def check_node(self, n):
x = var_to_skolem('__', Variable('X', n.sort)).suffix(str(n.sort))
y = var_to_skolem('__', Variable('Y', n.sort)).suffix(str(n.sort))
# print "x.sort: {}",format(x.sort)
self.solver.push()
s = self.solver
# if we have a witness we can show node is definite (present in all models)
wit = get_witness(n)
# print "checking: {}".format(n.fmla)
cube = substitute_clause(n.fmla, {'X': x})
# print "cube: {!r}".format(cube)
# print wit
# if wit != None:
## print "wit: {}, wit.sort: {}, x.sort: {}".format(wit,wit.sort,x.sort)
res = s_check_cube(s, cube,
(Atom(equals, [x, wit]) if wit != None else None))
## print"check cube: %s = %s" % (cube,res)
# res = s_check_cube(s,substitute_clause(n.fmla,{'X':x}))
# print "status: {}".format(res)
n.status = res
s_add(s, cube_to_z3(substitute_clause(n.fmla, {'X': x})))
s_add(s, cube_to_z3(substitute_clause(n.fmla, {'X': y})))
s_add(s, cube_to_z3([Literal(0, Atom(equals, [x, y]))]))
n.summary = s.check() != z3.unsat
self.solver.pop()
def check_edge(self,f1,f2,solver):
x = var_to_skolem('__',Variable('X',self.sorts[0])).suffix(str(self.sorts[0]))
y = var_to_skolem('__',Variable('Y',self.sorts[1])).suffix(str(self.sorts[1]))
solver.push()
s_add(solver,cube_to_z3(substitute_clause(f1,{'X':x})))
s_add(solver,cube_to_z3(substitute_clause(f2,{'X':y})))
# print "xsort: {}, ysort: {}".format(x.get_sort(),y.get_sort())
# print "rel_lit: {}, subs: {}".format(self.rel_lit,substitute_lit(self.rel_lit,{'X':x,'Y':y}))
res = s_check_cube(solver,[substitute_lit(self.rel_lit,{'X':x,'Y':y})])
solver.pop()
return res
def compile_concepts(self,concepts):
clauses = self.post.post_step(concepts)
# print "compile_concepts clauses={}".format(clauses)
vs = used_variables_in_order_clauses(clauses)
sksubs = dict((v.rep,var_to_skolem('__',Variable(v.rep,v.sort))) for v in vs)
clauses = substitute_clauses(clauses,sksubs)
# print "clauses: {}".format(clauses)
return clauses
def compile_concepts(self, concepts):
clauses = self.post.post_step(concepts)
# print "compile_concepts clauses={}".format(clauses)
vs = used_variables_in_order_clauses(clauses)
sksubs = dict(
(v.rep, var_to_skolem('__', Variable(v.rep, v.sort))) for v in vs)
clauses = substitute_clauses(clauses, sksubs)
# print "clauses: {}".format(clauses)
return clauses
def check_edge(self, f1, f2, solver):
x = var_to_skolem('__',
Variable('X',
self.sorts[0])).suffix(str(self.sorts[0]))
y = var_to_skolem('__',
Variable('Y',
self.sorts[1])).suffix(str(self.sorts[1]))
solver.push()
s_add(solver, cube_to_z3(substitute_clause(f1, {'X': x})))
s_add(solver, cube_to_z3(substitute_clause(f2, {'X': y})))
# print "xsort: {}, ysort: {}".format(x.get_sort(),y.get_sort())
# print "rel_lit: {}, subs: {}".format(self.rel_lit,substitute_lit(self.rel_lit,{'X':x,'Y':y}))
res = s_check_cube(solver,
[substitute_lit(self.rel_lit, {
'X': x,
'Y': y
})])
solver.pop()
return res
def emit_derived(header, impl, df, classname):
name = df.defines().name
sort = df.defines().sort
retval = il.Symbol("ret:val", sort)
vs = df.args[0].args
ps = [ilu.var_to_skolem('p:', v) for v in vs]
mp = dict(zip(vs, ps))
rhs = ilu.substitute_ast(df.args[1], mp)
action = ia.AssignAction(retval, rhs)
action.formal_params = ps
action.formal_returns = [retval]
emit_some_action(header, impl, name, action, classname)
def emit_derived(header,impl,df,classname):
name = df.defines().name
sort = df.defines().sort
retval = il.Symbol("ret:val",sort)
vs = df.args[0].args
ps = [ilu.var_to_skolem('p:',v) for v in vs]
mp = dict(zip(vs,ps))
rhs = ilu.substitute_ast(df.args[1],mp)
action = ia.AssignAction(retval,rhs)
action.formal_params = ps
action.formal_returns = [retval]
emit_some_action(header,impl,name,action,classname)
def check_node(self,n):
x = var_to_skolem('__',Variable('X',n.sort)).suffix(str(n.sort))
