mirror of
https://github.com/Z3Prover/z3
synced 2025-10-31 11:42:28 +00:00
786 lines
25 KiB
Python
786 lines
25 KiB
Python
from z3 import *
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import heapq
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import numpy
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import time
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import random
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verbose = True
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# Simplistic (and fragile) converter from
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# a class of Horn clauses corresponding to
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# a transition system into a transition system
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# representation as <init, trans, goal>
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# It assumes it is given three Horn clauses
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# of the form:
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# init(x) => Invariant(x)
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# Invariant(x) and trans(x,x') => Invariant(x')
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# Invariant(x) and goal(x) => Goal(x)
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# where Invariant and Goal are uninterpreted predicates
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class Horn2Transitions:
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def __init__(self):
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self.trans = True
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self.init = True
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self.inputs = []
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self.goal = True
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self.index = 0
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def parse(self, file):
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fp = Fixedpoint()
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goals = fp.parse_file(file)
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for r in fp.get_rules():
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if not is_quantifier(r):
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continue
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b = r.body()
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if not is_implies(b):
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continue
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f = b.arg(0)
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g = b.arg(1)
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if self.is_goal(f, g):
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continue
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if self.is_transition(f, g):
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continue
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if self.is_init(f, g):
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continue
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def is_pred(self, p, name):
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return is_app(p) and p.decl().name() == name
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def is_goal(self, body, head):
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if not self.is_pred(head, "Goal"):
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return False
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pred, inv = self.is_body(body)
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if pred is None:
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return False
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self.goal = self.subst_vars("x", inv, pred)
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self.goal = self.subst_vars("i", self.goal, self.goal)
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self.inputs += self.vars
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self.inputs = list(set(self.inputs))
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return True
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def is_body(self, body):
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if not is_and(body):
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return None, None
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fmls = [f for f in body.children() if self.is_inv(f) is None]
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inv = None
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for f in body.children():
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if self.is_inv(f) is not None:
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inv = f;
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break
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return And(fmls), inv
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def is_inv(self, f):
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if self.is_pred(f, "Invariant"):
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return f
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return None
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def is_transition(self, body, head):
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pred, inv0 = self.is_body(body)
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if pred is None:
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return False
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inv1 = self.is_inv(head)
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if inv1 is None:
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return False
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pred = self.subst_vars("x", inv0, pred)
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self.xs = self.vars
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pred = self.subst_vars("xn", inv1, pred)
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self.xns = self.vars
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pred = self.subst_vars("i", pred, pred)
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self.inputs += self.vars
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self.inputs = list(set(self.inputs))
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self.trans = pred
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return True
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def is_init(self, body, head):
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for f in body.children():
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if self.is_inv(f) is not None:
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return False
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inv = self.is_inv(head)
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if inv is None:
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return False
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self.init = self.subst_vars("x", inv, body)
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return True
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def subst_vars(self, prefix, inv, fml):
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subst = self.mk_subst(prefix, inv)
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self.vars = [ v for (k,v) in subst ]
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return substitute(fml, subst)
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def mk_subst(self, prefix, inv):
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self.index = 0
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if self.is_inv(inv) is not None:
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return [(f, self.mk_bool(prefix)) for f in inv.children()]
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else:
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vars = self.get_vars(inv)
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return [(f, self.mk_bool(prefix)) for f in vars]
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def mk_bool(self, prefix):
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self.index += 1
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return Bool("%s%d" % (prefix, self.index))
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def get_vars(self, f, rs=[]):
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if is_var(f):
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return z3util.vset(rs + [f], str)
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else:
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for f_ in f.children():
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rs = self.get_vars(f_, rs)
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return z3util.vset(rs, str)
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# Produce a finite domain solver.
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# The theory QF_FD covers bit-vector formulas
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# and pseudo-Boolean constraints.
