def mk_variant_assign_clauses(lhs, rhs):
n = lhs.rep
new_n = new(n)
args = lhs.args
dlhs = new_n(*sym_placeholders(n))
vs = dlhs.args
eqs = [
eq_atom(v, a) for (v, a) in zip(vs, args)
if not isinstance(a, Variable)
]
rn = dict((a.rep, v) for v, a in zip(vs, args) if isinstance(a, Variable))
drhs = substitute_ast(rhs, rn)
nondet = n.suffix("_nd").skolem()
if eqs:
nondet = Ite(And(*eqs), nondet, n(*dlhs.args))
lsort, rsort = lhs.sort, rhs.sort
fmlas = [
Iff(
pto(lsort, rsort)(dlhs, Variable('X', rsort)),
Equals(Variable('X', rsort), drhs))
]
for s in ivy_module.module.variants[lsort.name]:
if s != rsort:
fmlas.append(Not(pto(lsort, s)(dlhs, Variable('X', s))))
new_clauses = Clauses(fmlas, [Definition(dlhs, nondet)])
return new_clauses
def handle(self,action,env):
def my_is_skolem(x):
return tr.is_skolem(x) and x not in self.cnsts
def show_sym(v,decd,val):
if all(x in inv_env or not my_is_skolem(x) and
not tr.is_new(x) and x not in env for x in ilu.used_symbols_ast(decd)):
expr = ilu.rename_ast(decd,inv_env)
if not (expr in self.current and self.current[expr] == val):
print ' {} = {}'.format(expr,val)
self.current[expr] = val
if hasattr(action,'lineno'):
# print ' env: {}'.format('{'+','.join('{}:{}'.format(x,y) for x,y in env.iteritems())+'}')
inv_env = dict((y,x) for x,y in env.iteritems())
for v in self.aiger.inputs:
if v in self.decoder:
show_sym(v,self.decoder[v],self.aiger.get_sym(v))
rn = dict((x,tr.new(x)) for x in self.stvarset)
for v in self.aiger.latches:
if v in self.decoder:
decd = self.decoder[v]
show_sym(v,ilu.rename_ast(decd,rn),self.aiger.get_next_sym(v))
print ' {}{}'.format(action.lineno,action)
def action_update(self, domain, pvars):
lhs, rhs = self.args
n = lhs.rep
# Handle the hierarchical case
if n in domain.hierarchy:
asgns = [
postfix_atoms_ast(self, Atom(x, []))
for x in domain.hierarchy[n]
]
res = unzip_append(
[asgn.action_update(domain, pvars) for asgn in asgns])
return res
# If the lhs application is partial, make it total by adding parameters
xtra = len(lhs.rep.sort.dom) - len(lhs.args)
if xtra < 0:
raise IvyError(self,
"too many parameters in assignment to " + lhs.rep)
if xtra > 0:
extend = sym_placeholders(lhs.rep)[-xtra:]
extend = variables_distinct_list_ast(extend,
self) # get unused variables
lhs = add_parameters_ast(lhs, extend)
# Assignment of individual to a boolean is a special case
if is_individual_ast(rhs) and not is_individual_ast(lhs):
rhs = eq_atom(extend[-1],
add_parameters_ast(rhs, extend[0:-1]))
else:
rhs = add_parameters_ast(rhs, extend)
type_check(domain, rhs)
if is_individual_ast(lhs) != is_individual_ast(rhs):
# print type(lhs.rep)
# print str(lhs.rep)
# print type(lhs.rep.sort)
# print "lhs: %s: %s" % (lhs,type(lhs))
# print "rhs: %s: %s" % (rhs,type(rhs))
raise IvyError(self,
"sort mismatch in assignment to {}".format(lhs.rep))
new_n = new(n)
args = lhs.args
dlhs = new_n(*sym_placeholders(n))
vs = dlhs.args
eqs = [
eq_atom(v, a) for (v, a) in zip(vs, args)
if not isinstance(a, Variable)
]
rn = dict(
