, d))
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if (!s.try_eval(p, b.d))
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return false;
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a = b = c = 0;
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b.a = b.b = b.c = 0;
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return true;
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case 1: {
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pdd q = p, r = p, u = p, v = p;
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p.factor(x, 1, q, r);
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if (!extract_linear_form(q, y, a, b))
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if (!extract_linear_form(q, y, b.a, b.b))
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return false;
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if (a == 0) {
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c = 0;
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return eval_round(M, r, d);
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if (b.a == 0) {
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b.c = 0;
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return eval_round(M, r, b.d);
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}
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SASSERT(y != null_var);
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switch (r.degree(y)) {
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case 0:
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if (!eval_round(M, r, d))
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if (!eval_round(M, r, b.d))
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return false;
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c = 0;
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b.c = 0;
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return true;
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case 1:
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r.factor(y, 1, u, v);
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if (!eval_round(M, u, c))
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if (!eval_round(M, u, b.c))
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return false;
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if (!eval_round(M, v, d))
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if (!eval_round(M, v, b.d))
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return false;
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return true;
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default:
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@ -1592,10 +1592,11 @@ namespace polysat {
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* Update d such that -M < a*x*y0 + b*x + c*y0 + d < M for every value x_min <= x <= x_max, return x_split such that [x_min,x_split[ and [x_split,x_max] can fit into [0,M[
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* return false if there is no such d.
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*/
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bool saturation::adjust_bound(rational const& x_min, rational const& x_max, rational const& y0, rational const& M, rational const& a, rational const& b, rational const& c, rational& d, rational* x_split) {
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bool saturation::adjust_bound(rational const& x_min, rational const& x_max, rational const& y0, rational const& M,
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bilinear& b, rational* x_split) {
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SASSERT(x_min <= x_max);
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rational A = a*y0 + b;
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rational B = c*y0 + d;
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rational A = b.a*y0 + b.b;
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rational B = b.c*y0 + b.d;
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rational max = A >= 0 ? x_max * A + B : x_min * A + B;
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rational min = A >= 0 ? x_min * A + B : x_max * A + B;
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VERIFY(min <= max);
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@ -1610,7 +1611,7 @@ namespace polysat {
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rational offset = rational::zero();
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if (max < 0 || max >= M)
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offset = -M * floor(max / M);
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d += offset;
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b.d += offset;
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// If min + offset < 0, then [min,max] contains a multiple of M.
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if (min + offset < 0) {
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@ -1622,25 +1623,25 @@ namespace polysat {
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rational x = ceil((-offset-B) / A);
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// [x_min; x_split-1] maps to interval < 0
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// [x_split; x_max] maps to interval >= 0
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VERIFY(a*x*y0 + b*x + c*y0 + d >= 0);
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VERIFY(a*(x-1)*y0 + b*(x-1) + c*y0 + d < 0);
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VERIFY(b.eval(x, y0) >= 0);
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VERIFY(b.eval(x-1, y0) < 0);
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VERIFY(x_min <= x && x <= x_max);
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*x_split = x;
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} else {
