UnSAT core extraction with assumption variables

When a model is unsatisfiable, it can be desirable to get a better idea of which Boolean variables make it unsatisfiable. Commonly, these Boolean variables are ‘switches’ that turn constraints on, hence such Boolean variables can be used to get a better idea of which constraints make the model unsatisfiable.

In the SATisfiability literature, the Boolean variables of interests are called assumption variables and the solver will assume they are true. The subset of these variables that, when true, make the model unsatisfiable is called an unsatisfiable core.

Lazy Clause Generation solvers, like or-tools, are built on SAT solvers and hence can inherit the ability to define assumption variables and extract an unsatisfiable core.

Since version 8.2, or-tools supports declaring assumption variables, and extracting an unsat core. We also implement this functionality in CPMpy, using PySAT-like s.solve(assumptions=[...]) and s.get_core():

from cpmpy import *
from cpmpy.solvers import CPM_ortools

bv = boolvar(shape=3)
iv = intvar(0,9, shape=3)

# circular 'bigger then', UNSAT
m = Model(
    bv[0].implies(iv[0] > iv[1]),
    bv[1].implies(iv[1] > iv[2]),
    bv[2].implies(iv[2] > iv[0])

s = CPM_ortools(m)
print("core:", s.get_core())

This opens the door to more advanced use cases, such as Minimal Unsatisfiable Subsets and QuickXplain-like tools to help debugging. We welcome any examples or additions that use CPMpy in this way!! Here is one example: the MARCO algorithm for enumerating all MUS/MSSes. Here is another: a deletion-based MUS algorithm, made to also work for non-reifable (global) constraints. Also useful to debug unsatisfiable models!

One final caveat is that the or-tools Python interface is by design stateless. That means that, unlike in PySAT, calling s.solve(assumptions=bv) twice for a different bv array does NOT REUSE anything from the previous run: no warm-starting, no learnt clauses that are kept, no incrementality, so there will be some pre-processing overhead. If you know of another CP solver with a (Python) assumption interface that is incremental, let us know!!

A final-final note is that you can manually warm-start or-tools with a previously found solution with s.solution_hint(); see also the MARCO code linked above.