def functions_total_axioms(prog: syntax.Program) -> List[Expr]:
res = []
for func in prog.functions():
# TODO: generation of part of the formula duplicated from relaxation_action_def.
# TODO: would be best to beef up the expression-generation library
names: List[str] = []
params = []
for arg_sort in func.arity:
arg_sort_decl = syntax.get_decl_from_sort(arg_sort)
name = prog.scope.fresh(arg_sort_decl.name[0].upper(),
also_avoid=names)
names.append(name)
params.append(syntax.SortedVar(name, arg_sort))
ap_func = syntax.Apply(func.name,
tuple(syntax.Id(v.name) for v in params))
name = prog.scope.fresh('y', also_avoid=names)
ax = syntax.Forall(
tuple(params),
syntax.Exists((syntax.SortedVar(name, func.sort), ),
syntax.Eq(syntax.Id(name), ap_func)))
with prog.scope.n_states(1):
typechecker.typecheck_expr(prog.scope, ax, syntax.BoolSort)
res.append(ax)
return res
def as_onestate_formula(self, index: Optional[int] = None) -> Expr:
# TODO: move to class State, this shouldn't be here
assert self.num_states == 1 or index is not None, \
'to generate a onestate formula from a multi-state model, ' + \
'you must specify which state you want'
assert index is None or (0 <= index and index < self.num_states)
if index is None:
index = 0
if index not in self.onestate_formula_cache:
prog = syntax.the_program
mut_rel_interps = self.rel_interps[index]
mut_const_interps = self.const_interps[index]
mut_func_interps = self.func_interps[index]
vs: List[syntax.SortedVar] = []
ineqs: Dict[SortDecl, List[Expr]] = {}
rels: Dict[RelationDecl, List[Expr]] = {}
consts: Dict[ConstantDecl, Expr] = {}
funcs: Dict[FunctionDecl, List[Expr]] = {}
for sort in self.univs:
vs.extend(syntax.SortedVar(v, syntax.UninterpretedSort(sort.name))
for v in self.univs[sort])
u = [syntax.Id(v) for v in self.univs[sort]]
ineqs[sort] = [syntax.Neq(a, b) for a, b in combinations(u, 2)]
for R, l in chain(mut_rel_interps.items(), self.immut_rel_interps.items()):
rels[R] = []
for tup, ans in l.items():
e: Expr = (
syntax.AppExpr(R.name, tuple(syntax.Id(col) for col in tup))
if tup else syntax.Id(R.name)
)
rels[R].append(e if ans else syntax.Not(e))
for C, c in chain(mut_const_interps.items(), self.immut_const_interps.items()):
consts[C] = syntax.Eq(syntax.Id(C.name), syntax.Id(c))
for F, fl in chain(mut_func_interps.items(), self.immut_func_interps.items()):
funcs[F] = [
syntax.Eq(syntax.AppExpr(F.name, tuple(syntax.Id(col) for col in tup)),
syntax.Id(res))
for tup, res in fl.items()
]
# get a fresh variable, avoiding names of universe elements in vs
fresh = prog.scope.fresh('x', [v.name for v in vs])
e = syntax.Exists(tuple(vs), syntax.And(
*chain(*ineqs.values(), *rels.values(), consts.values(), *funcs.values(), (
syntax.Forall((syntax.SortedVar(fresh,
syntax.UninterpretedSort(sort.name)),),
syntax.Or(*(syntax.Eq(syntax.Id(fresh), syntax.Id(v))
for v in self.univs[sort])))
for sort in self.univs
))))
assert prog.scope is not None
with prog.scope.n_states(1):
typechecker.typecheck_expr(prog.scope, e, None)
self.onestate_formula_cache[index] = e
return self.onestate_formula_cache[index]
def relaxed_program(prog: syntax.Program) -> syntax.Program:
new_decls: List[syntax.Decl] = [d for d in prog.sorts()]
actives: Dict[syntax.SortDecl, syntax.RelationDecl] = {}
for sort in prog.sorts():
name = prog.scope.fresh('active_' + sort.name)
r = syntax.RelationDecl(name, arity=[syntax.UninterpretedSort(sort.name)],
mutable=True, derived=None, annotations=[])
actives[sort] = r
new_decls.append(r)