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rational x = floor((-offset-B) / A) + 1;
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// [x_min; x_split-1] maps to interval >= 0
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// [x_split; x_max] maps to interval < 0
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VERIFY(a*x*y0 + b*x + c*y0 + d < 0);
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VERIFY(a*(x-1)*y0 + b*(x-1) + c*y0 + d >= 0);
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VERIFY(b.eval(x, y0) < 0);
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VERIFY(b.eval(x-1, y0) >= 0);
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VERIFY(x_min <= x && x <= x_max);
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*x_split = x;
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}
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}
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VERIFY(-M < a*x_min*y0 + b*x_min + c*y0 + d);
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VERIFY(a*x_min*y0 + b*x_min + c*y0 + d < M);
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VERIFY(-M < a*x_max*y0 + b*x_max + c*y0 + d);
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VERIFY(a*x_max*y0 + b*x_max + c*y0 + d < M);
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VERIFY(-M < b.eval(x_min, y0));
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VERIFY(b.eval(x_min, y0) < M);
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VERIFY(-M < b.eval(x_max, y0));
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VERIFY(b.eval(x_max, y0) < M);
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return true;
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}
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@ -1648,14 +1649,15 @@ namespace polysat {
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* Based on a*x*y + b*x + c*y + d >= 0
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* update lower bound for y
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*/
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bool saturation::update_min(rational& y_min, rational const& x_min, rational const& x_max, rational const& a, rational const& b, rational const& c, rational const& d) {
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if (a == 0 && c == 0)
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bool saturation::update_min(rational& y_min, rational const& x_min, rational const& x_max,
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bilinear const& b) {
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if (b.a == 0 && b.c == 0)
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return true;
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rational x_bound;
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if (a >= 0 && b >= 0)
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if (b.a >= 0 && b.b >= 0)
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x_bound = x_min;
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else if (a <= 0 && b <= 0)
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else if (b.a <= 0 && b.b <= 0)
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x_bound = x_max;
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else
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return false;
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@ -1663,23 +1665,24 @@ namespace polysat {
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// a*x_bound*y + b*x_bound + c*y + d >= 0
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// (a*x_bound + c)*y >= -d - b*x_bound
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// if a*x_bound + c > 0
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rational A = a*x_bound + c;
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rational A = b.a*x_bound + b.c;
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if (A <= 0)
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return true;
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rational y1 = ceil((- d - b*x_bound)/A);
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rational y1 = ceil((- b.d - b.b*x_bound)/A);
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if (y1 > y_min)
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y_min = y1;
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return true;
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}
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bool saturation::update_max(rational& y_max, rational const& x_min, rational const& x_max, rational const& a, rational const& b, rational const& c, rational const& d) {
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if (a == 0 && c == 0)
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bool saturation::update_max(rational& y_max, rational const& x_min, rational const& x_max,
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bilinear const& b) {
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if (b.a == 0 && b.c == 0)
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return true;
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rational x_bound;
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if (a >= 0 && b >= 0)
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if (b.a >= 0 && b.b >= 0)
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x_bound = x_min;
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else if (a <= 0 && b <= 0)
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else if (b.a <= 0 && b.b <= 0)
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x_bound = x_max;
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else
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return false;
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@ -1687,10 +1690,10 @@ namespace polysat {