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# By default cardinality and pseudo-Boolean
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# constraints are converted to clauses. To override
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# this default for cardinality constraints
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# we set sat.cardinality.solver to True
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def fd_solver():
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s = SolverFor("QF_FD")
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s.set("sat.cardinality.solver", True)
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return s
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# negate, avoid double negation
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def negate(f):
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if is_not(f):
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return f.arg(0)
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else:
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return Not(f)
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def cube2clause(cube):
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return Or([negate(f) for f in cube])
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class State:
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def __init__(self, s):
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self.R = set([])
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self.solver = s
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def add(self, clause):
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if clause not in self.R:
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self.R |= { clause }
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self.solver.add(clause)
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def is_seq(f):
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return isinstance(f, list) or isinstance(f, tuple) or isinstance(f, AstVector)
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# Check if the initial state is bad
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def check_disjoint(a, b):
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s = fd_solver()
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s.add(a)
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s.add(b)
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return unsat == s.check()
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# Remove clauses that are subsumed
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def prune(R):
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removed = set([])
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s = fd_solver()
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for f1 in R:
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s.push()
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for f2 in R:
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if f2 not in removed:
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s.add(Not(f2) if f1.eq(f2) else f2)
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if s.check() == unsat:
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removed |= { f1 }
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s.pop()
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return R - removed
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# Quip variant of IC3
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must = True
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may = False
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class QLemma:
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def __init__(self, c):
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self.cube = c
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self.clause = cube2clause(c)
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self.bad = False
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def __hash__(self):
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return hash(tuple(set(self.cube)))
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def __eq__(self, qlemma2):
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if set(self.cube) == set(qlemma2.cube) and self.bad == qlemma2.bad:
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return True
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else:
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return False
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def __ne__():
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if not self.__eq__(self, qlemma2):
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return True
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else:
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return False
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class QGoal:
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def __init__(self, cube, parent, level, must, encounter):
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self.level = level
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self.cube = cube
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self.parent = parent
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self.must = must
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def __lt__(self, other):
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return self.level < other.level
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class QReach:
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# it is assumed that there is a single initial state
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# with all latches set to 0 in hardware design, so
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# here init will always give a state where all variable are set to 0
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def __init__(self, init, xs):
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self.xs = xs
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self.constant_xs = [Not(x) for x in self.xs]
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s = fd_solver()
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s.add(init)
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is_sat = s.check()
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assert is_sat == sat
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m = s.model()
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# xs is a list, "for" will keep the order when iterating
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self.states = numpy.array([[False for x in self.xs]]) # all set to False
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assert not numpy.max(self.states) # since all element is False, so maximum should be False
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# check if new state exists
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def is_exist(self, state):
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if state in self.states:
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return True
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return False
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def enumerate(self, i, state_b, state):
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while i < len(state) and state[i] not in self.xs:
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i += 1
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if i >= len(state):
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if state_b.tolist() not in self.states.tolist():
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self.states = numpy.append(self.states, [state_b], axis = 0)
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return state_b
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else:
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return None
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state_b[i] = False
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if self.enumerate(i+1, state_b, state) is not None:
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return state_b
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else:
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state_b[i] = True
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return self.enumerate(i+1, state_b, state)
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def is_full_state(self, state):
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for i in range(len(self.xs)):
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if state[i] in self.xs:
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return False
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return True
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def add(self, cube):
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state = self.cube2partial_state(cube)
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assert len(state) == len(self.xs)
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if not self.is_exist(state):
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return None
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if self.is_full_state(state):
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self.states = numpy.append(self.states, [state], axis = 0)
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else:
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# state[i] is instance, state_b[i] is boolean
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state_b = numpy.array(state)
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for i in range(len(state)): # state is of same length as self.xs
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# i-th literal in state hasn't been assigned value
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# init un-assigned literals in state_b as True
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# make state_b only contain boolean value
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if state[i] in self.xs:
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state_b[i] = True
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else:
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state_b[i] = is_true(state[i])
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if self.enumerate(0, state_b, state) is not None:
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lits_to_remove = set([negate(f) for f in list(set(cube) - set(self.constant_xs))])
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self.constant_xs = list(set(self.constant_xs) - lits_to_remove)
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return state
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return None
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def cube2partial_state(self, cube):
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s = fd_solver()
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s.add(And(cube))
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is_sat = s.check()
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assert is_sat == sat
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m = s.model()
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state = numpy.array([m.eval(x) for x in self.xs])
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return state
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def state2cube(self, s):
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result = copy.deepcopy(self.xs) # x1, x2, ...
