qe_arith.cpp

/*++
Copyright (c) 2015 Microsoft Corporation

--*/

#include "qe/mbp/mbp_arith.h"
#include "qe/qe.h"
#include "ast/rewriter/th_rewriter.h"
#include "parsers/smt2/smt2parser.h"
#include "ast/arith_decl_plugin.h"
#include "ast/reg_decl_plugins.h"
#include "ast/rewriter/arith_rewriter.h"
#include "ast/ast_pp.h"
#include "smt/smt_context.h"
#include "ast/expr_abstract.h"
#include "ast/rewriter/expr_safe_replace.h"

static expr_ref parse_fml(ast_manager& m, char const* str) {
    expr_ref result(m);
    cmd_context ctx(false, &m);
    ctx.set_ignore_check(true);
    std::ostringstream buffer;
    buffer << "(declare-const x Real)\n"
           << "(declare-const y Real)\n"
           << "(declare-const z Real)\n"
           << "(declare-const u Real)\n"
           << "(declare-const v Real)\n"
           << "(declare-const t Real)\n"
           << "(declare-const a Real)\n"
           << "(declare-const b Real)\n"
           << "(declare-const i Int)\n"
           << "(declare-const j Int)\n"
           << "(declare-const k Int)\n"
           << "(declare-const l Int)\n"
           << "(declare-const m Int)\n"
           << "(assert " << str << ")\n";
    std::istringstream is(buffer.str());
    VERIFY(parse_smt2_commands(ctx, is));
    result = ctx.assertions().get(0);
    return result;
}

static char const* example1 = "(and (<= x 3.0) (<= (* 3.0 x) y) (<= z y))";
static char const* example2 = "(and (<= z x) (<= x 3.0) (<= (* 3.0 x) y) (<= z y))";
static char const* example3 = "(and (<= z x) (<= x 3.0) (< (* 3.0 x) y) (<= z y))";
static char const* example4 = "(and (<= z x) (<= x 3.0) (not (>= (* 3.0 x) y)) (<= z y))";
static char const* example5 = "(and (<= y x) (<= z x) (<= x u) (<= x v) (<= x t))";
static char const* example7 = "(and (<= x y) (<= x z) (<= u x) (< v x))";
static char const* example8 = "(and  (<= (* 2 i) k) (<= j (* 3 i)))";

static char const* example6 = "(and (<= 0 (+ x z))\
     (>= y x) \
     (<= y x)\
     (<= (- u y) 0.0)\
     (>= x (+ v z))\
     (>= x 0.0)\
     (<= x 1.0))";

// phi[M] => result => E x . phi[x] 

static void test(app* var, expr_ref& fml) {

    ast_manager& m = fml.get_manager();
    smt_params params;
    params.m_model = true;

    symbol x_name(var->get_decl()->get_name());   
    sort* x_sort = var->get_sort();

    expr_ref pr(m);
    expr_ref_vector lits(m);
    flatten_and(fml, lits);

    model_ref md;
    {
        smt::context ctx(m, params);
        ctx.assert_expr(fml);
        lbool result = ctx.check();
        if (result != l_true) return;
        ctx.get_model(md);
    }    
    VERIFY(mbp::arith_project(*md, var, lits));
    pr = mk_and(lits);
   
    std::cout << "original:  " << mk_pp(fml, m) << "\n";
    std::cout << "projected: " << mk_pp(pr,  m) << "\n";

    // projection is consistent with model.
    VERIFY(md->is_true(pr));

    // projection implies E x. fml
    {
        qe::expr_quant_elim qelim(m, params);
        expr_ref result(m), efml(m);
        expr* x = var;
        expr_abstract(m, 0, 1, &x, fml, efml);
        efml = m.mk_exists(1, &x_sort, &x_name, efml);
        qelim(m.mk_true(), efml, result);

        smt::context ctx(m, params);
        ctx.assert_expr(pr);
        ctx.assert_expr(m.mk_not(result));
        std::cout << "exists: " << pr << " =>\n" << result << "\n";
        VERIFY(l_false == ctx.check());
    }    

    std::cout << "\n";
}



static void testR(char const *ex) {
    ast_manager m;
    reg_decl_plugins(m);
    arith_util a(m);
    expr_ref fml = parse_fml(m, ex);
    symbol x_name("x");
    sort_ref x_sort(a.mk_real(), m);
    app_ref var(m.mk_const(x_name, x_sort), m);
    test(var, fml);
}