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// a*x_bound*y + b*x_bound + c*y + d >= 0
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// (a*x_bound + c)*y >= -d - b*x_bound
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// if a*x_bound + c < 0
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rational A = a*x_bound + c;
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rational A = b.a*x_bound + b.c;
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if (A >= 0)
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return true;
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rational y1 = floor((- d - b*x_bound)/A);
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rational y1 = floor((- b.d - b.b*x_bound)/A);
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if (y1 < y_max)
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y_max = y1;
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return true;
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@ -1721,57 +1724,57 @@ namespace polysat {
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}
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}
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bool saturation::update_bounds_for_xs(rational const& x_min, rational const& x_max, rational& y_min, rational& y_max, rational const& y0, rational const& a1, rational const& b1, rational const& c1, rational const& dd1, rational const& a2, rational const& b2, rational const& c2, rational const& dd2, rational const& M, inequality const& a_l_b) {
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bool saturation::update_bounds_for_xs(rational const& x_min, rational const& x_max, rational& y_min, rational& y_max, rational const& y0, bilinear const& b1, bilinear const& b2, rational const& M, inequality const& a_l_b) {
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VERIFY(x_min <= x_max);
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rational d1 = dd1;
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if (a1*x_min*y0 + b1*x_min + c1*y0 + d1 < 0)
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rational d1 = b1.d;
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if (b1.eval(x_min, y0) < 0)
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d1 += M;
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rational d2 = dd2;
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if (a2*x_min*y0 + b2*x_min + c2*y0 + d2 < 0)
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rational d2 = b2.d;
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if (b2.eval(x_min, y0) < 0)
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d2 += M;
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IF_VERBOSE(2,
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verbose_stream() << "Adjusted for x in [" << x_min << "; " << x_max << "]\n";
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verbose_stream() << "p ... " << a1 << " " << b1 << " " << c1 << " " << d1 << "\n";
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verbose_stream() << "q ... " << a2 << " " << b2 << " " << c2 << " " << d2 << "\n";
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verbose_stream() << "p ... " << b1 << "\n";
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verbose_stream() << "q ... " << b2 << "\n";
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);
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// Precondition: forall x . x_min <= x <= x_max ==> p(x,y0) > q(x,y0)
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// check the endpoints
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VERIFY(a1*x_min*y0 + b1*x_min + c1*y0 + d1 >= a2*x_min*y0 + b2*x_min + c2*y0 + d2 + (a_l_b.is_strict() ? 0 : 1));
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VERIFY(a1*x_max*y0 + b1*x_max + c1*y0 + d1 >= a2*x_max*y0 + b2*x_max + c2*y0 + d2 + (a_l_b.is_strict() ? 0 : 1));
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VERIFY(b1.eval(x_min, y0) >= b2.eval(x_min, y0) + (a_l_b.is_strict() ? 0 : 1));
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VERIFY(b1.eval(x_max, y0) >= b2.eval(x_max, y0) + (a_l_b.is_strict() ? 0 : 1));
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if (!update_min(y_min, x_min, x_max, a1, b1, c1, d1))
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if (!update_min(y_min, x_min, x_max, b1))
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return false;
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if (!update_min(y_min, x_min, x_max, a2, b2, c2, d2))
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if (!update_min(y_min, x_min, x_max, b2))
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return false;
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//verbose_stream() << "min-max: x := v" << x << " [" << x_min << "," << x_max << "] y := v" << y << " [" << y_min << ", " << y_max << "] y0 " << y0 << "\n";
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VERIFY(y_min <= y0 && y0 <= y_max);
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if (!update_max(y_max, x_min, x_max, a1, b1, c1, d1))
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if (!update_max(y_max, x_min, x_max, b1))
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return false;
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if (!update_max(y_max, x_min, x_max, a2, b2, c2, d2))
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if (!update_max(y_max, x_min, x_max, b2))
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return false;
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//verbose_stream() << "min-max: x := v" << x << " [" << x_min << "," << x_max << "] y := v" << y << " [" << y_min << ", " << y_max << "] y0 " << y0 << "\n";
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VERIFY(y_min <= y0 && y0 <= y_max);
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// p < M iff -p > -M iff -p + M - 1 >= 0