# active relations initial conditions: always true
for sort in prog.sorts():
name = prog.scope.fresh(sort.name[0].upper())
expr = syntax.Forall([syntax.SortedVar(name, None)],
syntax.Apply(actives[sort].name, [syntax.Id(name)]))
new_decls.append(syntax.InitDecl(name=None, expr=expr))
for d in prog.decls:
if isinstance(d, syntax.SortDecl):
pass # already included above
elif isinstance(d, syntax.RelationDecl):
if d.derived_axiom is not None:
expr = syntax.relativize_quantifiers(actives, d.derived_axiom)
new_decls.append(syntax.RelationDecl(d.name, d.arity, d.mutable, expr,
d.annotations))
else:
new_decls.append(d)
elif isinstance(d, syntax.ConstantDecl):
new_decls.append(d)
elif isinstance(d, syntax.FunctionDecl):
new_decls.append(d)
elif isinstance(d, syntax.AxiomDecl):
new_decls.append(d)
elif isinstance(d, syntax.InitDecl):
new_decls.append(d)
elif isinstance(d, syntax.DefinitionDecl):
assert not isinstance(d.body, syntax.BlockStatement), \
"relax does not support transitions written in imperative syntax"
mods, expr = d.body
expr = syntax.relativize_quantifiers(actives, expr)
if d.is_public_transition:
guard = syntax.relativization_guard_for_binder(actives, d.binder)
expr = syntax.And(guard, expr)
new_decls.append(syntax.DefinitionDecl(d.is_public_transition, d.num_states, d.name,
params=d.binder.vs, body=(mods, expr)))
elif isinstance(d, syntax.InvariantDecl):
expr = syntax.relativize_quantifiers(actives, d.expr)
new_decls.append(syntax.InvariantDecl(d.name, expr=expr,
is_safety=d.is_safety, is_sketch=d.is_sketch))
else:
assert False, d
new_decls.append(relaxation_action_def(prog, actives=actives, fresh=True))
res = syntax.Program(new_decls)
res.resolve() # #sorrynotsorry
return res
def consts_exist_axioms(prog: syntax.Program) -> List[Expr]:
res = []
for c in prog.constants():
name = prog.scope.fresh('e_%s' % c.name)
ax = syntax.Exists((syntax.SortedVar(name, c.sort), ),
syntax.Eq(syntax.Id(c.name), syntax.Id(name)))
with prog.scope.n_states(1):
typechecker.typecheck_expr(prog.scope, ax, syntax.BoolSort)
res.append(ax)
return res
def _read_first_order_structure(
struct: FirstOrderStructure
) -> Tuple[List[syntax.SortedVar], # vs
Dict[SortDecl, List[Expr]], # ineqs
Dict[RelationDecl, List[Expr]], # rels
Dict[ConstantDecl, Expr], # consts
Dict[FunctionDecl, List[Expr]], # funcs
]:
vars_by_sort: Dict[SortDecl, List[syntax.SortedVar]] = {}
ineqs: Dict[SortDecl, List[Expr]] = {}
rels: Dict[RelationDecl, List[Expr]] = {}
consts: Dict[ConstantDecl, Expr] = {}
funcs: Dict[FunctionDecl, List[Expr]] = {}
for sort in struct.univs:
vars_by_sort[sort] = [
syntax.SortedVar(v, syntax.UninterpretedSort(sort.name))
for v in struct.univs[sort]
]
u = [syntax.Id(s) for s in struct.univs[sort]]
ineqs[sort] = [
syntax.Neq(a, b) for a, b in itertools.combinations(u, 2)
]
for R, l in struct.rel_interps.items():
rels[R] = []
for tup, ans in l.items():
e: Expr
if tup:
args: List[Expr] = []
for (col, col_sort) in zip(tup, R.arity):
assert isinstance(col_sort, syntax.UninterpretedSort)
assert col_sort.decl is not None
args.append(syntax.Id(col))
e = syntax.AppExpr(R.name, tuple(args))
else:
e = syntax.Id(R.name)
e = e if ans else syntax.Not(e)
rels[R].append(e)
for C, c in struct.const_interps.items():
e = syntax.Eq(syntax.Id(C.name), syntax.Id(c))
consts[C] = e
for F, fl in struct.func_interps.items():
funcs[F] = []
for tup, res in fl.items():
e = syntax.AppExpr(F.name,
tuple(syntax.Id(col) for col in tup))
e = syntax.Eq(e, syntax.Id(res))
funcs[F].append(e)