static void testI(char const *ex) {
    ast_manager m;
    reg_decl_plugins(m);
    arith_util a(m);
    expr_ref fml = parse_fml(m, ex);
    symbol x_name("i");
    sort_ref x_sort(a.mk_int(), m);
    app_ref var(m.mk_const(x_name, x_sort), m);
    test(var, fml);
}

static expr_ref_vector mk_ineqs(app* x, app* y, app_ref_vector const& nums) {
    ast_manager& m = nums.get_manager();
    arith_util a(m);
    expr_ref_vector result(m);
    for (unsigned i = 0; i < nums.size(); ++i) {
        expr_ref ax(a.mk_mul(nums[i], x), m);
        result.push_back(a.mk_le(ax, y));
        result.push_back(m.mk_not(a.mk_ge(ax, y)));
        result.push_back(m.mk_not(a.mk_gt(y, ax)));
        result.push_back(a.mk_lt(y, ax));
    }
    return result;
}

static app_ref generate_ineqs(ast_manager& m, sort* s, vector<expr_ref_vector>& cs, bool mods_too) {
    arith_util a(m);
    app_ref_vector vars(m), nums(m);
    vars.push_back(m.mk_const(symbol("x"), s));
    vars.push_back(m.mk_const(symbol("y"), s));
    vars.push_back(m.mk_const(symbol("z"), s));
    vars.push_back(m.mk_const(symbol("u"), s));
    vars.push_back(m.mk_const(symbol("v"), s));
    vars.push_back(m.mk_const(symbol("w"), s));
    nums.push_back(a.mk_numeral(rational(1),  s));
    nums.push_back(a.mk_numeral(rational(2),  s));
    nums.push_back(a.mk_numeral(rational(3),  s));
    
    app* x = vars[0].get();
    app* y = vars[1].get();
    // app* z = vars[2].get();
    // 
    // ax <= by, ax < by, not (ax >= by), not (ax > by)
    // 
    cs.push_back(mk_ineqs(x, vars[1].get(), nums));
    cs.push_back(mk_ineqs(x, vars[2].get(), nums));
    cs.push_back(mk_ineqs(x, vars[3].get(), nums));
    cs.push_back(mk_ineqs(x, vars[4].get(), nums));
    cs.push_back(mk_ineqs(x, vars[5].get(), nums));

    if (mods_too) {
        expr_ref_vector mods(m);
        expr_ref zero(a.mk_numeral(rational(0), s), m);
        mods.push_back(m.mk_true());
        for (unsigned j = 0; j < nums.size(); ++j) {
            mods.push_back(m.mk_eq(a.mk_mod(a.mk_add(a.mk_mul(nums[j].get(),x), y), nums[1].get()), zero));
        }
        cs.push_back(mods);
        mods.resize(1);
        for (unsigned j = 0; j < nums.size(); ++j) {
            mods.push_back(m.mk_eq(a.mk_mod(a.mk_add(a.mk_mul(nums[j].get(),x), y), nums[2].get()), zero));
        }
        cs.push_back(mods);
    }
    return app_ref(x, m);
}


static void test_c(app* x, expr_ref_vector const& c) {
    ast_manager& m = c.get_manager();
    expr_ref fml(m);
    fml = m.mk_and(c.size(), c.data());
    test(x, fml);
}

static void test_cs(app* x, expr_ref_vector& c, vector<expr_ref_vector> const& cs) {
    if (c.size() == cs.size()) {
        test_c(x, c);
        return;
    }
    expr_ref_vector const& c1 = cs[c.size()];
    for (unsigned i = 0; i < c1.size(); ++i) {
        c.push_back(c1[i]);
        test_cs(x, c, cs);
        c.pop_back();
    }
}


static void test_ineqs(ast_manager& m, sort* s, bool mods_too) {
    vector<expr_ref_vector> ineqs;
    app_ref x = generate_ineqs(m, s, ineqs, mods_too);
    expr_ref_vector cs(m);
    test_cs(x, cs, ineqs);
}

static void test_ineqs() {
    ast_manager m;
    reg_decl_plugins(m);
    arith_util a(m);
    sort* s_int = a.mk_int();
    sort* s_real = a.mk_real();
    test_ineqs(m, s_int, true);
    test_ineqs(m, s_real, false);
}

