z3-src 501.0.0

Source distribution of the Z3 SMT solver, for use as a build dependency
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261

/*++
  Copyright (c) 2020 Microsoft Corporation

  Module Name:

  lcube.cpp

  Abstract:

  Tests for the largest cube test of Bromberger and Weidenbach
  (Fast Cube Tests for LIA Constraint Solving, IJCAR 2016),
  implemented in int_cube::find_largest_cube().

  This file lives directly under src/test (not src/test/lp) so that the
  scripts/mk_make.py build, which only compiles the top-level test
  directory, links tst_lcube().

  Author:

  Lev Nachmanson (levnach)

  --*/

#include <initializer_list>
#include <iostream>
#include <utility>

#include "util/debug.h"
#include "util/params.h"
#include "math/lp/int_cube.h"
#include "math/lp/int_solver.h"
#include "math/lp/lar_solver.h"
#include "math/lp/numeric_pair.h"

namespace lp {

// tests for the largest cube test of Bromberger and Weidenbach
namespace lcube_test {

    struct ineq {
        vector<std::pair<mpq, unsigned>> m_coeffs;
        lconstraint_kind m_kind;
        mpq m_rs;
    };

    // builds for every inequality a term with the bound and solves
    static void setup(lar_solver& solver, const vector<ineq>& ineqs, svector<unsigned>* term_columns = nullptr) {
        unsigned term_ext = 1000;
        for (const auto& in : ineqs) {
            unsigned t = solver.add_term(in.m_coeffs, term_ext++);
            solver.add_var_bound(t, in.m_kind, in.m_rs);
            if (term_columns)
                term_columns->push_back(t);
        }
        auto st = solver.solve();
        VERIFY(st == lp_status::OPTIMAL || st == lp_status::FEASIBLE);
    }

    static void verify_model(const lar_solver& solver, const vector<ineq>& ineqs) {
        for (const auto& in : ineqs) {
            impq v;
            for (const auto& p : in.m_coeffs)
                v += solver.get_column_value(p.second) * p.first;
            switch (in.m_kind) {
            case lconstraint_kind::LE: VERIFY(v <= impq(in.m_rs)); break;
            case lconstraint_kind::GE: VERIFY(v >= impq(in.m_rs)); break;
            default: VERIFY(false);
            }
        }
    }

    static void verify_int_values(const lar_solver& solver, std::initializer_list<unsigned> vars) {
        for (unsigned j : vars)
            VERIFY(solver.get_column_value(j).is_int());
    }

    // The example of Bromberger and Weidenbach: 3x1 - x2 <= 0, -2x1 - x2 <= -2, -2x1 + x2 <= 1.
    // The largest cube is smaller than the unit cube, the rounded center is not a solution,
    // no coordinate flip repairs it (the only integer solution lies outside the lattice cell
    // of the center): expect undef and an intact solver state.
    static void test_paper_example_undef() {
        std::cout << "lcube: paper example, expecting undef\n";
        lar_solver solver;
        unsigned x1 = solver.add_named_var(0, true, "x1");
        unsigned x2 = solver.add_named_var(1, true, "x2");
        vector<ineq> ineqs;
        ineqs.push_back(ineq{{{mpq(3), x1}, {mpq(-1), x2}}, lconstraint_kind::LE, mpq(0)});
        ineqs.push_back(ineq{{{mpq(-2), x1}, {mpq(-1), x2}}, lconstraint_kind::LE, mpq(-2)});
        ineqs.push_back(ineq{{{mpq(-2), x1}, {mpq(1), x2}}, lconstraint_kind::LE, mpq(1)});
        setup(solver, ineqs);
        int_solver i_s(solver);
        solver.set_int_solver(&i_s);
        lia_move m = int_cube(i_s).find_largest_cube();
        std::cout << "lcube returned " << lia_move_to_string(m) << "\n";
        VERIFY(m == lia_move::undef);
        VERIFY(solver.ax_is_correct());
    }