vs = list(itertools.chain(*(vs for vs in vars_by_sort.values())))
return vs, ineqs, rels, consts, funcs
def relax_actives_action_chunk(scope: syntax.Scope, actives: Dict[syntax.SortDecl, syntax.RelationDecl]) \
-> Tuple[Tuple[syntax.ModifiesClause, ...], List[Expr]]:
new_mods = []
new_conjs = []
for sort, active_rel in actives.items():
name = scope.fresh(sort.name[0].upper())
ap = syntax.Apply(active_rel.name, (syntax.Id(name), ))
expr = syntax.Forall((syntax.SortedVar(name, None), ),
syntax.Implies(syntax.New(ap), ap))
new_conjs.append(expr)
new_mods.append(syntax.ModifiesClause(actives[sort].name))
return tuple(new_mods), new_conjs
def load_relaxed_trace_from_updr_cex(prog: Program, s: Solver) -> logic.Trace:
import xml.dom.minidom # type: ignore
collection = xml.dom.minidom.parse(
"paxos_derived_trace.xml").documentElement
components: List[syntax.TraceComponent] = []
xml_decls = reversed(collection.childNodes)
seen_first = False
for elm in xml_decls:
if isinstance(elm, xml.dom.minidom.Text): # type: ignore
continue
if elm.tagName == 'state':
diagram = parser.parse_expr(elm.childNodes[0].data)
typechecker.typecheck_expr(prog.scope, diagram, syntax.BoolSort)
assert isinstance(
diagram, syntax.QuantifierExpr) and diagram.quant == 'EXISTS'
active_clauses = [
relaxed_traces.active_var(v.name, str(v.sort))
for v in diagram.get_vs()
]
if not seen_first:
# restrict the domain to be subdomain of the diagram's existentials
seen_first = True
import itertools # type: ignore
for sort, vars in itertools.groupby(
diagram.get_vs(),
lambda v: v.sort): # TODO; need to sort first
free_var = syntax.SortedVar(
syntax.the_program.scope.fresh("v_%s" % str(sort)),
None)
# TODO: diagram simplification omits them from the exists somewhere
consts = list(
filter(lambda c: c.sort == sort, prog.constants()))
els: Sequence[Union[syntax.SortedVar, syntax.ConstantDecl]]
els = list(vars)
els += consts
restrict_domain = syntax.Forall(
(free_var, ),
syntax.Or(*(syntax.Eq(syntax.Id(free_var.name),
syntax.Id(v.name))
for v in els)))
active_clauses += [restrict_domain]
diagram_active = syntax.Exists(
diagram.get_vs(), syntax.And(diagram.body, *active_clauses))
typechecker.typecheck_expr(prog.scope, diagram_active,
syntax.BoolSort)
components.append(syntax.AssertDecl(expr=diagram_active))
elif elm.tagName == 'action':
action_name = elm.childNodes[0].data.split()[0]
tcall = syntax.TransitionCalls(
calls=[syntax.TransitionCall(target=action_name, args=None)])
components.append(syntax.TraceTransitionDecl(transition=tcall))
else:
assert False, "unknown xml tagName"
trace_decl = syntax.TraceDecl(components=components, sat=True)
migrated_trace = bmc_trace(
prog,
trace_decl,
s,
lambda s, ks: logic.check_solver(s, ks, minimize=True),
log=False)
assert migrated_trace is not None
import pickle
pickle.dump(migrated_trace, open("migrated_trace.p", "wb"))
return migrated_trace
def p_sortedvar_nosort(p: Any) -> None:
'sortedvar : id'
p[0] = syntax.SortedVar(p[1], p[1].value, None)
def p_sortedvar(p: Any) -> None:
'sortedvar : id COLON sort'
p[0] = syntax.SortedVar(p[1], p[1].value, p[3])
def relaxation_action_def(prog: syntax.Program,
actives: Optional[Dict[syntax.SortDecl, syntax.RelationDecl]]=None,
fresh: bool=True) \
-> syntax.DefinitionDecl:
decrease_name = (prog.scope.fresh('decrease_domain') if fresh else 'decrease_domain')
mods = []
conjs: List[Expr] = []
if actives is None:
actives = active_rel_by_sort(prog)
# a conjunct allowing each domain to decrease
for sort in prog.sorts():
