cnfc

A CNF compiler that generates compact DIMACS CNF encodings from higher-level primitives in Python. DIMACS CNF is the input format accepted by most SAT solvers.

For simpler problems, cnfc comes bundled with an efficient pure Python CDCL SAT solver and a Formula.Solve() method that returns a solution without needing to build/install a SAT solver.

For harder combinatorial problems that may take hours or days to solve, cnfc can also write DIMACS CNF files that you pass to an external solver of your choice. You get the flexibility to run your own preprocessing or cubing as an intermediate step, or hand the problem off to whichever solver works best for your problem.

Read on for an extended example or look at the examples in this repository to get started.

Example

Suppose you need to schedule 8 employees to cover two shifts a day (7 a.m. - 3 p.m. and 3 p.m. - 11 p.m.) for the whole week. Every shift needs to be staffed by two employees, one of which has to be a manager. Each employee has a few shifts where they can't work. You need to give everyone at least 3 shifts of work for the week but they can't go over 4. Employees can't work both the morning and night shift on the same day.

This is a collection of constraints that should be easy to solve with a SAT solver, but encoding them into a propositional formula can be tedious. Here's how to do it with cnfc:

from cnfc import *

employees = ['Homer', 'Hamza', 'Veronica', 'Lottie', 'Zakaria', 'Keeley', 'Farhan', 'Seamus']
managers = ['Homer', 'Hamza', 'Keeley', 'Farhan']
days = ['Sun','Mon','Tue','Wed','Thu','Fri','Sat']
hours = ['7-3','3-11']
shifts = [f'{day} {hour}' for day in days for hour in hours]

# Associate a boolean variable with each pairing of employee and shift.
formula = Formula()
varz = dict(((employee, shift), formula.AddVar(f'{employee} {shift}'))
            for employee in employees for shift in shifts)

# Every shift needs exactly two people scheduled.
for shift in shifts:
    scheduled = [varz[(employee, shift)] for employee in employees]
    formula.Add(NumTrue(*scheduled) == 2)

# Every shift needs a manager.
for shift in shifts:
    manager_on_shift = [varz[(manager, shift)] for manager in managers]
    formula.Add(Or(*manager_on_shift))

# People have shifts they can't work.
formula.Add(Not(varz[('Homer', 'Sun 7-3')]))
formula.Add(Not(varz[('Lottie', 'Tue 7-3')]))
formula.Add(Not(varz[('Lottie', 'Tue 3-11')]))
formula.Add(Not(varz[('Farhan', 'Fri 3-11')]))
formula.Add(Not(varz[('Homer', 'Sat 3-11')]))
formula.Add(Not(varz[('Hamza', 'Sat 3-11')]))
formula.Add(Not(varz[('Keeley', 'Sat 3-11')]))

# Each employee needs to work at least 3 shifts but no more than 4.
for employee in employees:
    employee_shifts = [varz[(employee,shift)] for shift in shifts]
    formula.Add(NumTrue(*employee_shifts) >= 3)
    formula.Add(NumTrue(*employee_shifts) <= 4)

# People can't work both the morning and night shift in a single day.
for employee in employees:
    for day in days:
        formula.Add(Not(And(varz[(employee, f'{day} 7-3')],
                            varz[(employee, f'{day} 3-11')])))

# This function will be called to print the final schedule once we've solved for
# it. The extra_args will be full descriptions of the shift staffings -- the
# same strings we used to name the variables in our calls to AddVar above.
def print_solution(sol, *extra_args):
    for shift_assignment in extra_args[0]:
        if sol[shift_assignment]:
            print(shift_assignment)

# Solve the formula, print a solution if we find one.
solution = formula.Solve()
if solution:
    shift_assignments = [f'{employee} {shift}' for shift in shifts for employee in employees]
    print_solution(solution, shift_assignments)
else:
    print('UNSATISFIABLE')

You can run the example above with:

uv run python examples/scheduling/scheduling.py

and it'll print a complete schedule like:

Zakaria Sun 7-3
Farhan Sun 7-3
Homer Sun 3-11
Seamus Sun 3-11
Hamza Mon 7-3
Seamus Mon 7-3
Zakaria Mon 3-11
Keeley Mon 3-11
Homer Tue 7-3
Zakaria Tue 7-3
Keeley Tue 3-11
Seamus Tue 3-11
Homer Wed 7-3
Seamus Wed 7-3
Lottie Wed 3-11
Keeley Wed 3-11
Veronica Thu 7-3
Farhan Thu 7-3
Lottie Thu 3-11
Keeley Thu 3-11
Hamza Fri 7-3
Farhan Fri 7-3
Homer Fri 3-11
Veronica Fri 3-11
Hamza Sat 7-3
Veronica Sat 7-3
Lottie Sat 3-11
Farhan Sat 3-11

You can verify that all of our constraints are satisfied. Right now, three of the four managers (Homer, Hamza, and Keeley) want the Saturday 3-11 shift off, so if we change the script to add another constraint with the last remaining manager (Farhan) asking for that shift off:

formula.Add(Not(varz[('Farhan', 'Sat 3-11')]))

and then re-run the solver and extractor, we should see:

UNSATISFIABLE

which tells us that there's no assignment of people to shifts that satisfies all of the criteria we've laid out.

Instead of solving in Python, you can instead generate a DIMACS CNF file and an "extractor" to convert the solution from a SAT solver into something meaningful to you:

# Write the resulting CNF file to /tmp/cnf.
with open('/tmp/cnf', 'w') as f:
    formula.WriteCNF(f)
# Write an extractor script to /tmp/extractor.py.
with open('/tmp/extractor.py', 'w') as f:
    shift_assignments = [f'{employee} {shift}' for shift in shifts for employee in employees]
    formula.WriteExtractor(f, print_solution, extra_args=[shift_assignments])

Adding the lines above instead of running formula.Solve() will generate a DIMACS CNF file (/tmp/cnf) and an extractor script (/tmp/extractor.py). You'll need a SAT solver like kissat or cadical to solve the CNF file.

To see the solution, run a modified version of the script above:

uv run python examples/scheduling/scheduling-external.py

then run a solver on the CNF file, saving the output:

$ kissat /tmp/cnf > /tmp/solver-output

and finally, run the extractor script on the CNF file and the output of the solver:

$ python3 /tmp/extractor.py /tmp/cnf /tmp/solver-output

You should see similar output here to the output from the earlier version of the script that solved the problem entirely in Python.

Runnable versions of these scripts are in the examples subdirectory of this repository.

Features

Arbitrary clauses can be built, composed, and added to formulas with:

• Familiar boolean operators And, Or, Not, If, Eq, Neq.
• Tuples that can be compared for equality, inequality, or lexicographic order.
• Non-negative Integers that can be added, multiplied, or compared (see examples/prime). %, //, and ** are also supported. Subtraction is supported as long as the result is non-negative (e.g., Integer(1) - Integer(2) == x is unsolvable.
• NumTrue and NumFalse for cardinality constraints (see examples/nqueens).
• RegexMatch to apply binary regular expressions to Tuples (see examples/nonagram).

Installation

cnfc is tested on these versions of Python 3. To install the latest stable release of cnfc, run:

pip install cnfc

Development

Install uv and run tests with:

uv run python -m unittest discover

To publish a new version to PyPI, bump the version in pyproject.toml and create a release in Github.