    // 3x1 - x2 <= 0, -2x1 - x2 <= -1, -2x1 + x2 <= 1: the largest cube has
    // edge 4/17 with the center (3/17, 1) that rounds to the solution (0, 1),
    // while the unit cube test fails: the largest cube test is stronger here.
    static void test_beats_unit_cube() {
        std::cout << "lcube: beating the unit cube test\n";
        lar_solver solver;
        unsigned x1 = solver.add_named_var(0, true, "x1");
        unsigned x2 = solver.add_named_var(1, true, "x2");
        vector<ineq> ineqs;
        ineqs.push_back(ineq{{{mpq(3), x1}, {mpq(-1), x2}}, lconstraint_kind::LE, mpq(0)});
        ineqs.push_back(ineq{{{mpq(-2), x1}, {mpq(-1), x2}}, lconstraint_kind::LE, mpq(-1)});
        ineqs.push_back(ineq{{{mpq(-2), x1}, {mpq(1), x2}}, lconstraint_kind::LE, mpq(1)});
        svector<unsigned> tcols;
        setup(solver, ineqs, &tcols);
        // move the solution to a fractional feasible point: the cube tests only
        // run when the current solution has fractional integer variables
        solver.set_column_value_test(x1, impq(mpq(1, 4)));
        solver.set_column_value_test(x2, impq(mpq(5, 4)));
        solver.set_column_value_test(tcols[0], impq(mpq(-1, 2)));
        solver.set_column_value_test(tcols[1], impq(mpq(-7, 4)));
        solver.set_column_value_test(tcols[2], impq(mpq(3, 4)));
        int_solver i_s(solver);
        solver.set_int_solver(&i_s);
        lia_move m = int_cube(i_s)();
        std::cout << "unit cube returned " << lia_move_to_string(m) << "\n";
        VERIFY(m == lia_move::undef);
        m = int_cube(i_s).find_largest_cube();
        std::cout << "lcube returned " << lia_move_to_string(m) << "\n";
        VERIFY(m == lia_move::sat);
        verify_int_values(solver, {x1, x2});
        verify_model(solver, ineqs);
    }

    // 9/10 <= x + y + r <= 11/10, -11/10 <= x - y + r <= 1/10, 0 <= r <= 1/10,
    // x, y integer, r real. The real variable keeps the terms and their bounds
    // non-integer. A fractional center, e.g. (1/2, 1/2), rounds to an infeasible
    // point that is repaired by flipping one coordinate: expect sat.
    static void test_flip_repair() {
        std::cout << "lcube: rounding repair\n";
        lar_solver solver;
        unsigned x = solver.add_named_var(0, true, "x");
        unsigned y = solver.add_named_var(1, true, "y");
        unsigned r = solver.add_named_var(2, false, "r");
        solver.add_var_bound(r, lconstraint_kind::GE, mpq(0));
        solver.add_var_bound(r, lconstraint_kind::LE, mpq(1, 10));
        vector<ineq> ineqs;
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(1), y}, {mpq(1), r}}, lconstraint_kind::GE, mpq(9, 10)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(1), y}, {mpq(1), r}}, lconstraint_kind::LE, mpq(11, 10)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(-1), y}, {mpq(1), r}}, lconstraint_kind::GE, mpq(-11, 10)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(-1), y}, {mpq(1), r}}, lconstraint_kind::LE, mpq(1, 10)});
        setup(solver, ineqs);
        int_solver i_s(solver);
        solver.set_int_solver(&i_s);
        lia_move m = int_cube(i_s).find_largest_cube();
        std::cout << "lcube returned " << lia_move_to_string(m)
                  << ", flip successes: " << solver.settings().stats().m_lcube_flip_success << "\n";
        VERIFY(m == lia_move::sat);
        verify_int_values(solver, {x, y});
        verify_model(solver, ineqs);
    }

    // 3x + 5y >= 7 alone has infinite lattice width: the edge length is
    // unbounded and any cube center of edge >= 1 rounds to a solution.
    static void test_infinite_lattice_width() {
        std::cout << "lcube: infinite lattice width\n";
        lar_solver solver;
        unsigned x = solver.add_named_var(0, true, "x");
        unsigned y = solver.add_named_var(1, true, "y");
        vector<ineq> ineqs;
        ineqs.push_back(ineq{{{mpq(3), x}, {mpq(5), y}}, lconstraint_kind::GE, mpq(7)});
        setup(solver, ineqs);
        int_solver i_s(solver);
        solver.set_int_solver(&i_s);
        lia_move m = int_cube(i_s).find_largest_cube();
        std::cout << "lcube returned " << lia_move_to_string(m) << "\n";
        VERIFY(m == lia_move::sat);
        verify_int_values(solver, {x, y});
        verify_model(solver, ineqs);
    }