static void test2(char const *ex) {
    smt_params params;
    params.m_model = true;
    ast_manager m;
    reg_decl_plugins(m);
    arith_util a(m);
    expr_ref fml = parse_fml(m, ex);
    app_ref_vector vars(m);
    expr_ref_vector lits(m);
    vars.push_back(m.mk_const(symbol("x"), a.mk_real()));
    vars.push_back(m.mk_const(symbol("y"), a.mk_real()));
    vars.push_back(m.mk_const(symbol("z"), a.mk_real()));
    flatten_and(fml, lits);

    smt::context ctx(m, params);
    ctx.push();
    ctx.assert_expr(fml);
    lbool result = ctx.check();
    VERIFY(result == l_true);
    ref<model> md;
    ctx.get_model(md);  
    ctx.pop(1);
    
    std::cout << mk_pp(fml, m) << "\n";

    expr_ref pr1(m), pr2(m), fml2(m);
    expr_ref_vector bound(m);
    ptr_vector<sort> sorts;
    svector<symbol> names;
    for (unsigned i = 0; i < vars.size(); ++i) {
        bound.push_back(vars[i].get());
        names.push_back(vars[i]->get_decl()->get_name());
        sorts.push_back(vars[i]->get_sort());
    }
    expr_abstract(m, 0, bound.size(), bound.data(), fml, fml2);
    fml2 = m.mk_exists(bound.size(), sorts.data(), names.data(), fml2);
    qe::expr_quant_elim qe(m, params);
    for (unsigned i = 0; i < vars.size(); ++i) {
        VERIFY(mbp::arith_project(*md, vars[i].get(), lits));
    }
    pr1 = mk_and(lits);
    qe(m.mk_true(), fml2, pr2);
    std::cout << mk_pp(pr1, m) << "\n";
    std::cout << mk_pp(pr2, m) << "\n";

    expr_ref npr2(m);
    npr2 = m.mk_not(pr2);
    ctx.push();
    ctx.assert_expr(pr1);
    ctx.assert_expr(npr2);
    VERIFY(l_false == ctx.check());
    ctx.pop(1);       
}

typedef opt::model_based_opt::var var_t;

static void mk_var(unsigned x, app_ref& v) {
    ast_manager& m = v.get_manager();
    arith_util a(m);
    std::ostringstream strm;
    strm << "v" << x;
    v = m.mk_const(symbol(strm.str()), a.mk_real());
}

static void mk_term(vector<var_t> const& vars, rational const& coeff, app_ref& term) {
    ast_manager& m = term.get_manager();
    expr_ref_vector ts(m);
    arith_util a(m);

    for (unsigned i = 0; i < vars.size(); ++i) {
        app_ref var(m);
        mk_var(vars[i].m_id, var);
        rational coeff = vars[i].m_coeff;
        ts.push_back(a.mk_mul(a.mk_numeral(coeff, false), var));
    }
    ts.push_back(a.mk_numeral(coeff, a.mk_real()));
    term = a.mk_add(ts.size(), ts.data());    
}

static void add_random_ineq(
    expr_ref_vector& fmls, 
    opt::model_based_opt& mbo,
    random_gen& r,
    svector<int>  const& values,
    unsigned max_vars,
    unsigned max_coeff) 
{
    ast_manager& m = fmls.get_manager();
    arith_util a(m);

    unsigned num_vars = values.size();
    uint_set used_vars;
    vector<var_t> vars;
    int value = 0;
    for (unsigned i = 0; i < max_vars; ++i) {
        unsigned x = r(num_vars);
        if (used_vars.contains(x)) {
            continue;
        }
        used_vars.insert(x);
        int coeff = r(max_coeff + 1);
        if (coeff == 0) {
            continue;
        }
        unsigned sign = r(2);
        coeff = sign == 0 ? coeff : -coeff;
        vars.push_back(var_t(x, rational(coeff)));
        value += coeff*values[x];
    }
    unsigned abs_value = value < 0 ? - value : value;
    // value + k <= 0
    // k <= - value
    // range for k is 2*|value|
    // k <= - value - range
    opt::ineq_type rel = opt::t_le;

    int coeff = 0;
    if (r(4) == 0) {
        rel = opt::t_eq;
        coeff = -value;
    }
    else {
        if (abs_value > 0) {
            coeff = -value - r(2*abs_value);
        }
        else {
            coeff = 0;
        }
        if (coeff != -value && r(3) == 0) {
            rel = opt::t_lt;
        }   
    }
    expr_ref fml(m);
    app_ref t1(m);
    app_ref t2(a.mk_numeral(rational(0), a.mk_real()), m);
    mk_term(vars, rational(coeff), t1);
    switch (rel) {
    case opt::t_eq:
        fml = m.mk_eq(t1, t2);
        break;
    case opt::t_lt:
        fml = a.mk_lt(t1, t2);
        break;
    case opt::t_le:
        fml = a.mk_le(t1, t2);
        break;
    case opt::t_mod:
        NOT_IMPLEMENTED_YET();        
        break;        
    }
    fmls.push_back(fml);
    mbo.add_constraint(vars, rational(coeff), rel);
}

