bindings to lgl (a SAT solver)
Lingeling is an efficient SAT solver written by Armin Biere in pure C. This package provides efficient Python bindings to Lingeling on the C level, i.e., when importing pylgl, the lingeling solver becomes part of the Python process itself. For ease of deployment, the lingeling source files are included in this project. These files have been extracted from lingeling version bbe-6fe9691-170131.
The pylgl module has two functions solve and itersolve, both of which take an iterable of clauses as an argument. Each clause is itself represented as an iterable of (non-zero) integers.
- The function solve returns one of the following:
- one solution (a list of integers)
- the string “UNSAT” (when the clauses are unsatisfiable)
- the string “UNKNOWN” (when a solution could not be determined within the propagation limit)
The function itersolve returns an iterator over solutions. When the propagation limit is specified, exhausting the iterator may not yield all possible solutions.
- Both functions take the following keyword arguments:
- vars: number of variables (integer)
- verbose: the verbosity level (integer)
- simplify: enable simplification (integer)
- seed: random number generator seed (integer)
- randec: enable random decisions order (integer)
- randecint: random decision order interval (integer)
- randphase: enable random decision phases (integer)
- randphaseint: random decision phases interval (integer)
Let us consider the following clauses, represented using the DIMACS cnf format:
p cnf 5 3 1 -5 4 0 -1 5 3 4 0 -3 -4 0
Here, we have 5 variables and 3 clauses, the first clause being (x1 or not x5 or x4). Note that the variable x2 is not used in any of the clauses, which means that for each solution with x2 = True, we must also have a solution with x2 = False. In Python, each clause is most conveniently represented as a list of integers. Naturally, it makes sense to represent each solution also as a list of integers, where the sign corresponds to the Boolean value (+ for True and - for False) and the absolute value corresponds to ith variable:
>>> import pylgl >>> cnf = [[1, -5, 4], [-1, 5, 3, 4], [-3, -4]] >>> pylgl.solve(cnf) [1, -2, -3, -4, 5]
This solution translates to: x1 = x5 = True, x2 = x3 = x4 = False
To find all solutions, use itersolve:
>>> for sol in pylgl.itersolve(cnf): ... print sol ... [1, -2, -3, -4, 5] [1, -2, -3, 4, -5] [1, -2, -3, 4, 5] ... >>> len(list(pylgl.itersolve(cnf))) 18
In this example, there are a total of 18 possible solutions, which had to be an even number because x2 was left unspecified in the clauses.
The fact that itersolve returns an iterator, makes it very elegant and efficient for many types of operations. For example, using the itertools module from the standard library, here is how one would construct a list of (up to) 3 solutions:
>>> import itertools >>> list(itertools.islice(pylgl.itersolve(cnf), 3)) [[1, -2, -3, -4, 5], [1, -2, -3, 4, -5], [1, -2, -3, 4, 5]]
Implementation of itersolve
How does one go from having found one solution to another solution? The answer is surprisingly simple. One adds the inverse of the already found solution as a new clause. This new clause ensures that another solution is searched for, as it excludes the already found solution. Here is basically a pure Python implementation of itersolve in terms of solve:
def py_itersolve(clauses): # don't use this function! while True: # (it is only here to explain things) sol = pylgl.solve(clauses) if isinstance(sol, list): yield sol clauses.append([-x for x in sol]) else: # no more solutions -- stop iteration return
This implementation has several problems. Firstly, it is quite slow as pylgl.solve has to convert the list of clauses over and over and over again. Secondly, after calling py_itersolve the list of clauses will be modified. In pylgl, itersolve is implemented on the C level, making use of the lingeling C interface (which makes it much, much faster than the naive Python implementation above).
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