(a.rep, v) for v, a in zip(vs, args) if isinstance(a, Variable))
drhs = substitute_ast(rhs, rn)
if eqs:
drhs = Ite(And(*eqs), drhs, n(*dlhs.args))
new_clauses = Clauses([], [Definition(dlhs, drhs)])
# print "assign new_clauses = {}".format(new_clauses)
return ([n], new_clauses, false_clauses())
def action_update(self,domain,pvars):
lit = self.args[0]
n = lit.atom.relname
new_n = new(n)
args = lit.atom.args
vs = sym_placeholders(n)
eqs = [Atom(equals,[v,a]) for (v,a) in zip(vs,args) if not isinstance(a,Variable)]
new_clauses = And(*([Or(sign(lit.polarity,Atom(new_n,vs)),sign(1-lit.polarity,Atom(n,vs))),
sign(lit.polarity,Atom(new_n,args))] +
[Or(*([sign(0,Atom(new_n,vs)),sign(1,Atom(n,vs))] + [eq])) for eq in eqs] +
[Or(*([sign(1,Atom(new_n,vs)),sign(0,Atom(n,vs))] + [eq])) for eq in eqs]))
new_clauses = formula_to_clauses(new_clauses)
return ([n], new_clauses, false_clauses())
def mk_assign_clauses(lhs,rhs):
n = lhs.rep
new_n = new(n)
args = lhs.args
dlhs = new_n(*sym_placeholders(n))
vs = dlhs.args
eqs = [eq_atom(v,a) for (v,a) in zip(vs,args) if not isinstance(a,Variable)]
rn = dict((a.rep,v) for v,a in zip(vs,args) if isinstance(a,Variable))
drhs = substitute_ast(rhs,rn)
if eqs:
drhs = Ite(And(*eqs),drhs,n(*dlhs.args))
new_clauses = Clauses([],[Definition(dlhs,drhs)])
return new_clauses
def action_update(self,domain,pvars):
lit = self.args[0]
n = lit.atom.relname
new_n = new(n)
args = lit.atom.args
vs = sym_placeholders(n)
eqs = [Atom(equals,[v,a]) for (v,a) in zip(vs,args) if not isinstance(a,Variable)]
new_clauses = And(*([Or(sign(lit.polarity,Atom(new_n,vs)),sign(1-lit.polarity,Atom(n,vs))),
sign(lit.polarity,Atom(new_n,args))] +
[Or(*([sign(0,Atom(new_n,vs)),sign(1,Atom(n,vs))] + [eq])) for eq in eqs] +
[Or(*([sign(1,Atom(new_n,vs)),sign(0,Atom(n,vs))] + [eq])) for eq in eqs]))
new_clauses = formula_to_clauses(new_clauses)
return ([n], new_clauses, false_clauses())
def action_update(self,domain,pvars):
lhs = self.args[0]
n = lhs.rep
new_n = new(n)
args = lhs.args
vs = [Variable("X%d" % i,s) for i,s in enumerate(n.sort.dom)]
eqs = [eq_atom(v,a) for (v,a) in zip(vs,args) if not isinstance(a,Variable)]
if is_atom(lhs):
clauses = And(*([Or(Not(Atom(new_n,vs)),Atom(n,vs),eq) for eq in eqs] +
[Or(Atom(new_n,vs),Not(Atom(n,vs)),eq) for eq in eqs]))
elif is_individual_ast(lhs.rep):
clauses = And(*[Or(eq_atom(type(lhs)(new_n,vs),type(lhs)(n,vs)),eq) for eq in eqs])
else: # TODO: ???
clauses = And()
clauses = formula_to_clauses(clauses)
return ([n], clauses, false_clauses())
def action_update(self,domain,pvars):
lhs = type_ast(domain,self.args[0])
n = lhs.rep
new_n = new(n)
args = lhs.args
vs = [Variable("X%d" % i,s) for i,s in enumerate(n.sort.dom)]
eqs = [eq_atom(v,a) for (v,a) in zip(vs,args) if not isinstance(a,Variable)]
if is_atom(lhs):
clauses = And(*([Or(Not(Atom(new_n,vs)),Atom(n,vs),eq) for eq in eqs] +
[Or(Atom(new_n,vs),Not(Atom(n,vs)),eq) for eq in eqs]))
elif is_individual_ast(lhs.rep):
clauses = And(*[Or(eq_atom(type(lhs)(new_n,vs),type(lhs)(n,vs)),eq) for eq in eqs])
else: # TODO: ???