y = var_to_skolem('__',Variable('Y',n.sort)).suffix(str(n.sort))
# print "x.sort: {}",format(x.sort)
self.solver.push()
s = self.solver
# if we have a witness we can show node is definite (present in all models)
wit = get_witness(n)
# print "checking: {}".format(n.fmla)
cube = substitute_clause(n.fmla,{'X':x})
# print "cube: {!r}".format(cube)
# print wit
# if wit != None:
## print "wit: {}, wit.sort: {}, x.sort: {}".format(wit,wit.sort,x.sort)
res = s_check_cube(s,cube,(Atom(equals,[x,wit]) if wit != None else None))
## print"check cube: %s = %s" % (cube,res)
# res = s_check_cube(s,substitute_clause(n.fmla,{'X':x}))
# print "status: {}".format(res)
n.status = res
s_add(s,cube_to_z3(substitute_clause(n.fmla,{'X':x})))
s_add(s,cube_to_z3(substitute_clause(n.fmla,{'X':y})))
s_add(s,cube_to_z3([Literal(0,Atom(equals,[x,y]))]))
n.summary = s.check() != z3.unsat
self.solver.pop()
def check_edge(self,f1,solver):
x = var_to_skolem('__',Variable('X',self.sorts[0])).suffix(str(self.sorts[0]))
solver.push()
s_add(solver,cube_to_z3(substitute_clause(f1,{'X':x})))
# print "xsort: {}, ysort: {}".format(x.get_sort(),y.get_sort())
# print "rel_lit: {}, subs: {}".format(self.rel_lit,substitute_lit(self.rel_lit,{'X':x,'Y':y}))
f = self.fmla
vals = [Constant(Symbol(y,f.sort.rng)) for y in f.sort.rng.defines()]
status = 'undef'
for v in vals:
if s_check_fmla(solver,self.test_lit(x,v).atom) == z3.unsat:
status = v
break
solver.pop()
return status
def check_edge(self, f1, solver):
x = var_to_skolem('__',
Variable('X',
self.sorts[0])).suffix(str(self.sorts[0]))
solver.push()
s_add(solver, cube_to_z3(substitute_clause(f1, {'X': x})))
# print "xsort: {}, ysort: {}".format(x.get_sort(),y.get_sort())
# print "rel_lit: {}, subs: {}".format(self.rel_lit,substitute_lit(self.rel_lit,{'X':x,'Y':y}))
f = self.fmla
vals = [Constant(Symbol(y, f.sort.rng)) for y in f.sort.rng.defines()]
status = 'undef'
for v in vals:
if s_check_fmla(solver, self.test_lit(x, v).atom) == z3.unsat:
status = v
break
solver.pop()
return status
def to_aiger(mod,ext_act):
erf = il.Symbol('err_flag',il.find_sort('bool'))
errconds = []
add_err_flag_mod(mod,erf,errconds)
# we use a special state variable __init to indicate the initial state
ext_acts = [mod.actions[x] for x in sorted(mod.public_actions)]
ext_act = ia.EnvAction(*ext_acts)
init_var = il.Symbol('__init',il.find_sort('bool'))
init = add_err_flag(ia.Sequence(*([a for n,a in mod.initializers]+[ia.AssignAction(init_var,il.And())])),erf,errconds)
action = ia.Sequence(ia.AssignAction(erf,il.Or()),ia.IfAction(init_var,ext_act,init))
# get the invariant to be proved, replacing free variables with
# skolems. First, we apply any proof tactics.
pc = ivy_proof.ProofChecker(mod.axioms,mod.definitions,mod.schemata)
pmap = dict((lf.id,p) for lf,p in mod.proofs)
conjs = []
for lf in mod.labeled_conjs:
if lf.id in pmap:
proof = pmap[lf.id]
subgoals = pc.admit_proposition(lf,proof)
conjs.extend(subgoals)
else:
conjs.append(lf)
invariant = il.And(*[il.drop_universals(lf.formula) for lf in conjs])
# iu.dbg('invariant')
skolemizer = lambda v: ilu.var_to_skolem('__',il.Variable(v.rep,v.sort))
vs = ilu.used_variables_in_order_ast(invariant)
sksubs = dict((v.rep,skolemizer(v)) for v in vs)
invariant = ilu.substitute_ast(invariant,sksubs)
invar_syms = ilu.used_symbols_ast(invariant)
# compute the transition relation
stvars,trans,error = action.update(mod,None)
# print 'action : {}'.format(action)
# print 'annotation: {}'.format(trans.annot)
annot = trans.annot
# match_annotation(action,annot,MatchHandler())
indhyps = [il.close_formula(il.Implies(init_var,lf.formula)) for lf in mod.labeled_conjs]
# trans = ilu.and_clauses(trans,indhyps)
# save the original symbols for trace
orig_syms = ilu.used_symbols_clauses(trans)
orig_syms.update(ilu.used_symbols_ast(invariant))