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if (!update_min(y_min, x_min, x_max, -a1, -b1, -c1, -d1 + M - 1))
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if (!update_min(y_min, x_min, x_max, -b1 + (M - 1)))
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return false;
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if (!update_min(y_min, x_min, x_max, -a2, -b2, -c2, -d2 + M - 1))
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if (!update_min(y_min, x_min, x_max, -b2 + (M - 1)))
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return false;
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if (!update_max(y_max, x_min, x_max, -a1, -b1, -c1, -d1 + M - 1))
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if (!update_max(y_max, x_min, x_max, -b1 + (M - 1)))
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return false;
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if (!update_max(y_max, x_min, x_max, -a2, -b2, -c2, -d2 + M - 1))
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if (!update_max(y_max, x_min, x_max, -b2 + (M - 1)))
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return false;
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VERIFY(y_min <= y0 && y0 <= y_max);
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// p <= q or p < q is false
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// so p > q or p >= q
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// p - q - 1 >= 0 or p - q >= 0
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// min-max for p - q - 1 or p - q are non-negative
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if (!update_min(y_min, x_min, x_max, a1 - a2, b1 - b2, c1 - c2, d1 - d2 - (a_l_b.is_strict() ? 0 : 1)))
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if (!update_min(y_min, x_min, x_max, b1 - b2 - (a_l_b.is_strict() ? 0 : 1)))
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return false;
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if (!update_max(y_max, x_min, x_max, a1 - a2, b1 - b2, c1 - c2, d1 - d2 - (a_l_b.is_strict() ? 0 : 1)))
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if (!update_max(y_max, x_min, x_max, b1 - b2 - (a_l_b.is_strict() ? 0 : 1)))
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return false;
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return true;
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}
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@ -1791,10 +1794,10 @@ namespace polysat {
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return false;
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pvar y = null_var;
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rational a1, a2, b1, b2, c1, c2, d1, d2;
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if (!extract_bilinear_form(x, p, y, a1, b1, c1, d1))
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bilinear b1, b2;
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if (!extract_bilinear_form(x, p, y, b1))
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return false;
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if (!extract_bilinear_form(x, q, y, a2, b2, c2, d2))
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if (!extract_bilinear_form(x, q, y, b2))
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return false;
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if (y == null_var)
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return false;
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@ -1835,15 +1838,15 @@ namespace polysat {
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verbose_stream() << "\n";
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verbose_stream() << "x_min " << x_min << " x_max " << x_max << "\n";
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verbose_stream() << "v" << y << " " << y0 << "\n";
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verbose_stream() << p << " ... " << a1 << " " << b1 << " " << c1 << " " << d1 << "\n";
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verbose_stream() << q << " ... " << a2 << " " << b2 << " " << c2 << " " << d2 << "\n");
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verbose_stream() << p << " ... " << b1 << "\n";
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verbose_stream() << q << " ... " << b2 << "\n");
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rational x_sp1 = x_min;
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rational x_sp2 = x_min;
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if (!adjust_bound(x_min, x_max, y0, M, a1, b1, c1, d1, &x_sp1))
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if (!adjust_bound(x_min, x_max, y0, M, b1, &x_sp1))
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return false;
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if (!adjust_bound(x_min, x_max, y0, M, a2, b2, c2, d2, &x_sp2))
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if (!adjust_bound(x_min, x_max, y0, M, b2, &x_sp2))
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return false;
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if (x_sp1 > x_sp2)
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@ -1852,8 +1855,8 @@ namespace polysat {
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IF_VERBOSE(2,
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verbose_stream() << "Adjusted\n";
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verbose_stream() << p << " ... " << a1 << " " << b1 << " " << c1 << " " << d1 << "\n";
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verbose_stream() << q << " ... " << a2 << " " << b2 << " " << c2 << " " << d2 << "\n";
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verbose_stream() << p << " ... " << b1 << "\n";
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verbose_stream() << q << " ... " << b2 << "\n";