name = prog.scope.fresh(sort.name[0].upper())
ap = syntax.Apply(actives[sort].name, [syntax.Id(None, name)])
expr = syntax.Forall([syntax.SortedVar(None, name, None)],
syntax.Implies(ap, syntax.Old(ap)))
conjs.append(expr)
mods.append(syntax.ModifiesClause(None, actives[sort].name))
# constants are active
for const in prog.constants():
conjs.append(syntax.Apply(actives[syntax.get_decl_from_sort(const.sort)].name,
[syntax.Id(None, const.name)]))
# functions map active to active
for func in prog.functions():
names: List[str] = []
func_conjs = []
for arg_sort in func.arity:
arg_sort_decl = syntax.get_decl_from_sort(arg_sort)
name = prog.scope.fresh(arg_sort_decl.name[0].upper(),
also_avoid=names)
names.append(name)
func_conjs.append(syntax.Apply(actives[arg_sort_decl].name, [syntax.Id(None, name)]))
ap_func = syntax.Old(syntax.Apply(func.name, [syntax.Id(None, name) for name in names]))
active_func = syntax.Apply(actives[syntax.get_decl_from_sort(func.sort)].name, [ap_func])
conjs.append(syntax.Forall([syntax.SortedVar(None, name, None) for name in names],
syntax.Implies(syntax.And(*func_conjs), active_func)))
# (relativized) axioms hold after relaxation
for axiom in prog.axioms():
if not syntax.is_universal(axiom.expr):
conjs.append(syntax.relativize_quantifiers(actives, axiom.expr))
# derived relations have the same interpretation on the active domain
for rel in prog.derived_relations():
names = []
rel_conjs = []
for arg_sort in rel.arity:
arg_sort_decl = syntax.get_decl_from_sort(arg_sort)
name = prog.scope.fresh(arg_sort_decl.name[0].upper(),
also_avoid=names)
names.append(name)
rel_conjs.append(syntax.Apply(actives[arg_sort_decl].name, [syntax.Id(None, name)]))
ap_rel = syntax.Apply(rel.name, [syntax.Id(None, name) for name in names])
conjs.append(syntax.Forall([syntax.SortedVar(None, name, None) for name in names],
syntax.Implies(syntax.And(*rel_conjs),
syntax.Iff(ap_rel, syntax.Old(ap_rel)))))
return syntax.DefinitionDecl(None, public=True, twostate=True, name=decrease_name,
params=[], body=(mods, syntax.And(*conjs)))
def derived_rels_candidates_from_trace(trns: Trace, more_traces: List[Trace],
max_conj_size: int, max_free_vars: int) -> List[Tuple[List[syntax.SortedVar],Expr]]:
first_relax_idx = first_relax_step_idx(trns)
pre_relax_state = trns.as_state(first_relax_idx)
post_relax_state = trns.as_state(first_relax_idx + 1)
assert pre_relax_state.univs == post_relax_state.univs
# relaxed elements
relaxed_elements = []
for sort, univ in pre_relax_state.univs.items():
active_rel_name = 'active_' + sort.name # TODO: de-duplicate
pre_active_interp = dict_val_from_rel_name(active_rel_name, pre_relax_state.rel_interp)
post_active_interp = dict_val_from_rel_name(active_rel_name, post_relax_state.rel_interp)
pre_active_elements = [tup[0] for (tup, b) in pre_active_interp if b]
post_active_elements = [tup[0] for (tup, b) in post_active_interp if b]
assert set(post_active_elements).issubset(set(pre_active_elements))
for relaxed_elem in utils.OrderedSet(pre_active_elements) - set(post_active_elements):
relaxed_elements.append((sort, relaxed_elem))
# pre-relaxation step facts concerning at least one relaxed element (other to be found by UPDR)
relevant_facts: List[Union[RelationFact,FunctionFact,InequalityFact]] = []
for rel, rintp in pre_relax_state.rel_interp.items():
for rfact in rintp:
(elms, polarity) = rfact
relation_fact = RelationFact(rel, elms, polarity)
if set(relation_fact.involved_elms()) & set(ename for (_, ename) in relaxed_elements):