    // 0 <= x + 2y + r <= 8, 0 <= x - 2y + r <= 8: the maximal edge length is
    // 8/3 >= 1, so the rounded center is guaranteed to be a solution.
    static void test_edge_at_least_one() {
        std::cout << "lcube: edge length at least 1\n";
        lar_solver solver;
        unsigned x = solver.add_named_var(0, true, "x");
        unsigned y = solver.add_named_var(1, true, "y");
        unsigned r = solver.add_named_var(2, false, "r");
        solver.add_var_bound(r, lconstraint_kind::GE, mpq(0));
        solver.add_var_bound(r, lconstraint_kind::LE, mpq(1, 10));
        vector<ineq> ineqs;
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(2), y}, {mpq(1), r}}, lconstraint_kind::GE, mpq(0)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(2), y}, {mpq(1), r}}, lconstraint_kind::LE, mpq(8)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(-2), y}, {mpq(1), r}}, lconstraint_kind::GE, mpq(0)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(-2), y}, {mpq(1), r}}, lconstraint_kind::LE, mpq(8)});
        setup(solver, ineqs);
        int_solver i_s(solver);
        solver.set_int_solver(&i_s);
        lia_move m = int_cube(i_s).find_largest_cube();
        std::cout << "lcube returned " << lia_move_to_string(m) << "\n";
        VERIFY(m == lia_move::sat);
        verify_int_values(solver, {x, y});
        verify_model(solver, ineqs);
    }

    // runs the flip-repair instance through int_solver::check() with the
    // lp.lcube parameter set and the cube period lowered to 1: verifies the
    // dispatch and the parameter plumbing
    static void test_dispatch() {
        std::cout << "lcube: dispatch through int_solver::check\n";
        lar_solver solver;
        params_ref p;
        p.set_bool("lcube", true);
        solver.settings().updt_params(p);
        VERIFY(solver.settings().lcube());
        solver.settings().m_int_find_cube_period = 1;
        unsigned x = solver.add_named_var(0, true, "x");
        unsigned y = solver.add_named_var(1, true, "y");
        unsigned r = solver.add_named_var(2, false, "r");
        solver.add_var_bound(r, lconstraint_kind::GE, mpq(0));
        solver.add_var_bound(r, lconstraint_kind::LE, mpq(1, 10));
        vector<ineq> ineqs;
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(1), y}, {mpq(1), r}}, lconstraint_kind::GE, mpq(9, 10)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(1), y}, {mpq(1), r}}, lconstraint_kind::LE, mpq(11, 10)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(-1), y}, {mpq(1), r}}, lconstraint_kind::GE, mpq(-11, 10)});
        ineqs.push_back(ineq{{{mpq(1), x}, {mpq(-1), y}, {mpq(1), r}}, lconstraint_kind::LE, mpq(1, 10)});
        svector<unsigned> tcols;
        setup(solver, ineqs, &tcols);
        // a fractional feasible point, so that check() does not return sat right away
        solver.set_column_value_test(x, impq(mpq(1, 2)));
        solver.set_column_value_test(y, impq(mpq(1, 2)));
        solver.set_column_value_test(r, impq(mpq(1, 10)));
        solver.set_column_value_test(tcols[0], impq(mpq(11, 10)));
        solver.set_column_value_test(tcols[1], impq(mpq(11, 10)));
        solver.set_column_value_test(tcols[2], impq(mpq(1, 10)));
        solver.set_column_value_test(tcols[3], impq(mpq(1, 10)));
        int_solver i_s(solver);
        solver.set_int_solver(&i_s);
        explanation ex;
        lia_move m = i_s.check(&ex);
        std::cout << "check returned " << lia_move_to_string(m)
                  << ", lcube calls: " << solver.settings().stats().m_lcube_calls << "\n";
        VERIFY(m == lia_move::sat);
        VERIFY(solver.settings().stats().m_lcube_calls >= 1);
        verify_int_values(solver, {x, y});
        verify_model(solver, ineqs);
    }

    static void run() {
        test_paper_example_undef();
        test_beats_unit_cube();
        test_flip_repair();
        test_infinite_lattice_width();
        test_edge_at_least_one();
        test_dispatch();
        std::cout << "lcube tests passed\n";
    }
} // namespace lcube_test
}  // namespace lp

void tst_lcube() {
    lp::lcube_test::run();
}