How to get z3 to return multiple unsat cores, multiple satisfying assignments

Question

I am working on a component of a research tool; I am interested in retrieving (for QF_LRA)

-multiple (minimal or otherwise) UNSAT cores and
-multiple SAT assignments

I have checked the forum for earlier discussions on this topic e.g., How to get different unsat cores when using z3 on logic QF_LRA

They refer to the z3 Python tutorial(s) e.g, http://rise4fun.com/Z3Py/tutorial/musmss

which seems to be offline for now. I have tried other suggestions of github etc to find the mentioned tutorial, but have had no luck.

I am using the z3 Java API; but happy to switch to alternatives.

Answer 1

Here is the tutorial. You can find more information on MARCO from Mark Liffiton's web pages.

Enumeration of Minimal Unsatisfiable Cores and Maximal Satisfying Subsets

This tutorial illustrates how to use Z3 for extracting all minimal unsatisfiable cores together with all maximal satisfying subsets.

Origin

The algorithm that we describe next represents the essence of the core extraction procedure by Liffiton and Malik and independently by Previti and Marques-Silva:

Enumerating Infeasibility: Finding Multiple MUSes Quickly
Mark H. Liffiton and Ammar Malik
in Proc. 10^th International Conference on Integration of Artificial Intelligence (AI) and Operations Research (OR) techniques in Constraint Programming (CPAIOR-2013), 160-175, May 2013.

Partial MUS Enumeration
Alessandro Previti, Joao Marques-Silva in Proc. AAAI-2013 July 2013

Z3py Features

This implementation contains no tuning. It was contributed by Mark Liffiton and it is a simplification of one of the versions available from his Marco Polo Web site. Code for eMUS is also available. The example illustrates the following features of Z3's Python-based API:

1. Using assumptions to track unsatisfiable cores.
2. Using multiple solvers and passing constraints between them.
3. Calling the C-based API from Python. Not all API functions are supported over the Python wrappers. This example shows how to get a unique integer identifier of an AST, which can be used as a key in a hash-table.

Idea of the Algorithm

The main idea of the algorithm is to maintain two logical contexts and exchange information between them:

1. The MapSolver is used to enumerate sets of clauses that are not already supersets of an existing unsatisfiable core and not already a subset of a maximal satisfying assignment. The MapSolver uses one unique atomic predicate per soft clause, so it enumerates sets of atomic predicates. For each minimal unsatisfiable core, say, represented by predicates p₁, p₂, p₅, the MapSolver contains the clause ¬ p₁ ∨ ¬ p₂ ∨ ¬ p₅. For each maximal satisfiable subset, say, represented by predicats p₂, p₃, p₅, the MapSolver contains a clause corresponding to the disjunction of all literals not in the maximal satisfiable subset, p₁ ∨ p₄ ∨ p₆.
2. The SubsetSolver contains a set of soft clauses (clauses with the unique indicator atom occurring negated). The MapSolver feeds it a set of clauses (the indicator atoms). Recall that these are not already a superset of an existing minimal unsatisfiable core, or a subset of a maximal satisfying assignment. If asserting these atoms makes the SubsetSolver context infeasible, then it finds a minimal unsatisfiable subset corresponding to these atoms. If asserting the atoms is consistent with the SubsetSolver, then it extends this set of atoms maximally to a satisfying set.

from Z3 import *

def main():
    x, y = Reals('x y')
    constraints = [x > 2, x < 1, x < 0, Or(x + y > 0, y < 0), Or(y >= 0, x >= 0), Or(y < 0, x < 0), Or(y > 0, x < 0)]
    csolver = SubsetSolver(constraints)
    msolver = MapSolver(n=csolver.n)
    for orig, lits in enumerate_sets(csolver, msolver):
        output = "%s %s" % (orig, lits)
        print(output)



def get_id(x):
    return Z3_get_ast_id(x.ctx.ref(),x.as_ast())

def MkOr(clause):
    if clause == []:
       return False
    else:
       return Or(clause)

SubsetSolver:

class SubsetSolver:
   constraints = []
   n = 0
   s = Solver()
   varcache = {}
   idcache = {}

   def __init__(self, constraints):
       self.constraints = constraints
       self.n = len(constraints)
       for i in range(self.n):
           self.s.add(Implies(self.c_var(i), constraints[i]))

   def c_var(self, i):
       if i not in self.varcache:
          v = Bool(str(self.constraints[abs(i)]))
          self.idcache[get_id(v)] = abs(i)
          if i >= 0:
             self.varcache[i] = v
          else:
             self.varcache[i] = Not(v)
       return self.varcache[i]

   def check_subset(self, seed):
       assumptions = self.to_c_lits(seed)
       return (self.s.check(assumptions) == sat)

   def to_c_lits(self, seed):
       return [self.c_var(i) for i in seed]

   def complement(self, aset):
       return set(range(self.n)).difference(aset)

   def seed_from_core(self):
       core = self.s.unsat_core()
       return [self.idcache[get_id(x)] for x in core]

   def shrink(self, seed):       
       current = set(seed)
       for i in seed:
          if i not in current:
             continue
          current.remove(i)
          if not self.check_subset(current):
             current = set(self.seed_from_core())
          else:
             current.add(i)
       return current

   def grow(self, seed):
       current = seed
       for i in self.complement(current):
           current.append(i)
           if not self.check_subset(current):
              current.pop()
       return current

MapSolver:

class MapSolver:
    def __init__(self, n):
        """Initialization.
           Args:
            n: The number of constraints to map.
        """
       self.solver = Solver()
       self.n = n
       self.all_n = set(range(n))  # used in complement fairly frequently

   def next_seed(self):
       """Get the seed from the current model, if there is one. 
            Returns:
            A seed as an array of 0-based constraint indexes.
       """
       if self.solver.check() == unsat:
            return None
       seed = self.all_n.copy()  # default to all True for "high bias"
       model = self.solver.model()
       for x in model:
           if is_false(model[x]):
              seed.remove(int(x.name()))
       return list(seed)

   def complement(self, aset):
       """Return the complement of a given set w.r.t. the set of mapped constraints."""
       return self.all_n.difference(aset)

   def block_down(self, frompoint):
       """Block down from a given set."""
       comp = self.complement(frompoint)
       self.solver.add( MkOr( [Bool(str(i)) for i in comp] ) )

   def block_up(self, frompoint):
       """Block up from a given set."""
       self.solver.add( MkOr( [Not(Bool(str(i))) for i in frompoint] ) )



  def enumerate_sets(csolver, map):
      """Basic MUS/MCS enumeration, as a simple example."""
      while True:
        seed = map.next_seed()
        if seed is None:
           return
        if csolver.check_subset(seed):
           MSS = csolver.grow(seed)
           yield ("MSS", csolver.to_c_lits(MSS))
           map.block_down(MSS)
        else:
           MUS = csolver.shrink(seed)
           yield ("MUS", csolver.to_c_lits(MUS))
           map.block_up(MUS)

   main()