# TODO: get the axioms (or maybe only the ground ones?)
# axioms = mod.background_theory()
# rn = dict((sym,tr.new(sym)) for sym in stvars)
# next_axioms = ilu.rename_clauses(axioms,rn)
# return ilu.and_clauses(axioms,next_axioms)
funs = set()
for df in trans.defs:
funs.update(ilu.used_symbols_ast(df.args[1]))
for fmla in trans.fmlas:
funs.update(ilu.used_symbols_ast(fmla))
# funs = ilu.used_symbols_clauses(trans)
funs.update(ilu.used_symbols_ast(invariant))
funs = set(sym for sym in funs if il.is_function_sort(sym.sort))
iu.dbg('[str(fun) for fun in funs]')
# Propositionally abstract
# step 1: get rid of definitions of non-finite symbols by turning
# them into constraints
new_defs = []
new_fmlas = []
for df in trans.defs:
if len(df.args[0].args) == 0 and is_finite_sort(df.args[0].sort):
new_defs.append(df)
else:
fmla = df.to_constraint()
new_fmlas.append(fmla)
trans = ilu.Clauses(new_fmlas+trans.fmlas,new_defs)
# step 2: get rid of ite's over non-finite sorts, by introducing constraints
cnsts = []
new_defs = [elim_ite(df,cnsts) for df in trans.defs]
new_fmlas = [elim_ite(fmla,cnsts) for fmla in trans.fmlas]
trans = ilu.Clauses(new_fmlas+cnsts,new_defs)
# step 3: eliminate quantfiers using finite instantiations
from_asserts = il.And(*[il.Equals(x,x) for x in ilu.used_symbols_ast(il.And(*errconds)) if
tr.is_skolem(x) and not il.is_function_sort(x.sort)])
iu.dbg('from_asserts')
invar_syms.update(ilu.used_symbols_ast(from_asserts))
sort_constants = mine_constants(mod,trans,il.And(invariant,from_asserts))
sort_constants2 = mine_constants2(mod,trans,invariant)
print '\ninstantiations:'
trans,invariant = Qelim(sort_constants,sort_constants2)(trans,invariant,indhyps)
# print 'after qe:'
# print 'trans: {}'.format(trans)
# print 'invariant: {}'.format(invariant)
# step 4: instantiate the axioms using patterns
# We have to condition both the transition relation and the
# invariant on the axioms, so we define a boolean symbol '__axioms'
# to represent the axioms.
axs = instantiate_axioms(mod,stvars,trans,invariant,sort_constants,funs)
ax_conj = il.And(*axs)
ax_var = il.Symbol('__axioms',ax_conj.sort)
ax_def = il.Definition(ax_var,ax_conj)
invariant = il.Implies(ax_var,invariant)
trans = ilu.Clauses(trans.fmlas+[ax_var],trans.defs+[ax_def])
# step 5: eliminate all non-propositional atoms by replacing with fresh booleans
# An atom with next-state symbols is converted to a next-state symbol if possible
stvarset = set(stvars)
prop_abs = dict() # map from atoms to proposition variables
global prop_abs_ctr # sigh -- python lameness
prop_abs_ctr = 0 # counter for fresh symbols
new_stvars = [] # list of fresh symbols
# get the propositional abstraction of an atom
def new_prop(expr):
res = prop_abs.get(expr,None)
if res is None:
prev = prev_expr(stvarset,expr,sort_constants)
if prev is not None:
# print 'stvar: old: {} new: {}'.format(prev,expr)
pva = new_prop(prev)
res = tr.new(pva)
new_stvars.append(pva)
prop_abs[expr] = res # prevent adding this again to new_stvars
else:
global prop_abs_ctr
res = il.Symbol('__abs[{}]'.format(prop_abs_ctr),expr.sort)
# print '{} = {}'.format(res,expr)
prop_abs[expr] = res
prop_abs_ctr += 1
return res
# propositionally abstract an expression
global mk_prop_fmlas
mk_prop_fmlas = []
def mk_prop_abs(expr):
if il.is_quantifier(expr) or len(expr.args) > 0 and any(not is_finite_sort(a.sort) for a in expr.args):
return new_prop(expr)
return expr.clone(map(mk_prop_abs,expr.args))
# apply propositional abstraction to the transition relation
new_defs = map(mk_prop_abs,trans.defs)
new_fmlas = [mk_prop_abs(il.close_formula(fmla)) for fmla in trans.fmlas]
# find any immutable abstract variables, and give them a next definition
def my_is_skolem(x):
res = tr.is_skolem(x) and x not in invar_syms
return res
def is_immutable_expr(expr):
res = not any(my_is_skolem(sym) or tr.is_new(sym) or sym in stvarset for sym in ilu.used_symbols_ast(expr))
return res
for expr,v in prop_abs.iteritems():
if is_immutable_expr(expr):
new_stvars.append(v)
print 'new state: {}'.format(expr)
new_defs.append(il.Definition(tr.new(v),v))
trans = ilu.Clauses(new_fmlas+mk_prop_fmlas,new_defs)
# apply propositional abstraction to the invariant
invariant = mk_prop_abs(invariant)
