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()