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for i in range(len(self.xs)):
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if not s[i]:
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result[i] = Not(result[i])
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return result
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def intersect(self, cube):
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state = self.cube2partial_state(cube)
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mask = True
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for i in range(len(self.xs)):
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if is_true(state[i]) or is_false(state[i]):
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mask = (self.states[:, i] == state[i]) & mask
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intersects = numpy.reshape(self.states[mask], (-1, len(self.xs)))
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if intersects.size > 0:
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return And(self.state2cube(intersects[0])) # only need to return one single intersect
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return None
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class Quip:
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def __init__(self, init, trans, goal, x0, inputs, xn):
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self.x0 = x0
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self.inputs = inputs
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self.xn = xn
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self.init = init
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self.bad = goal
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self.trans = trans
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self.min_cube_solver = fd_solver()
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self.min_cube_solver.add(Not(trans))
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self.goals = []
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s = State(fd_solver())
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s.add(init)
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s.solver.add(trans) # check if a bad state can be reached in one step from current level
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self.states = [s]
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self.s_bad = fd_solver()
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self.s_good = fd_solver()
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self.s_bad.add(self.bad)
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self.s_good.add(Not(self.bad))
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self.reachable = QReach(self.init, x0)
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self.frames = [] # frames is a 2d list, each row (representing level) is a set containing several (clause, bad) pairs
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self.count_may = 0
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def next(self, f):
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if is_seq(f):
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return [self.next(f1) for f1 in f]
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return substitute(f, zip(self.x0, self.xn))
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def prev(self, f):
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if is_seq(f):
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return [self.prev(f1) for f1 in f]
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return substitute(f, zip(self.xn, self.x0))
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def add_solver(self):
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s = fd_solver()
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s.add(self.trans)
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self.states += [State(s)]
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def R(self, i):
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return And(self.states[i].R)
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def value2literal(self, m, x):
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value = m.eval(x)
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if is_true(value):
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return x
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if is_false(value):
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return Not(x)
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return None
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def values2literals(self, m, xs):
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p = [self.value2literal(m, x) for x in xs]
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return [x for x in p if x is not None]
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def project0(self, m):
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return self.values2literals(m, self.x0)
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def projectI(self, m):
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return self.values2literals(m, self.inputs)
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def projectN(self, m):
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return self.values2literals(m, self.xn)
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# Block a cube by asserting the clause corresponding to its negation
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def block_cube(self, i, cube):
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self.assert_clause(i, cube2clause(cube))
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# Add a clause to levels 1 until i
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def assert_clause(self, i, clause):
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for j in range(1, i + 1):
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self.states[j].add(clause)
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assert str(self.states[j].solver) != str([False])
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# minimize cube that is core of Dual solver.
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# this assumes that props & cube => Trans
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# which means props & cube can only give us a Tr in Trans,
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# and it will never make !Trans sat
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def minimize_cube(self, cube, inputs, lits):
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# min_cube_solver has !Trans (min_cube.solver.add(!Trans))
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is_sat = self.min_cube_solver.check(lits + [c for c in cube] + [i for i in inputs])
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assert is_sat == unsat
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# unsat_core gives us some lits which make Tr sat,
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# so that we can ignore other lits and include more states
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core = self.min_cube_solver.unsat_core()
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assert core
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return [c for c in core if c in set(cube)]
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# push a goal on a heap
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def push_heap(self, goal):
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heapq.heappush(self.goals, (goal.level, goal))
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# make sure cube to be blocked excludes all reachable states
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def check_reachable(self, cube):
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s = fd_solver()
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for state in self.reachable.states:
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s.push()
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r = self.reachable.state2cube(state)
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s.add(And(self.prev(r)))
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s.add(self.prev(cube))
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is_sat = s.check()
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s.pop()
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if is_sat == sat:
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# if sat, it means the cube to be blocked contains reachable states
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# so it is an invalid cube
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return False
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# if all fail, is_sat will be unsat
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return True
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# Rudimentary generalization:
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# If the cube is already unsat with respect to transition relation
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# extract a core (not necessarily minimal)
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# otherwise, just return the cube.
