Comparison Operators

PyQBPP supports two types of operators for creating constraints:

  • Equality operator: f == n, where f is an expression and n is an integer.
  • Range operator: between(f, l, u), where f is an expression and l and u ($l\leq u$) are integers.

These operators return an ExprExpr object that attains the minimum value of 0 if and only if the corresponding constraints are satisfied.

Equality operator

The equality operator f == n creates the following expression:

\[(f-n)^2\]

This expression attains the minimum value of 0 if and only if the equality $f=n$ is satisfied.

The following program searches for all solutions satisfying $a+2b+3c=3$ using the Exhaustive Solver:

from pyqbpp import var, ExhaustiveSolver

a = var("a")
b = var("b")
c = var("c")
f = a + 2 * b + 3 * c == 3
f.simplify_as_binary()
print("f =", f)
print("body =", f.body)

solver = ExhaustiveSolver(f)
sols = solver.search_optimal_solutions()
for sol in sols:
    print(f"a={sol.get(a)}, b={sol.get(b)}, c={sol.get(c)}, "
          f"f={sol.eval(f)}, body={sol.eval(f.body)}")

In this program, f internally holds two expressions:

  • f: $(a+2b+3c-3)^2$, which attains the minimum value of 0 if the equality is satisfied.
  • f.body: the left-hand side of the equality, $a+2b+3c$.

This program produces the following output:

f = 9 -5*a -8*b -9*c +4*a*b +6*a*c +12*b*c
body = a +2*b +3*c
a=0, b=0, c=1, f=0, body=3
a=1, b=1, c=0, f=0, body=3

Notes on Supported Equality Forms

PyQBPP supports the equality operator only in the following form:

  • expression == integer

The following forms are not supported:

  • integer == expression
  • expression1 == expression2

Instead of expression1 == expression2, you can rewrite the constraint as:

  • expression1 - expression2 == 0

Range operator

The range operator between(f, l, u) creates an expression that attains the minimum value of 0 if and only if the constraint $l\leq f\leq u$ is satisfied.

NOTE Unlike the C++ version which uses the syntax l <= f <= u, PyQBPP uses the function between(f, l, u) for range constraints on expressions.

The following program demonstrates the use of the range operator:

from pyqbpp import var, between, ExhaustiveSolver

a = var("a")
b = var("b")
c = var("c")
f = between(4 * a + 9 * b + 15 * c, 5, 14)
f.simplify_as_binary()

solver = ExhaustiveSolver(f)
sols = solver.search_optimal_solutions()
for sol in sols:
    print(f"a={sol.get(a)}, b={sol.get(b)}, c={sol.get(c)}, "
          f"f={sol.eval(f)}, body={sol.eval(f.body)}")

This program searches for solutions satisfying the constraint $5\leq 4a+9b+15c \leq 14$ and produces the following output:

a=0, b=1, c=0, f=0, body=9
a=1, b=1, c=0, f=0, body=13