relevant_facts.append(relation_fact)
for func, fintp in pre_relax_state.func_interp.items():
for ffact in fintp:
(els_params, els_res) = ffact
function_fact = FunctionFact(func, els_params, els_res)
if set(function_fact.involved_elms()) & set(ename for (_, ename) in relaxed_elements):
relevant_facts.append(function_fact)
for sort, elm in relaxed_elements: # other inequalities presumably handled by UPDR
for other_elm in pre_relax_state.univs[sort]:
if other_elm == elm:
continue
relevant_facts.append(InequalityFact(elm, other_elm))
# facts blocking this specific relaxation step
diff_conjunctions = []
candidates_cache: Set[str] = set()
for fact_lst in itertools.combinations(relevant_facts, max_conj_size):
elements = utils.OrderedSet(itertools.chain.from_iterable(fact.involved_elms() for fact in fact_lst))
relaxed_elements_relevant = [elm for (_, elm) in relaxed_elements if elm in elements]
vars_from_elm = dict((elm, syntax.SortedVar(None, syntax.the_program.scope.fresh("v%d" % i), None))
for (i, elm) in enumerate(elements))
parameter_elements = elements - set(relaxed_elements_relevant)
if len(parameter_elements) > max_free_vars:
continue
conjuncts = [fact.as_expr(lambda elm: vars_from_elm[elm].name) for fact in fact_lst]
# for elm, var in vars_from_elm.items():
# TODO: make the two loops similar
for elm in relaxed_elements_relevant:
var = vars_from_elm[elm]
sort = pre_relax_state.element_sort(elm)
active_element_conj = syntax.Apply('active_%s' % sort.name, [syntax.Id(None, var.name)])
conjuncts.append(active_element_conj)
derived_relation_formula = syntax.Exists([vars_from_elm[elm]
for (_, elm) in relaxed_elements
if elm in vars_from_elm],
syntax.And(*conjuncts))
if str(derived_relation_formula) in candidates_cache:
continue
candidates_cache.add(str(derived_relation_formula))
if closing_qa_cycle(syntax.the_program, [pre_relax_state.element_sort(elm) for elm in parameter_elements],
[pre_relax_state.element_sort(elm) for elm in relaxed_elements_relevant]):
# adding the derived relation would close a quantifier alternation cycle, discard the candidate
continue
# if trns.eval_double_vocab(diffing_formula, first_relax_idx):
if is_rel_blocking_relax(trns, first_relax_idx,
([(vars_from_elm[elm], pre_relax_state.element_sort(elm).name) for elm in parameter_elements],
derived_relation_formula)):
# if all(trs.eval_double_vocab(diffing_formula, first_relax_step_idx(trs)) for trs in more_traces):
diff_conjunctions.append(([vars_from_elm[elm] for elm in parameter_elements],
derived_relation_formula))
return diff_conjunctions
def p_sortedvar_nosort(p: Any) -> None:
'sortedvar : id'
id_tok: Token = p[1]
p[0] = syntax.SortedVar(id_tok.value, None, span=span_from_tok(id_tok))
def p_sortedvar(p: Any) -> None:
'sortedvar : id COLON sort'
id_tok: Token = p[1]
sort: syntax.UninterpretedSort = p[3]
p[0] = syntax.SortedVar(id_tok.value, sort, span=loc_join(id_tok, sort.span))
def relaxed_program(prog: syntax.Program) -> syntax.Program:
new_decls: List[syntax.Decl] = [d for d in prog.sorts()]
actives: Dict[syntax.SortDecl, syntax.RelationDecl] = {}
for sort in prog.sorts():
name = prog.scope.fresh('active_' + sort.name)
r = syntax.RelationDecl(name,
arity=(syntax.UninterpretedSort(sort.name), ),
mutable=True)
actives[sort] = r
new_decls.append(r)
# active relations initial conditions: always true
for sort in prog.sorts():
name = prog.scope.fresh(sort.name[0].upper())
expr = syntax.Forall((syntax.SortedVar(name, None), ),
syntax.Apply(actives[sort].name,