clauses = And()
clauses = formula_to_clauses(clauses)
return ([n], clauses, false_clauses())
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
def action_update(self, domain, pvars):
lhs, rhs = self.args
n = lhs.rep
# Handle the hierarchical case
if n in domain.hierarchy:
asgns = [postfix_atoms_ast(self, Atom(x, [])) for x in domain.hierarchy[n]]
res = unzip_append([asgn.action_update(domain, pvars) for asgn in asgns])
return res
# If the lhs application is partial, make it total by adding parameters
xtra = len(lhs.rep.sort.dom) - len(lhs.args)
if xtra < 0:
raise IvyError(self, "too many parameters in assignment to " + lhs.rep)
if xtra > 0:
extend = sym_placeholders(lhs.rep)[-xtra:]
extend = variables_distinct_list_ast(extend, self) # get unused variables
lhs = add_parameters_ast(lhs, extend)
# Assignment of individual to a boolean is a special case
if is_individual_ast(rhs) and not is_individual_ast(lhs):
rhs = eq_atom(extend[-1], add_parameters_ast(rhs, extend[0:-1]))
else:
rhs = add_parameters_ast(rhs, extend)
type_check(domain, rhs)
if is_individual_ast(lhs) != is_individual_ast(rhs):
# print type(lhs.rep)
# print str(lhs.rep)
# print type(lhs.rep.sort)
# print "lhs: %s: %s" % (lhs,type(lhs))
# print "rhs: %s: %s" % (rhs,type(rhs))
raise IvyError(self, "sort mismatch in assignment to {}".format(lhs.rep))
new_n = new(n)
args = lhs.args
dlhs = new_n(*sym_placeholders(n))
vs = dlhs.args
eqs = [eq_atom(v, a) for (v, a) in zip(vs, args) if not isinstance(a, Variable)]
rn = dict((a.rep, v) for v, a in zip(vs, args) if isinstance(a, Variable))
drhs = substitute_ast(rhs, rn)
if eqs:
drhs = Ite(And(*eqs), drhs, n(*dlhs.args))
new_clauses = Clauses([], [Definition(dlhs, drhs)])
# print "assign new_clauses = {}".format(new_clauses)
return ([n], new_clauses, false_clauses())
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 forward_clauses(clauses, inflex):
return lu.rename_clauses(
clauses,
dict((x, tr.new(x)) for x in lu.used_symbols_clauses(clauses)
if x != '=' and x not in inflex))
print "inflex = {}".format(inflex)
# get all the skolem symbols
all_syms = sig.symbols
skolems = set(sym for sym in all_syms if tr.is_skolem(sym))
print "skolems = {}".format(skolems)
# make frames explicit
updates = [tr.frame_update(u, flex, sig) for u in updates]
for u in updates:
print u
# locals are flexible and skolem symbols (next values unused for skolems)
ns = sv.native_symbol
#lsyms = [(ns(sym),ns(tr.new(sym))) for sym in chain(flex,skolems)]
lsyms = [(ns(sym), ns(tr.new(sym))) for sym in chain(flex)]
# globals are the inflexible symbols
gsyms = [ns(sym) for sym in inflex]
print "lsyms = {}".format(lsyms)
print "gsyms = {}".format(gsyms)
init = state.clauses
init = forward_clauses(init, inflex)
print "init = {}".format(init)
error = forward_clauses(error, inflex)
print "error = {}".format(error)
axioms += forward_clauses(axioms, inflex)
def forward_clauses(clauses,inflex):
return lu.rename_clauses(clauses, dict((x,tr.new(x)) for x in lu.used_symbols_clauses(clauses)
if x != '=' and x not in inflex))
print "inflex = {}".format(inflex)
# get all the skolem symbols
all_syms = sig.symbols
skolems = set(sym for sym in all_syms if tr.is_skolem(sym))
print "skolems = {}".format(skolems)
# make frames explicit
updates = [tr.frame_update(u,flex,sig) for u in updates]
for u in updates:
print u
# locals are flexible and skolem symbols (next values unused for skolems)
ns = sv.native_symbol
#lsyms = [(ns(sym),ns(tr.new(sym))) for sym in chain(flex,skolems)]
lsyms = [(ns(sym),ns(tr.new(sym))) for sym in chain(flex)]
# globals are the inflexible symbols
gsyms = [ns(sym) for sym in inflex]
print "lsyms = {}".format(lsyms)
print "gsyms = {}".format(gsyms)
init = state.clauses
init = forward_clauses(init,inflex)
print "init = {}".format(init)
error = forward_clauses(error,inflex)
print "error = {}".format(error)
axioms += forward_clauses(axioms,inflex)