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def generalize(self, cube, f):
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s = self.states[f - 1].solver
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if unsat == s.check(cube):
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core = s.unsat_core()
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if self.check_reachable(core):
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return core, f
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return cube, f
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def valid_reachable(self, level):
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s = fd_solver()
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s.add(self.init)
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for i in range(level):
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s.add(self.trans)
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for state in self.reachable.states:
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s.push()
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s.add(And(self.next(self.reachable.state2cube(state))))
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print self.reachable.state2cube(state)
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print s.check()
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s.pop()
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def lemmas(self, level):
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return [(l.clause, l.bad) for l in self.frames[level]]
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# whenever a new reachable state is found, we use it to mark some existing lemmas as bad lemmas
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def mark_bad_lemmas(self, new):
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s = fd_solver()
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reset = False
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for frame in self.frames:
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for lemma in frame:
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s.push()
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s.add(lemma.clause)
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is_sat = s.check(new)
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if is_sat == unsat:
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reset = True
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lemma.bad = True
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s.pop()
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if reset:
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self.states = [self.states[0]]
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for i in range(1, len(self.frames)):
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self.add_solver()
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for lemma in self.frames[i]:
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if not lemma.bad:
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self.states[i].add(lemma.clause)
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# prev & tras -> r', such that r' intersects with cube
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def add_reachable(self, prev, cube):
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s = fd_solver()
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s.add(self.trans)
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s.add(prev)
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s.add(self.next(And(cube)))
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is_sat = s.check()
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assert is_sat == sat
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m = s.model()
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new = self.projectN(m)
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state = self.reachable.add(self.prev(new)) # always add as non-primed
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if state is not None: # if self.states do not have new state yet
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self.mark_bad_lemmas(self.prev(new))
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# Check if the negation of cube is inductive at level f