static void test_maximize(opt::model_based_opt& mbo, ast_manager& m, unsigned num_vars, expr_ref_vector const& fmls, app* t) {
    mbp::arith_project_plugin plugin(m);
    model mdl(m);    
    arith_util a(m);
    for (unsigned i = 0; i < num_vars; ++i) {
        app_ref var(m);
        mk_var(i, var);
        rational val = mbo.get_value(i);
        mdl.register_decl(var->get_decl(), a.mk_numeral(val, false));
    }
    expr_ref ge(m), gt(m);
    opt::inf_eps value1 = plugin.maximize(fmls, mdl, t, ge, gt);
    opt::inf_eps value2 = mbo.maximize();    
    std::cout << "optimal: " << value1 << " " << value2 << "\n";
    mbo.display(std::cout);
}
                        
static void check_random_ineqs(random_gen& r, ast_manager& m, unsigned num_vars, unsigned max_value, unsigned num_ineqs, unsigned max_vars, unsigned max_coeff) {
    opt::model_based_opt mbo;
    expr_ref_vector fmls(m);

    svector<int> values;
    for (unsigned i = 0; i < num_vars; ++i) {
        values.push_back(r(max_value + 1));
        mbo.add_var(rational(values.back()));
    }
    for (unsigned i = 0; i < num_ineqs; ++i) {
        add_random_ineq(fmls, mbo, r, values, max_vars, max_coeff);
    }

    vector<var_t> vars;
    vars.reset();
    vars.push_back(var_t(0, rational(2)));
    vars.push_back(var_t(1, rational(-2)));
    mbo.set_objective(vars, rational(0));
       

    mbo.display(std::cout);
    app_ref t(m);
    mk_term(vars, rational(0), t);
    
    test_maximize(mbo, m, num_vars, fmls, t);
    
    for (unsigned i = 0; i < values.size(); ++i) {
        std::cout << i << ": " << values[i] << " -> " << mbo.get_value(i) << "\n";
    }
}

static void check_random_ineqs() {
    random_gen r(1);
    ast_manager m;
    reg_decl_plugins(m);    

    for (unsigned i = 0; i < 100; ++i) {
        check_random_ineqs(r, m, 4, 5, 5, 3, 6);
    }
}

static void test_project() {
    ast_manager m;
    reg_decl_plugins(m);    
    mbp::arith_project_plugin plugin(m);    
    arith_util a(m);
    app_ref_vector vars(m);
    expr_ref_vector lits(m), ds(m);
    model mdl(m);    
    app_ref x(m), y(m), z(m), u(m);
    x = m.mk_const(symbol("x"), a.mk_int());
    y = m.mk_const(symbol("y"), a.mk_int());
    z = m.mk_const(symbol("z"), a.mk_int());
    u = m.mk_const(symbol("u"), a.mk_int());
    func_decl_ref f(m);
    sort* int_sort = a.mk_int();
    f = m.mk_func_decl(symbol("f"), 1, &int_sort, int_sort);

    // test non-projection
    mdl.register_decl(x->get_decl(), a.mk_int(0));
    mdl.register_decl(y->get_decl(), a.mk_int(1));
    mdl.register_decl(z->get_decl(), a.mk_int(2));
    mdl.register_decl(u->get_decl(), a.mk_int(3));
    func_interp* fi = alloc(func_interp, m, 1);
    expr_ref_vector nums(m);
    nums.push_back(a.mk_int(0));
    nums.push_back(a.mk_int(1));
    nums.push_back(a.mk_int(2));
    fi->insert_new_entry(nums.data(),   a.mk_int(1));
    fi->insert_new_entry(nums.data()+1, a.mk_int(2));
    fi->insert_new_entry(nums.data()+2, a.mk_int(3));
    fi->set_else(a.mk_int(10));
    mdl.register_decl(f, fi);
    vars.reset();
    lits.reset();
    vars.push_back(x);
    lits.push_back(x <= app_ref(m.mk_app(f, (expr*)x), m));
    lits.push_back(x < y);
    plugin(mdl, vars, lits);
    std::cout << lits << "\n";