# create next-state symbols for atoms in the invariant (is this needed?)
rn = dict((sym,tr.new(sym)) for sym in stvars)
mk_prop_abs(ilu.rename_ast(invariant,rn)) # this is to pick up state variables from invariant
# update the state variables by removing the non-finite ones and adding the fresh state booleans
stvars = [sym for sym in stvars if is_finite_sort(sym.sort)] + new_stvars
# iu.dbg('trans')
# iu.dbg('stvars')
# iu.dbg('invariant')
# exit(0)
# For each state var, create a variable that corresponds to the input of its latch
# Also, havoc all the state bits except the init flag at the initial time. This
# is needed because in aiger, all latches start at 0!
def fix(v):
return v.prefix('nondet')
def curval(v):
return v.prefix('curval')
def initchoice(v):
return v.prefix('initchoice')
stvars_fix_map = dict((tr.new(v),fix(v)) for v in stvars)
stvars_fix_map.update((v,curval(v)) for v in stvars if v != init_var)
trans = ilu.rename_clauses(trans,stvars_fix_map)
# iu.dbg('trans')
new_defs = trans.defs + [il.Definition(ilu.sym_inst(tr.new(v)),ilu.sym_inst(fix(v))) for v in stvars]
new_defs.extend(il.Definition(curval(v),il.Ite(init_var,v,initchoice(v))) for v in stvars if v != init_var)
trans = ilu.Clauses(trans.fmlas,new_defs)
# Turn the transition constraint into a definition
cnst_var = il.Symbol('__cnst',il.find_sort('bool'))
new_defs = list(trans.defs)
new_defs.append(il.Definition(tr.new(cnst_var),fix(cnst_var)))
new_defs.append(il.Definition(fix(cnst_var),il.Or(cnst_var,il.Not(il.And(*trans.fmlas)))))
stvars.append(cnst_var)
trans = ilu.Clauses([],new_defs)
# Input are all the non-defined symbols. Output indicates invariant is false.
# iu.dbg('trans')
def_set = set(df.defines() for df in trans.defs)
def_set.update(stvars)
# iu.dbg('def_set')
used = ilu.used_symbols_clauses(trans)
used.update(ilu.symbols_ast(invariant))
inputs = [sym for sym in used if
sym not in def_set and not il.is_interpreted_symbol(sym)]
fail = il.Symbol('__fail',il.find_sort('bool'))
outputs = [fail]
# iu.dbg('trans')
# make an aiger
aiger = Encoder(inputs,stvars,outputs)
comb_defs = [df for df in trans.defs if not tr.is_new(df.defines())]
invar_fail = il.Symbol('invar__fail',il.find_sort('bool')) # make a name for invariant fail cond
comb_defs.append(il.Definition(invar_fail,il.Not(invariant)))
aiger.deflist(comb_defs)
for df in trans.defs:
if tr.is_new(df.defines()):
aiger.set(tr.new_of(df.defines()),aiger.eval(df.args[1]))
miter = il.And(init_var,il.Not(cnst_var),il.Or(invar_fail,il.And(fix(erf),il.Not(fix(cnst_var)))))
aiger.set(fail,aiger.eval(miter))
# aiger.sub.debug()
# make a decoder for the abstract propositions
decoder = dict((y,x) for x,y in prop_abs.iteritems())
for sym in aiger.inputs + aiger.latches:
if sym not in decoder and sym in orig_syms:
decoder[sym] = sym
cnsts = set(sym for syms in sort_constants.values() for sym in syms)
return aiger,decoder,annot,cnsts,action,stvarset
