Theory of Bit-Vectors

examples/api/cpp/bitvectors.cpp 

  1 /******************************************************************************
  2  * Top contributors (to current version):
  3  *   Liana Hadarean, Aina Niemetz, Mathias Preiner
  4  *
  5  * This file is part of the cvc5 project.
  6  *
  7  * Copyright (c) 2009-2022 by the authors listed in the file AUTHORS
  8  * in the top-level source directory and their institutional affiliations.
  9  * All rights reserved.  See the file COPYING in the top-level source
 10  * directory for licensing information.
 11  * ****************************************************************************
 12  *
 13  * A simple demonstration of the solving capabilities of the cvc5
 14  * bit-vector solver.
 15  *
 16  */
 17 
 18 #include <cvc5/cvc5.h>
 19 
 20 #include <iostream>
 21 
 22 using namespace std;
 23 using namespace cvc5;
 24 
 25 int main()
 26 {
 27   Solver slv;
 28   slv.setLogic("QF_BV");  // Set the logic
 29 
 30   // The following example has been adapted from the book A Hacker's Delight by
 31   // Henry S. Warren.
 32   //
 33   // Given a variable x that can only have two values, a or b. We want to
 34   // assign to x a value other than the current one. The straightforward code
 35   // to do that is:
 36   //
 37   //(0) if (x == a ) x = b;
 38   //    else x = a;
 39   //
 40   // Two more efficient yet equivalent methods are:
 41   //
 42   //(1) x = a ⊕ b ⊕ x;
 43   //
 44   //(2) x = a + b - x;
 45   //
 46   // We will use cvc5 to prove that the three pieces of code above are all
 47   // equivalent by encoding the problem in the bit-vector theory.
 48 
 49   // Creating a bit-vector type of width 32
 50   Sort bitvector32 = slv.mkBitVectorSort(32);
 51 
 52   // Variables
 53   Term x = slv.mkConst(bitvector32, "x");
 54   Term a = slv.mkConst(bitvector32, "a");
 55   Term b = slv.mkConst(bitvector32, "b");
 56 
 57   // First encode the assumption that x must be equal to a or b
 58   Term x_eq_a = slv.mkTerm(EQUAL, {x, a});
 59   Term x_eq_b = slv.mkTerm(EQUAL, {x, b});
 60   Term assumption = slv.mkTerm(OR, {x_eq_a, x_eq_b});
 61 
 62   // Assert the assumption
 63   slv.assertFormula(assumption);
 64 
 65   // Introduce a new variable for the new value of x after assignment.
 66   Term new_x = slv.mkConst(bitvector32, "new_x");  // x after executing code (0)
 67   Term new_x_ =
 68       slv.mkConst(bitvector32, "new_x_");  // x after executing code (1) or (2)
 69 
 70   // Encoding code (0)
 71   // new_x = x == a ? b : a;
 72   Term ite = slv.mkTerm(ITE, {x_eq_a, b, a});
 73   Term assignment0 = slv.mkTerm(EQUAL, {new_x, ite});
 74 
 75   // Assert the encoding of code (0)
 76   cout << "Asserting " << assignment0 << " to cvc5 " << endl;
 77   slv.assertFormula(assignment0);
 78   cout << "Pushing a new context." << endl;
 79   slv.push();
 80 
 81   // Encoding code (1)
 82   // new_x_ = a xor b xor x
 83   Term a_xor_b_xor_x = slv.mkTerm(BITVECTOR_XOR, {a, b, x});
 84   Term assignment1 = slv.mkTerm(EQUAL, {new_x_, a_xor_b_xor_x});
 85 
 86   // Assert encoding to cvc5 in current context;
 87   cout << "Asserting " << assignment1 << " to cvc5 " << endl;
 88   slv.assertFormula(assignment1);
 89   Term new_x_eq_new_x_ = slv.mkTerm(EQUAL, {new_x, new_x_});
 90 
 91   cout << " Check sat assuming: " << new_x_eq_new_x_.notTerm() << endl;
 92   cout << " Expect UNSAT. " << endl;
 93   cout << " cvc5: " << slv.checkSatAssuming(new_x_eq_new_x_.notTerm()) << endl;
 94   cout << " Popping context. " << endl;
 95   slv.pop();
 96 
 97   // Encoding code (2)
 98   // new_x_ = a + b - x
 99   Term a_plus_b = slv.mkTerm(BITVECTOR_ADD, {a, b});
100   Term a_plus_b_minus_x = slv.mkTerm(BITVECTOR_SUB, {a_plus_b, x});
101   Term assignment2 = slv.mkTerm(EQUAL, {new_x_, a_plus_b_minus_x});
102 
103   // Assert encoding to cvc5 in current context;
104   cout << "Asserting " << assignment2 << " to cvc5 " << endl;
105   slv.assertFormula(assignment2);
106 
107   cout << " Check sat assuming: " << new_x_eq_new_x_.notTerm() << endl;
108   cout << " Expect UNSAT. " << endl;
109   cout << " cvc5: " << slv.checkSatAssuming(new_x_eq_new_x_.notTerm()) << endl;
110 
111   Term x_neq_x = slv.mkTerm(EQUAL, {x, x}).notTerm();
112   std::vector<Term> v{new_x_eq_new_x_, x_neq_x};
113   Term query = slv.mkTerm(AND, {v});
114   cout << " Check sat assuming: " << query.notTerm() << endl;
115   cout << " Expect SAT. " << endl;
116   cout << " cvc5: " << slv.checkSatAssuming(query.notTerm()) << endl;
117 
118   // Assert that a is odd
119   Op extract_op = slv.mkOp(BITVECTOR_EXTRACT, {0, 0});
120   Term lsb_of_a = slv.mkTerm(extract_op, {a});
121   cout << "Sort of " << lsb_of_a << " is " << lsb_of_a.getSort() << endl;
122   Term a_odd = slv.mkTerm(EQUAL, {lsb_of_a, slv.mkBitVector(1u, 1u)});
123   cout << "Assert " << a_odd << endl;
124   cout << "Check satisfiability." << endl;
125   slv.assertFormula(a_odd);
126   cout << " Expect sat. " << endl;
127   cout << " cvc5: " << slv.checkSat() << endl;
128   return 0;
129 }