def __init__(self, clauses, model, vocab): # iu.dbg('clauses') self.clauses = clauses self.model = model self.vocab = vocab self.current = dict() mod_clauses = islv.clauses_model_to_clauses(clauses, model=model, numerals=True) # iu.dbg('mod_clauses') self.eqs = defaultdict(list) for fmla in mod_clauses.fmlas: if lg.is_eq(fmla): lhs, rhs = fmla.args if lg.is_app(lhs): self.eqs[lhs.rep].append(fmla) elif isinstance(fmla, lg.Not): app = fmla.args[0] if lg.is_app(app): self.eqs[app.rep].append(lg.Equals(app, lg.Or())) else: if lg.is_app(fmla): self.eqs[fmla.rep].append(lg.Equals(fmla, lg.And())) # for sym in vocab: # if not itr.is_new(sym) and not itr.is_skolem(sym): # self.show_sym(sym,sym) self.started = False self.renaming = dict() print print 'Trace follows...' print 80 * '*'
def __init__(self, clauses, model, vocab, top_level=True): TraceBase.__init__(self) self.clauses = clauses self.model = model self.vocab = vocab self.top_level = top_level if clauses is not None: ignore = lambda s: islv.solver_name(s) == None mod_clauses = islv.clauses_model_to_clauses(clauses, model=model, numerals=True, ignore=ignore) self.eqs = defaultdict(list) for fmla in mod_clauses.fmlas: if lg.is_eq(fmla): lhs, rhs = fmla.args if lg.is_app(lhs): self.eqs[lhs.rep].append(fmla) elif isinstance(fmla, lg.Not): app = fmla.args[0] if lg.is_app(app): self.eqs[app.rep].append(lg.Equals(app, lg.Or())) else: if lg.is_app(fmla): self.eqs[fmla.rep].append(lg.Equals(fmla, lg.And()))
def __init__(self, clauses, model, vocab, top_level=True): art.AnalysisGraph.__init__(self) self.clauses = clauses self.model = model self.vocab = vocab mod_clauses = islv.clauses_model_to_clauses(clauses, model=model, numerals=True) self.eqs = defaultdict(list) for fmla in mod_clauses.fmlas: if lg.is_eq(fmla): lhs, rhs = fmla.args if lg.is_app(lhs): self.eqs[lhs.rep].append(fmla) elif isinstance(fmla, lg.Not): app = fmla.args[0] if lg.is_app(app): self.eqs[app.rep].append(lg.Equals(app, lg.Or())) else: if lg.is_app(fmla): self.eqs[fmla.rep].append(lg.Equals(fmla, lg.And())) self.last_action = None self.sub = None self.returned = None self.top_level = top_level
def elim_ite(expr,cnsts): if isinstance(expr,il.Ite): global ite_ctr c,x,y = expr.args if not is_finite_sort(x.sort): v = il.Symbol('__ite[{}]'.format(ite_ctr),x.sort) ite_ctr += 1 cnsts.append(il.Ite(elim_ite(c,cnsts),elim_ite(il.Equals(v,x),cnsts),elim_ite(il.Equals(v,y),cnsts))) return v if il.is_eq(expr): v,e = expr.args if isinstance(e,il.Ite): c,x,y = e.args if not is_finite_sort(x.sort): return il.Ite(elim_ite(c,cnsts),elim_ite(il.Equals(v,x),cnsts),elim_ite(il.Equals(v,y),cnsts)) return expr.clone([elim_ite(e,cnsts) for e in expr.args])
def aiger_witness_to_ivy_trace(aiger,witnessfilename,action,stvarset,ext_act,annot,consts,decoder): with open(witnessfilename,'r') as f: res = f.readline().strip() if res != '1': badwit() tr = None aiger.sub.reset() lines = [] for line in f: if line.endswith('\n'): line = line[:-1] lines.append(line) print '\nCounterexample follows:' print 80*'-' current = dict() count = 0 for line in lines: if tr: print '' cols = line.split(' ') # iu.dbg('cols') if len(cols) != 4: badwit() pre,inp,out,post = cols aiger.sub.step(inp) count += 1 if count == len(lines): invar_fail = il.Symbol('invar__fail',il.find_sort('bool')) if il.is_true(aiger.get_sym(invar_fail)): break # print 'inputs:' # for v in aiger.inputs: # if v in decoder: # print ' {} = {}'.format(decoder[v],aiger.get_sym(v)) print 'path:' match_annotation(action,annot,AigerMatchHandler(aiger,decoder,consts,stvarset,current)) aiger.sub.next() post = aiger.sub.latch_vals() # use this, since file can be wrong! stvals = [] stmap = aiger.get_state(post) # iu.dbg('stmap') current = dict() for v in aiger.latches: # last two are used for encoding if v in decoder and v.name != '__init': val = stmap[v] if val is not None: stvals.append(il.Equals(decoder[v],val)) current[decoder[v]] = val print 'state:' for stval in stvals: print ' {}'.format(stval) if not tr: tr = IvyMCTrace(stvals) # first transition is initialization else: tr.add_state(stvals,ext_act) # remainder are exported actions print 80*'-' if tr is None: badwit() return tr
def sort_size_constraint(sort,size): if isinstance(sort,ivy_logic.UninterpretedSort): syms = [ivy_logic.Symbol('__'+sort.name+'$'+str(i),sort) for i in range(size)] v = ivy_logic.Variable('X'+sort.name,sort) res = ivy_logic.Or(*[ivy_logic.Equals(v,s) for s in syms]) # print "sort_size_constraint : {}".format(res) return res return ivy_logic.And()
def let_tactic(self,decls,proof): cond = il.And(*[il.Equals(x,y) for x,y in proof.args]) subgoal = ia.LabeledFormula(decls[0].label,il.Implies(cond,decls[0].formula)) if not hasattr(decls[0],'lineno'): print 'has no line number: {}'.format(decls[0]) exit(1) subgoal.lineno = decls[0].lineno return attrib_goals(proof,[subgoal]) + decls[1:]
def let_tactic(self, decls, proof): cond = il.And(*[il.Equals(x, y) for x, y in proof.args]) subgoal = ia.LabeledFormula(decls[0].label, il.Implies(cond, decls[0].formula)) subgoal.lineno = decls[0].lineno return attrib_goals(proof, [subgoal]) + decls[1:]
def bdv(v): """ Return a formula bounding a variable of ubninterpreted sort """ eqs = [ ivy_logic.Equals(v, reps[c.rep]) for c in h.sort_universe(v.sort) ] return ivy_logic.Or(*eqs)
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 let_tactic(self,decls,proof): cond = il.And(*[il.Equals(x,y) for x,y in proof.args]) return [ia.LabeledFormula(decls[0].label, il.Implies(cond,decls[0].formula))] + decls[1:]