def apply_conj_proofs(mod):
# Apply any proof tactics to the conjs to get the conj_subgoals.
pc = ivy_proof.ProofChecker(mod.labeled_axioms + mod.assumed_invariants,
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)
subgoals = map(ivy_compiler.theorem_to_property, subgoals)
conjs.extend(subgoals)
else:
conjs.append(lf)
mod.conj_subgoals = conjs
def check_temporals():
mod = im.module
props = mod.labeled_props
pmap = dict((prop.id, p) for prop, p in mod.proofs)
pc = ivy_proof.ProofChecker(mod.labeled_axioms + mod.assumed_invariants,
mod.definitions, mod.schemata)
for prop in props:
print '\n The following temporal property is being proved:\n'
print pretty_lf(prop)
if prop.temporal:
proof = pmap.get(prop.id, None)
model = itmp.normal_program_from_module(im.module)
subgoal = prop.clone(
[prop.args[0],
itmp.TemporalModels(model, prop.args[1])])
subgoals = [subgoal]
subgoals = pc.admit_proposition(prop, proof, subgoals)
check_subgoals(subgoals)
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
def check_isolate():
mod = im.module
if mod.isolate_proof is not None:
pc = ivy_proof.ProofChecker(
mod.labeled_axioms + mod.assumed_invariants, mod.definitions,
mod.schemata)
model = itmp.normal_program_from_module(im.module)
prop = ivy_ast.LabeledFormula(ivy_ast.Atom('safety'), lg.And())
subgoal = ivy_ast.LabeledFormula(ivy_ast.Atom('safety'),
itmp.TemporalModels(model, lg.And()))
# print 'subgoal = {}'.format(subgoal)
subgoals = [subgoal]
subgoals = pc.admit_proposition(prop, mod.isolate_proof, subgoals)
check_subgoals(subgoals)
return
ifc.check_fragment()
with im.module.theory_context():
global check_lineno
check_lineno = act.checked_assert.get()
if check_lineno == "":
check_lineno = None
# print 'check_lineno: {}'.format(check_lineno)
check = not opt_summary.get()
subgoalmap = dict((x.id, y) for x, y in im.module.subgoals)
axioms = [m for m in mod.labeled_axioms if m.id not in subgoalmap]
schema_instances = [
m for m in mod.labeled_axioms if m.id in subgoalmap
]
if axioms:
print "\n The following properties are assumed as axioms:"
for lf in axioms:
print pretty_lf(lf)
if mod.definitions:
print "\n The following definitions are used:"
for lf in mod.definitions:
print pretty_lf(lf)
if (mod.labeled_props
or schema_instances) and not checked_action.get():
print "\n The following properties are to be checked:"
if check:
for lf in schema_instances:
print pretty_lf(lf) + " [proved by axiom schema]"
ag = ivy_art.AnalysisGraph()
clauses1 = lut.true_clauses(annot=act.EmptyAnnotation())
pre = itp.State(value=clauses1)
props = [x for x in im.module.labeled_props if not x.temporal]
props = [
p for p in props if not (p.id in subgoalmap and p.explicit)
]
fcs = ([(ConjAssumer if prop.assumed or prop.id in subgoalmap
else ConjChecker)(prop) for prop in props])
check_fcs_in_state(mod, ag, pre, fcs)
else:
for lf in schema_instances + mod.labeled_props:
print pretty_lf(lf)
# after checking properties, make them axioms, except temporals
im.module.labeled_axioms.extend(p for p in im.module.labeled_props
if not p.temporal)
im.module.update_theory()
if mod.labeled_inits:
print "\n The following properties are assumed initially:"
for lf in mod.labeled_inits:
print pretty_lf(lf)
if mod.labeled_conjs:
print "\n The inductive invariant consists of the following conjectures:"
for lf in mod.labeled_conjs:
print pretty_lf(lf)
apply_conj_proofs(mod)
if mod.isolate_info is not None and mod.isolate_info.implementations:
print "\n The following action implementations are present:"
for mixer, mixee, action in sorted(
mod.isolate_info.implementations, key=lambda x: x[0]):
print " {}implementation of {}".format(
pretty_lineno(action), mixee)
if mod.isolate_info is not None and mod.isolate_info.monitors:
print "\n The following action monitors are present:"
for mixer, mixee, action in sorted(mod.isolate_info.monitors,
key=lambda x: x[0]):
print " {}monitor of {}".format(pretty_lineno(action),