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def is_inductive(self, f, cube):
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s = self.states[f - 1].solver
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s.push()
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s.add(self.prev(Not(And(cube))))
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is_sat = s.check(cube)
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if is_sat == sat:
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m = s.model()
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s.pop()
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if is_sat == sat:
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cube = self.next(self.minimize_cube(self.project0(m), self.projectI(m), self.projectN(m)))
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elif is_sat == unsat:
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cube, f = self.generalize(cube, f)
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cube = self.next(cube)
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return cube, f, is_sat
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# Determine if there is a cube for the current state
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|
# that is potentially reachable.
|
|
def unfold(self, level):
|
|
core = []
|
|
self.s_bad.push()
|
|
R = self.R(level)
|
|
self.s_bad.add(R) # check if current frame intersects with bad states, no trans
|
|
is_sat = self.s_bad.check()
|
|
if is_sat == sat:
|
|
m = self.s_bad.model()
|
|
cube = self.project0(m)
|
|
props = cube + self.projectI(m)
|
|
self.s_good.push()
|
|
self.s_good.add(R)
|
|
is_sat2 = self.s_good.check(props)
|
|
assert is_sat2 == unsat
|
|
core = self.s_good.unsat_core()
|
|
assert core
|
|
core = [c for c in core if c in set(cube)]
|
|
self.s_good.pop()
|
|
self.s_bad.pop()
|
|
return is_sat, core
|
|
|
|
# A state s0 and level f0 such that
|
|
# not(s0) is f0-1 inductive
|
|
def quip_blocked(self, s0, f0):
|
|
self.push_heap(QGoal(self.next(s0), None, f0, must, 0))
|
|
while self.goals:
|
|
f, g = heapq.heappop(self.goals)
|
|
sys.stdout.write("%d." % f)
|
|
if not g.must:
|
|
self.count_may -= 1
|
|
sys.stdout.flush()
|
|
if f == 0:
|
|
if g.must:
|
|
s = fd_solver()
|
|
s.add(self.init)
|
|
s.add(self.prev(g.cube))
|
|
# since init is a complete assignment, so g.cube must equal to init in sat solver
|
|
assert is_sat == s.check()
|
|
if verbose:
|
|
print("")
|
|
return g
|
|
self.add_reachable(self.init, g.parent.cube)
|
|
continue
|
|
|
|
r0 = self.reachable.intersect(self.prev(g.cube))
|
|
if r0 is not None:
|
|
if g.must:
|
|
if verbose:
|
|
print ""
|
|
s = fd_solver()
|
|
s.add(self.trans)
|
|
# make it as a concrete reachable state
|
|
# intersect returns an And(...), so use children to get cube list
|
|
g.cube = r0.children()
|
|
while True:
|
|
is_sat = s.check(self.next(g.cube))
|
|
assert is_sat == sat
|
|
r = self.next(self.project0(s.model()))
|
|
r = self.reachable.intersect(self.prev(r))
|
|
child = QGoal(self.next(r.children()), g, 0, g.must, 0)
|
|
g = child
|
|
if not check_disjoint(self.init, self.prev(g.cube)):
|
|
# g is init, break the loop
|
|
break
|
|
init = g
|
|
while g.parent is not None:
|
|
g.parent.level = g.level + 1
|
|
g = g.parent
|
|
return init
|
|
if g.parent is not None:
|
|
self.add_reachable(r0, g.parent.cube)
|
|
continue
|
|
|
|
cube = None
|
|
is_sat = sat
|
|
f_1 = len(self.frames) - 1
|
|
while f_1 >= f:
|
|
for l in self.frames[f_1]:
|
|
if not l.bad and len(l.cube) > 0 and set(l.cube).issubset(g.cube):
|
|
cube = l.cube
|
|
is_sat == unsat
|
|
break
|
|
f_1 -= 1
|
|
if cube is None:
|
|
cube, f_1, is_sat = self.is_inductive(f, g.cube)
|
|
if is_sat == unsat:
|
|
self.frames[f_1].add(QLemma(self.prev(cube)))
|
|
self.block_cube(f_1, self.prev(cube))
|
|
if f_1 < f0:
|
|
# learned clause might also be able to block same bad states in higher level
|
|
if set(list(cube)) != set(list(g.cube)):
|
|
self.push_heap(QGoal(cube, None, f_1 + 1, may, 0))
|
|
self.count_may += 1
|
|
else:
|
|
# re-queue g.cube in higher level, here g.parent is simply for tracking down the trace when output.
|
|
self.push_heap(QGoal(g.cube, g.parent, f_1 + 1, g.must, 0))
|