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// verbose_stream() << "p(x_min,y0) = " << (a1*x_min*y0 + b1*x_min + c1*y0 + d1) << "\n";
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// verbose_stream() << "q(x_min,y0) = " << (a2*x_min*y0 + b2*x_min + c2*y0 + d2) << "\n";
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// verbose_stream() << "p(x_max,y0) = " << (a1*x_max*y0 + b1*x_max + c1*y0 + d1) << "\n";
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@ -1861,13 +1864,13 @@ namespace polysat {
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);
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rational y_min(0), y_max(M-1);
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if (x_min != x_sp1 && !update_bounds_for_xs(x_min, x_sp1-1, y_min, y_max, y0, a1, b1, c1, d1, a2, b2, c2, d2, M, a_l_b))
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if (x_min != x_sp1 && !update_bounds_for_xs(x_min, x_sp1-1, y_min, y_max, y0, b1, b2, M, a_l_b))
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return false;
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// IF_VERBOSE(0, verbose_stream() << "min-max: x := v" << x << " [" << x_min << "," << x_max << "] y := v" << y << " [" << y_min << ", " << y_max << "] y0 " << y0 << "\n");
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if (x_sp1 != x_sp2 && !update_bounds_for_xs(x_sp1, x_sp2-1, y_min, y_max, y0, a1, b1, c1, d1, a2, b2, c2, d2, M, a_l_b))
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if (x_sp1 != x_sp2 && !update_bounds_for_xs(x_sp1, x_sp2-1, y_min, y_max, y0, b1, b2, M, a_l_b))
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return false;
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// IF_VERBOSE(0, verbose_stream() << "min-max: x := v" << x << " [" << x_min << "," << x_max << "] y := v" << y << " [" << y_min << ", " << y_max << "] y0 " << y0 << "\n");
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if (!update_bounds_for_xs(x_sp2, x_max, y_min, y_max, y0, a1, b1, c1, d1, a2, b2, c2, d2, M, a_l_b))
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if (!update_bounds_for_xs(x_sp2, x_max, y_min, y_max, y0, b1, b2, M, a_l_b))
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return false;
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IF_VERBOSE(1, verbose_stream() << "min-max: x := v" << x << " [" << x_min << "," << x_max << "] y := v" << y << " [" << y_min << ", " << y_max << "] y0 " << y0 << "\n");
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@ -18,6 +18,55 @@ Author:
|
|||
|
||||
namespace polysat {
|
||||
|
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struct bilinear {
|
||||
rational a, b, c, d;
|
||||
|
||||
|
||||
rational eval(rational const& x, rational const& y) const {
|
||||
return a*x*y + b*x + c*y + d;
|
||||
}
|
||||
|
||||
bilinear operator-() const {
|
||||
bilinear r(*this);
|
||||
r.a = -r.a;
|
||||
r.b = -r.b;
|
||||
r.c = -r.c;
|
||||
r.d = -r.d;
|
||||
return r;
|
||||
}
|
||||
|
||||
bilinear operator-(bilinear const& other) const {
|
||||
bilinear r(*this);
|
||||
r.a -= other.a;
|
||||
r.b -= other.b;
|
||||
r.c -= other.c;
|
||||
r.d -= other.d;
|
||||
return r;
|
||||
}
|
||||
|
||||
bilinear operator+(rational const& d) const {
|
||||
bilinear r(*this);
|
||||
r.d += d;
|
||||
return r;
|
||||
}
|
||||
|
||||
bilinear operator-(rational const& d) const {
|
||||
bilinear r(*this);
|
||||
r.d -= d;
|
||||
return r;
|
||||
}
|
||||
|
||||
bilinear operator-(int d) const {
|
||||
bilinear r(*this);
|
||||
r.d -= d;
|
||||
return r;
|
||||
}
|
||||
};
|
||||
|
||||
inline std::ostream& operator<<(std::ostream& out, bilinear const& b) {
|
||||
return out << b.a << "*x*y + " << b.b << "*x + " << b.c << "*y + " << b.d;
|
||||
}
|
||||
|
||||
/**
|
||||
* Introduce lemmas that derive new (simpler) constraints from the current conflict and partial model.
|
||||
*/
|
||||
|
|
@ -75,11 +124,16 @@ namespace polysat {
|
|||
rational round(rational const& M, rational const& x);
|
||||
bool eval_round(rational const& M, pdd const& p, rational& r);
|
||||
bool extract_linear_form(pdd const& q, pvar& y, rational& a, rational& b);
|
||||
bool extract_bilinear_form(pvar x, pdd const& p, pvar& y, rational& a, rational& b, rational& c, rational& d);
|
||||
bool adjust_bound(rational const& x_min, rational const& x_max, rational const& y0, rational const& M, rational const& a, rational const& b, rational const& c, rational& d, rational* x_split);
|
||||
bool update_min(rational& y_min, rational const& x_min, rational const& x_max, rational const& a, rational const& b, rational const& c, rational const& d);
|
||||
bool update_max(rational& y_max, rational const& x_min, rational const& x_max, rational const& a, rational const& b, rational const& c, rational const& d);
|
||||
bool update_bounds_for_xs(rational const& x_min, rational const& x_max, rational& y_min, rational& y_max, rational const& y0, rational const& a1, rational const& b1, rational const& c1, rational const& d1, rational const& a2, rational const& b2, rational const& c2, rational const& d2, rational const& M, inequality const& a_l_b);
|
||||
bool extract_bilinear_form(pvar x, pdd const& p, pvar& y, bilinear& b);
|
||||
bool adjust_bound(rational const& x_min, rational const& x_max, rational const& y0, rational const& M,
|
||||
bilinear& b, rational* x_split);
|
||||
bool update_min(rational& y_min, rational const& x_min, rational const& x_max,
|
||||
bilinear const& b);
|
||||
bool update_max(rational& y_max, rational const& x_min, rational const& x_max,
|
||||
bilinear const& b);
|
||||
bool update_bounds_for_xs(rational const& x_min, rational const& x_max, rational& y_min, rational& y_max,
|
||||
rational const& y0, bilinear const& b1, bilinear const& b2,
|
||||
rational const& M, inequality const& a_l_b);
|
||||
void fix_values(pvar x, pvar y, pdd const& p);
|
||||
void fix_values(pvar y, pdd const& p);
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue