def get_unstratified_funs(assumes, asserts, macros):
vu = il.VariableUniqifier()
def vupair(p):
return (vu(p[0]), p[1])
assumes = map(vupair, assumes)
asserts = map(vupair, asserts)
macros = map(vupair, macros)
strat_map = create_strat_map(assumes, asserts, macros)
# for f,g in macros:
# print 'macro: {}'.format(f)
arcs = list(get_sort_arcs(assumes + macros, asserts, strat_map))
sccs = get_sort_sccs(arcs)
scc_map = dict((name, idx) for idx, scc in enumerate(sccs) for name in scc)
scc_arcs = [[] for x in sccs]
unstrat = set()
for ds, rng, ast in arcs:
if scc_map[ds] == scc_map[rng]:
scc_arcs[scc_map[ds]].append(ast)
for y in strat_map.values():
find(y).variables.update(y.variables)
fun_sccs = [(x, y) for x, y in zip(sccs, scc_arcs)
if y and any(len(n.variables) > 0 for n in x)]
arc_map = defaultdict(list)
for x, y, z in arcs:
arc_map[x].append(y)
for scc in sccs:
for n in scc:
for m in arc_map[n]:
m.variables.update(n.variables)
# print 'sccs:'
# for scc in sccs:
# print [str(x) for x in scc]
# show_strat_map(strat_map)
bad_interpreted = set()
for x, y in strat_map.iteritems():
y = find(y)
if isinstance(x, tuple) and (il.is_interpreted_symbol(x[0])
or x[0].name == '='):
if any(v in universally_quantified_variables and v.sort == x[0].
sort.dom[x[1]] and il.has_infinite_interpretation(v.sort)
for v in y.variables):
bad_interpreted.add(x[0])
return fun_sccs, bad_interpreted
def get_unstratified_funs(assumes,asserts,macros):
vu = il.VariableUniqifier()
def vupair(p):
return (vu(p[0]),p[1])
assumes = map(vupair,assumes)
asserts = map(vupair,asserts)
macros = map(vupair,macros)
strat_map = create_strat_map(assumes,asserts,macros)
# for f,g in macros:
# print 'macro: {}'.format(f)
arcs = list(get_sort_arcs(assumes+macros,asserts,strat_map))
sccs = get_sort_sccs(arcs)
scc_map = dict((name,idx) for idx,scc in enumerate(sccs) for name in scc)
scc_arcs = [[] for x in sccs]
unstrat = set()
for ds,rng,ast in arcs:
if scc_map[ds] == scc_map[rng]:
scc_arcs[scc_map[ds]].append(ast)
for y in strat_map.values():
find(y).variables.update(y.variables)
fun_sccs = [(x,y) for x,y in zip(sccs,scc_arcs)
if y and any(len(n.variables) > 0 for n in x)]
arc_map = defaultdict(list)
for x,y,z in arcs:
arc_map[x].append(y)
for scc in sccs:
for n in scc:
for m in arc_map[n]:
m.variables.update(n.variables)
# print 'sccs:'
# for scc in sccs:
# print [str(x) for x in scc]
# show_strat_map(strat_map)
bad_interpreted = set()
for x,y in strat_map.iteritems():
y = find(y)
if isinstance(x,tuple) and (il.is_interpreted_symbol(x[0]) or x[0].name == '='):
if any(v in universally_quantified_variables and
v.sort == x[0].sort.dom[x[1]] and
il.has_infinite_interpretation(v.sort) for v in y.variables):
bad_interpreted.add(x[0])
return fun_sccs, bad_interpreted
def get_sort_arcs(assumes,asserts,strat_map):
# for sym in il.all_symbols():
# name = sym.name
# sort = sym.sort
# rng = sort.rng
# if il.is_uninterpreted_sort(rng):
# for ds in sort.dom:
# if il.is_uninterpreted_sort(ds):
# yield (ds,rng,sym)
# show_strat_map(strat_map)
for func,node in list(strat_map.iteritems()):
if isinstance(func,tuple) and not il.is_interpreted_symbol(func[0]):
yield (find(node),find(strat_map[func[0]]),func[0])
for fmla,ast in assumes + asserts:
for a in get_qa_arcs(fmla,ast,True,list(lu.free_variables(fmla)),strat_map):
yield a
for fmla,ast in asserts:
for a in get_qa_arcs(fmla,ast,False,[],strat_map):
yield a
def get_sort_arcs(assumes,asserts,strat_map):
# for sym in il.all_symbols():
# name = sym.name
# sort = sym.sort
# rng = sort.rng
# if il.is_uninterpreted_sort(rng):
# for ds in sort.dom:
# if il.is_uninterpreted_sort(ds):
# yield (ds,rng,sym)
# show_strat_map(strat_map)
for func,node in list(strat_map.iteritems()):
if isinstance(func,tuple) and not il.is_interpreted_symbol(func[0]):
yield (find(node),find(strat_map[func[0]]),func[0])
for fmla,ast in assumes + asserts:
for a in get_qa_arcs(fmla,ast,True,list(lu.free_variables(fmla)),strat_map):
yield a
for fmla,ast in asserts:
for a in get_qa_arcs(fmla,ast,False,[],strat_map):
yield a
def map_fmla(lineno, fmla, pol):
""" Add all of the subterms of `fmla` to the stratification graph. """
global universally_quantified_variables
global macro_var_map
global macro_dep_map
global macro_map
global macro_val_map
global strat_map
global arcs
if il.is_binder(fmla):
return map_fmla(lineno, fmla.body, pol)
if il.is_variable(fmla):
if fmla in universally_quantified_variables:
if fmla not in strat_map:
res = UFNode()
strat_map[fmla] = res
return strat_map[fmla], set()
node, vs = macro_var_map.get(fmla,
None), macro_dep_map.get(fmla, set())
return node, vs
reses = [
map_fmla(lineno, f, il.polar(fmla, pos, pol))
for pos, f in enumerate(fmla.args)
]
nodes, uvs = iu.unzip_pairs(reses)
all_uvs = iu.union_of_list(uvs)
all_uvs.update(n for n in nodes if n is not None)
if il.is_eq(fmla):
if not il.is_interpreted_sort(fmla.args[0].sort):
S_sigma = strat_map[il.Symbol('=', fmla.args[0])]
for x, uv in zip(nodes, uvs):
if x is not None:
unify(x, S_sigma)
arcs.extend((v, S_sigma, fmla, lineno) for v in uv)
else:
check_interpreted(fmla, nodes, uvs, lineno, pol)
return None, all_uvs
if il.is_ite(fmla):
# S_sigma = strat_map[il.Symbol('=',fmla.args[1])]
# for x,uv in zip(nodes[1:],uvs[1:]):
# if x is not None:
# unify(x,S_sigma)
# arcs.extend((v,S_sigma,fmla,lineno) for v in uv)
# TODO: treat ite as pseudo-macro: does this work?
if nodes[1] and nodes[2]:
unify(*nodes[1:])
return nodes[1] or nodes[2], all_uvs
if il.is_app(fmla):
func = fmla.rep
if not il.is_interpreted_symbol(func):
if func in macro_value_map:
return macro_value_map[func]
if func in macro_map:
defn, lf = macro_map[func]
res = map_fmla(lf.lineno, defn.rhs(), None)
macro_value_map[func] = res
return res
for idx, node in enumerate(nodes):
anode = strat_map[(func, idx)]
if node is not None:
unify(anode, node)
arcs.extend((v, anode, fmla, lineno) for v in uvs[idx])
else:
check_interpreted(fmla, nodes, uvs, lineno, pol)
return None, all_uvs
return None, all_uvs
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