|
if not g.must:
|
|
self.count_may += 1
|
|
else:
|
|
# qcube is a predecessor of g
|
|
qcube = QGoal(cube, g, f_1 - 1, g.must, 0)
|
|
if not g.must:
|
|
self.count_may += 1
|
|
self.push_heap(qcube)
|
|
|
|
if verbose:
|
|
print("")
|
|
return None
|
|
|
|
# Check if there are two states next to each other that have the same clauses.
|
|
def is_valid(self):
|
|
i = 1
|
|
inv = None
|
|
while True:
|
|
# self.states[].R contains full lemmas
|
|
# self.frames[] contains delta-encoded lemmas
|
|
while len(self.states) <= i+1:
|
|
self.add_solver()
|
|
while len(self.frames) <= i+1:
|
|
self.frames.append(set())
|
|
duplicates = set([])
|
|
for l in self.frames[i+1]:
|
|
if l in self.frames[i]:
|
|
duplicates |= {l}
|
|
self.frames[i] = self.frames[i] - duplicates
|
|
pushed = set([])
|
|
for l in (self.frames[i] - self.frames[i+1]):
|
|
if not l.bad:
|
|
s = self.states[i].solver
|
|
s.push()
|
|
s.add(self.next(Not(l.clause)))
|
|
s.add(l.clause)
|
|
is_sat = s.check()
|
|
s.pop()
|
|
if is_sat == unsat:
|
|
self.frames[i+1].add(l)
|
|
self.states[i+1].add(l.clause)
|
|
pushed |= {l}
|
|
self.frames[i] = self.frames[i] - pushed
|
|
if (not (self.states[i].R - self.states[i+1].R)
|
|
and len(self.states[i].R) != 0):
|
|
inv = prune(self.states[i].R)
|
|
F_inf = self.frames[i]
|
|
j = i + 1
|
|
while j < len(self.states):
|
|
for l in F_inf:
|
|
self.states[j].add(l.clause)
|
|
j += 1
|
|
self.frames[len(self.states)-1] = F_inf
|
|
self.frames[i] = set([])
|
|
break
|
|
elif (len(self.states[i].R) == 0
|
|
and len(self.states[i+1].R) == 0):
|
|
break
|
|
i += 1
|
|
|
|
if inv is not None:
|
|
self.s_bad.push()
|
|
self.s_bad.add(And(inv))
|
|
is_sat = self.s_bad.check()
|
|
if is_sat == unsat:
|
|
self.s_bad.pop()
|
|
return And(inv)
|
|
self.s_bad.pop()
|
|
return None
|
|
|
|
def run(self):
|
|
if not check_disjoint(self.init, self.bad):
|
|
return "goal is reached in initial state"
|
|
level = 0
|
|
while True:
|
|
inv = self.is_valid() # self.add_solver() here
|
|
if inv is not None:
|
|
return inv
|
|
is_sat, cube = self.unfold(level)
|
|
if is_sat == unsat:
|
|
level += 1
|
|
if verbose:
|
|
print("Unfold %d" % level)
|
|
sys.stdout.flush()
|
|
elif is_sat == sat:
|
|
cex = self.quip_blocked(cube, level)
|
|
if cex is not None:
|
|
return cex
|
|
else:
|
|
return is_sat
|
|
|
|
def test(file):
|
|
h2t = Horn2Transitions()
|
|
h2t.parse(file)
|
|
if verbose:
|
|
print("Test file: %s") % file
|
|
mp = Quip(h2t.init, h2t.trans, h2t.goal, h2t.xs, h2t.inputs, h2t.xns)
|
|
start_time = time.time()
|
|
result = mp.run()
|
|
end_time = time.time()
|
|
if isinstance(result, QGoal):
|
|
g = result
|
|
if verbose:
|
|
print("Trace")
|
|
while g:
|
|
if verbose:
|
|
print(g.level, g.cube)
|
|
g = g.parent
|
|
print("--- used %.3f seconds ---" % (end_time - start_time))
|
|
validate(mp, result, mp.trans)
|
|
return
|
|
if isinstance(result, ExprRef):
|
|
if verbose:
|
|
print("Invariant:\n%s " % result)
|
|
print("--- used %.3f seconds ---" % (end_time - start_time))
|
|
validate(mp, result, mp.trans)
|
|
return
|
|
print(result)
|
|
|
|
def validate(var, result, trans):
|
|
if isinstance(result, QGoal):
|
|
g = result
|
|
s = fd_solver()
|
|
s.add(trans)
|
|
while g.parent is not None:
|
|
s.push()
|
|
s.add(var.prev(g.cube))
|
|
s.add(var.next(g.parent.cube))
|
|
assert sat == s.check()
|
|
s.pop()
|
|
g = g.parent
|
|
if verbose:
|
|
print "--- validation succeed ----"
|
|
return
|
|
if isinstance(result, ExprRef):
|
|
inv = result
|
|
s = fd_solver()
|
|
s.add(trans)
|
|
s.push()
|
|
s.add(var.prev(inv))
|
|
s.add(Not(var.next(inv)))
|
|
assert unsat == s.check()
|
|
s.pop()
|
|
cube = var.prev(var.init)
|
|
step = 0
|
|
while True:
|
|
step += 1
|
|
# too many steps to reach invariant
|
|
if step > 1000:
|
|
if verbose:
|
|
print "--- validation failed --"
|
|
return
|
|
if not check_disjoint(var.prev(cube), var.prev(inv)):
|
|
# reach invariant
|
|
break
|
|
s.push()
|
|
s.add(cube)
|
|
assert s.check() == sat
|
|
cube = var.projectN(s.model())
|
|
s.pop()
|
|
if verbose:
|
|
print "--- validation succeed ----"
|
|
return
|
|
|
|
|
|
|
|
test("data/horn1.smt2")
|
|
test("data/horn2.smt2")
|
|
test("data/horn3.smt2")
|
|
test("data/horn4.smt2")
|
|
test("data/horn5.smt2")
|
|
# test("data/horn6.smt2") # not able to finish
|