    // test not-equals
    vars.reset();
    lits.reset();
    vars.push_back(x);
    lits.push_back(m.mk_not(m.mk_eq(x, y)));
    plugin(mdl, vars, lits);
    std::cout << lits << "\n";

    // test negation of distinct using bound variables
    mdl.register_decl(x->get_decl(), a.mk_int(0));
    mdl.register_decl(y->get_decl(), a.mk_int(1));
    mdl.register_decl(z->get_decl(), a.mk_int(0));
    mdl.register_decl(u->get_decl(), a.mk_int(6));
    vars.reset();
    lits.reset();
    ds.reset();
    vars.push_back(x);
    vars.push_back(y);
    ds.push_back(x);
    ds.push_back(y);
    ds.push_back(z + 2);
    ds.push_back(u);
    ds.push_back(z);
    lits.push_back(m.mk_not(m.mk_distinct(ds.size(), ds.data())));
    plugin(mdl, vars, lits);
    std::cout << lits << "\n";

    // test negation of distinct, not using bound variables
    mdl.register_decl(x->get_decl(), a.mk_int(0));
    mdl.register_decl(y->get_decl(), a.mk_int(1));
    mdl.register_decl(z->get_decl(), a.mk_int(0));
    mdl.register_decl(u->get_decl(), a.mk_int(6));
    vars.reset();
    lits.reset();
    ds.reset();
    vars.push_back(x);
    vars.push_back(y);
    ds.push_back(x);
    ds.push_back(y);
    ds.push_back(z + 2);
    ds.push_back(u);
    ds.push_back(z + 10);
    ds.push_back(u + 4);
    lits.push_back(m.mk_not(m.mk_distinct(ds.size(), ds.data())));
    plugin(mdl, vars, lits);
    std::cout << lits << "\n";


    // test distinct
    mdl.register_decl(x->get_decl(), a.mk_int(0));
    mdl.register_decl(y->get_decl(), a.mk_int(1));
    mdl.register_decl(z->get_decl(), a.mk_int(0));
    mdl.register_decl(u->get_decl(), a.mk_int(6));
    vars.reset();
    lits.reset();
    ds.reset();
    vars.push_back(x);
    vars.push_back(y);
    ds.push_back(x);
    ds.push_back(y);
    ds.push_back(z + 2);
    ds.push_back(u);
    lits.push_back(m.mk_distinct(ds.size(), ds.data()));
    plugin(mdl, vars, lits);
    std::cout << lits << "\n";

    // equality over modulus
    mdl.register_decl(y->get_decl(), a.mk_int(4));
    mdl.register_decl(z->get_decl(), a.mk_int(8));
    lits.reset();
    vars.reset();
    vars.push_back(y);
    lits.push_back(m.mk_eq(a.mk_mod(y, a.mk_int(3)), a.mk_int(1)));
    lits.push_back(m.mk_eq(2*y, z));
    plugin(mdl, vars, lits);
    std::cout << lits << "\n";

    // inequality test
    mdl.register_decl(x->get_decl(), a.mk_int(0));
    mdl.register_decl(y->get_decl(), a.mk_int(1));
    mdl.register_decl(z->get_decl(), a.mk_int(0));
    mdl.register_decl(u->get_decl(), a.mk_int(6));
    vars.reset();
    lits.reset();
    vars.push_back(x);
    vars.push_back(y);
    lits.push_back(z <= (x + (2*y)));
    lits.push_back(2*x < u + 3);
    lits.push_back(2*y <= u);
    plugin(mdl, vars, lits);
    std::cout << lits << "\n";

    // non-unit equalities
    mdl.register_decl(x->get_decl(), a.mk_int(1));
    mdl.register_decl(z->get_decl(), a.mk_int(2));
    mdl.register_decl(u->get_decl(), a.mk_int(3));
    mdl.register_decl(y->get_decl(), a.mk_int(4));
    lits.reset();
    vars.reset();
    vars.push_back(x);
    lits.push_back(m.mk_eq(2*x, z));
    lits.push_back(m.mk_eq(3*x, u));
    plugin(mdl, vars, lits);
    std::cout << lits << "\n";


}


void tst_qe_arith() {
    test_project();
    return;
    check_random_ineqs();
    return;
//    enable_trace("qe");
    testI(example8);    
    testR(example7);
    test_ineqs();
    return;
    testR(example1);
    testR(example2);
    testR(example3);
    testR(example4);
    testR(example5);
    return;
    test2(example6);
    return;
}