def is_skolem(self, sym):
res = itr.is_skolem(sym) and not (sym.name.startswith('__')
and sym.name[2:3].isupper())
# if not res and self.top_level:
# name = sym.name
# res = name.startswith('loc:') or name.startswith('fml:')
return res
def handle(self,action,env):
# iu.dbg('env')
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())
if not self.started:
for sym in self.vocab:
if sym not in env and not itr.is_new(sym) and not itr.is_skolem(sym):
self.show_sym(sym,sym)
self.started = True
for sym,renamed_sym in env.iteritems():
if not itr.is_new(sym) and not itr.is_skolem(sym):
self.show_sym(sym,renamed_sym)
print '{}{}'.format(action.lineno,action)
def mine_constants(mod,trans,invariant):
res = defaultdict(list)
for c in ilu.used_symbols_ast(invariant):
if not il.is_function_sort(c.sort) and tr.is_skolem(c):
res[c.sort].append(c)
# iu.dbg('res')
return res
def prev_expr(stvarset,expr,sort_constants):
if any(sym in stvarset or tr.is_skolem(sym) and not sym in sort_constants[sym.sort]
for sym in ilu.symbols_ast(expr)):
return None
news = [sym for sym in ilu.used_symbols_ast(expr) if tr.is_new(sym)]
if news:
rn = dict((sym,tr.new_of(sym)) for sym in news)
return ilu.rename_ast(expr,rn)
return None
def get_predicates(self,clauses):
# print "get_predicates: {}".format(clauses)
d = self.parent_state.domain
sig = d.sig
urs = [x for x in used_unary_relations_clauses(clauses) if not is_skolem(x)]
cs = [x for x in used_constants_clauses(clauses)
if not is_skolem(x) and not has_enumerated_sort(sig,x) and not x.is_numeral()]
ufs = [x for x in used_unary_functions_clauses(clauses)
if not is_skolem(x) and has_enumerated_sort(sig,x)]
nrs = [x for x,arity in d.relations.iteritems() if arity == 0]
union_to_list(urs,[x for x,arity in d.relations.iteritems() if arity == 1])
union_to_list(cs,[x for x,arity in d.functions.iteritems()
if arity == 0 and not has_enumerated_sort(sig,x)])
union_to_list(ufs,[x for x,arity in d.functions.iteritems()
if arity == 1 and has_enumerated_sort(sig,x)])
# print "ufs: {}".format(ufs)
ccs = [Constant(c) for c in cs]
# print "sorts: {}".format([(c,c.get_sort()) for c in ccs])
return ([Literal(1,Atom(c,[])) for c in nrs] +
[Literal(1,Atom(equals,[Variable("X",c.get_sort()),c])) for c in ccs] +
[Literal(1,Atom(r,[Variable("X",r.sort.dom[0])])) for r in urs] +
[(App(f,Variable('X',f.sort.dom[0])),[Constant(Symbol(x,f.sort.rng)) for x in f.sort.rng.defines()]) for f in ufs])
def get_predicates(self, clauses):
# print "get_predicates: {}".format(clauses)
d = self.parent_state.domain
sig = d.sig
urs = [
x for x in used_unary_relations_clauses(clauses)
if not is_skolem(x)
]
cs = [
x for x in used_constants_clauses(clauses) if not is_skolem(x)
and not has_enumerated_sort(sig, x) and not x.is_numeral()
]
ufs = [
x for x in used_unary_functions_clauses(clauses)
if not is_skolem(x) and has_enumerated_sort(sig, x)
]
nrs = [x for x, arity in d.relations.iteritems() if arity == 0]
union_to_list(
urs, [x for x, arity in d.relations.iteritems() if arity == 1])
union_to_list(cs, [
x for x, arity in d.functions.iteritems()
if arity == 0 and not has_enumerated_sort(sig, x)
])
union_to_list(ufs, [
x for x, arity in d.functions.iteritems()
if arity == 1 and has_enumerated_sort(sig, x)
])
# print "ufs: {}".format(ufs)
ccs = [Constant(c) for c in cs]
# print "sorts: {}".format([(c,c.get_sort()) for c in ccs])
return ([Literal(1, Atom(c, [])) for c in nrs] + [
Literal(1, Atom(equals, [Variable("X", c.get_sort()), c]))
for c in ccs
] + [Literal(1, Atom(r, [Variable("X", r.sort.dom[0])]))
for r in urs] + [(App(f, Variable('X', f.sort.dom[0])), [
Constant(Symbol(x, f.sort.rng)) for x in f.sort.rng.defines()
]) for f in ufs])
def is_skolem(self, sym):
res = itr.is_skolem(sym) and not (sym.name.startswith('__')
and sym.name[2:3].isupper())
return res
def end(self):
for sym in self.vocab:
if not itr.is_new(sym) and not itr.is_skolem(sym):
self.show_sym(sym,sym)
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 my_is_skolem(x):
return tr.is_skolem(x) and x not in self.cnsts
def my_is_skolem(x):
res = tr.is_skolem(x) and x not in invar_syms
return res
axioms = state.domain.background_theory(state.in_scope)
error = tr.reverse_image([], err_act)
print "error = {}".format(error)
# get actions as updates
updates = [action.update(ag.domain, state.in_scope) for action in actions]
# get all the flexible (updated) variables
flex = set(sym for u in updates for sym in u[0])
print "flex = {}".format(flex)
# inflex is all non-skolem symbols in signature that are not updated
all_syms = sig.symbols
inflex = set(
sym for sym in all_syms
if sym not in flex and not tr.is_new(sym) and not tr.is_skolem(sym))
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)]
err_act = ag.actions["error"].update(ag.domain,state.in_scope)
axioms = state.domain.background_theory(state.in_scope)
error = tr.reverse_image([],axioms,err_act)
print "error = {}".format(error)
# get actions as updates
updates = [action.update(ag.domain,state.in_scope) for action in actions]
# get all the flexible (updated) variables
flex = set(sym for u in updates for sym in u[0])
print "flex = {}".format(flex)
# inflex is all non-skolem symbols in signature that are not updated
all_syms = sig.symbols
inflex = set(sym for sym in all_syms if sym not in flex and not tr.is_new(sym) and not tr.is_skolem(sym))
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)]