mixee)
# if mod.actions:
# print "\n The following actions are present:"
# for actname,action in sorted(mod.actions.iteritems()):
# print " {}{}".format(pretty_lineno(action),actname)
if mod.initializers:
print "\n The following initializers are present:"
for actname, action in sorted(mod.initializers,
key=lambda x: x[0]):
print " {}{}".format(pretty_lineno(action), actname)
if mod.labeled_conjs and not checked_action.get():
print "\n Initialization must establish the invariant"
if check:
with itp.EvalContext(check=False):
ag = ivy_art.AnalysisGraph(initializer=lambda x: None)
check_conjs_in_state(mod, ag, ag.states[0])
else:
print ''
if mod.initializers:
print "\n Any assertions in initializers must be checked",
if check:
ag = ivy_art.AnalysisGraph(initializer=lambda x: None)
fail = itp.State(expr=itp.fail_expr(ag.states[0].expr))
check_safety_in_state(mod, ag, fail)
checked_actions = get_checked_actions()
if checked_actions and mod.labeled_conjs:
print "\n The following set of external actions must preserve the invariant:"
for actname in sorted(checked_actions):
action = act.env_action(actname)
print " {}{}".format(pretty_lineno(action), actname)
if check:
ag = ivy_art.AnalysisGraph()
pre = itp.State()
pre.clauses = get_conjs(mod)
with itp.EvalContext(check=False): # don't check safety
# post = ag.execute(action, pre, None, actname)
post = ag.execute(action, pre)
check_conjs_in_state(mod, ag, post, indent=12)
else:
print ''
callgraph = defaultdict(list)
for actname, action in mod.actions.iteritems():
for called_name in action.iter_calls():
callgraph[called_name].append(actname)
some_assumps = False
for actname, action in mod.actions.iteritems():
assumptions = [
sub for sub in action.iter_subactions()
if isinstance(sub, act.AssumeAction)
]
if assumptions:
if not some_assumps:
print "\n The following program assertions are treated as assumptions:"
some_assumps = True
callers = callgraph[actname]
if actname in mod.public_actions:
callers.append("the environment")
prettyname = actname[4:] if actname.startswith(
'ext:') else actname
prettycallers = [
c[4:] if c.startswith('ext:') else c for c in callers
]
print " in action {} when called from {}:".format(
prettyname, ','.join(prettycallers))
for sub in assumptions:
print " {}assumption".format(pretty_lineno(sub))
tried = set()
some_guarants = False
for actname, action in mod.actions.iteritems():
guarantees = [
sub for sub in action.iter_subactions()
if isinstance(sub, (act.AssertAction, act.Ranking))
]
if check_lineno is not None:
guarantees = [
sub for sub in guarantees if sub.lineno == check_lineno
]
if guarantees:
if not some_guarants:
print "\n The following program assertions are treated as guarantees:"
some_guarants = True
callers = callgraph[actname]
if actname in mod.public_actions:
callers.append("the environment")
prettyname = actname[4:] if actname.startswith(
'ext:') else actname
prettycallers = [
c[4:] if c.startswith('ext:') else c for c in callers
]
print " in action {} when called from {}:".format(
prettyname, ','.join(prettycallers))
roots = set(iu.reachable([actname], lambda x: callgraph[x]))
for sub in guarantees:
print " {}guarantee".format(pretty_lineno(sub)),
if check and any(
r in roots and (r, sub.lineno) not in tried
for r in checked_actions):
print_dots()
old_checked_assert = act.checked_assert.get()
act.checked_assert.value = sub.lineno
some_failed = False
for root in checked_actions:
if root in roots:
tried.add((root, sub.lineno))
action = act.env_action(root)
ag = ivy_art.AnalysisGraph()
pre = itp.State()
pre.clauses = get_conjs(mod)
with itp.EvalContext(check=False):
post = ag.execute(action, prestate=pre)
fail = itp.State(expr=itp.fail_expr(post.expr))
if not check_safety_in_state(
mod, ag, fail, report_pass=False):
some_failed = True
break
if not some_failed:
print 'PASS'
act.checked_assert.value = old_checked_assert
else:
print ""
check_temporals()