(syntax.Id(name), )))
new_decls.append(syntax.InitDecl(name=None, expr=expr))
for d in prog.decls:
if isinstance(d, syntax.SortDecl):
pass # already included above
elif isinstance(d, syntax.RelationDecl):
if d.derived_axiom is not None:
expr = syntax.relativize_quantifiers(actives, d.derived_axiom)
new_decls.append(
syntax.RelationDecl(d.name,
d.arity,
d.mutable,
expr,
annotations=d.annotations))
else:
new_decls.append(d)
elif isinstance(d, syntax.ConstantDecl):
new_decls.append(d)
elif isinstance(d, syntax.FunctionDecl):
new_decls.append(d)
elif isinstance(d, syntax.AxiomDecl):
new_decls.append(d)
elif isinstance(d, syntax.InitDecl):
new_decls.append(d)
elif isinstance(d, syntax.DefinitionDecl):
relativized_def = relativize_decl(d,
actives,
prog.scope,
inline_relax_actives=False)
new_decls.append(relativized_def)
elif isinstance(d, syntax.InvariantDecl):
expr = syntax.relativize_quantifiers(actives, d.expr)
new_decls.append(
syntax.InvariantDecl(d.name,
expr=expr,
is_safety=d.is_safety,
is_sketch=d.is_sketch))
else:
assert False, d
new_decls.append(relaxation_action_def(prog, actives=actives, fresh=True))
res = syntax.Program(new_decls)
typechecker.typecheck_program(res) # #sorrynotsorry
return res
def relaxation_action_def(prog: syntax.Program,
actives: Optional[Dict[syntax.SortDecl,
syntax.RelationDecl]] = None,
fresh: bool = True) -> syntax.DefinitionDecl:
decrease_name = (prog.scope.fresh('decrease_domain')
if fresh else 'decrease_domain')
mods: Tuple[syntax.ModifiesClause, ...] = ()
conjs: List[Expr] = []
if actives is None:
actives = active_rel_by_sort(prog)
# a conjunct allowing each domain to decrease
new_mods, new_conjs = relax_actives_action_chunk(prog.scope, actives)
mods += new_mods
conjs += new_conjs
# constants are active
for const in prog.constants():
conjs.append(
syntax.New(
syntax.Apply(
actives[syntax.get_decl_from_sort(const.sort)].name,
(syntax.Id(const.name), ))))
# functions map active to active
for func in prog.functions():
names: List[str] = []
func_conjs = []
for arg_sort in func.arity:
arg_sort_decl = syntax.get_decl_from_sort(arg_sort)
name = prog.scope.fresh(arg_sort_decl.name[0].upper(),
also_avoid=names)
names.append(name)
func_conjs.append(
syntax.New(
syntax.Apply(actives[arg_sort_decl].name,
(syntax.Id(name), ))))
ap_func = syntax.Apply(func.name,
tuple(syntax.Id(name) for name in names))
name = prog.scope.fresh('y', also_avoid=names)
active_func = syntax.Let(
syntax.SortedVar(name, func.sort), ap_func,
syntax.New(
syntax.Apply(
actives[syntax.get_decl_from_sort(func.sort)].name,
(syntax.Id(name), ))))
conjs.append(
syntax.Forall(
tuple(syntax.SortedVar(name, None) for name in names),
syntax.Implies(syntax.And(*func_conjs), active_func)))
# (relativized) axioms hold after relaxation
for axiom in prog.axioms():
if not syntax.is_universal(axiom.expr):
conjs.append(syntax.relativize_quantifiers(actives, axiom.expr))
# derived relations have the same interpretation on the active domain
for rel in prog.derived_relations():
names = []
rel_conjs = []
for arg_sort in rel.arity:
arg_sort_decl = syntax.get_decl_from_sort(arg_sort)
name = prog.scope.fresh(arg_sort_decl.name[0].upper(),
also_avoid=names)
names.append(name)
rel_conjs.append(
syntax.Apply(actives[arg_sort_decl].name, (syntax.Id(name), )))
ap_rel = syntax.Apply(rel.name,
tuple(syntax.Id(name) for name in names))
conjs.append(
syntax.Forall(
tuple(syntax.SortedVar(name, None) for name in names),
syntax.Implies(syntax.And(*rel_conjs),
syntax.Iff(syntax.New(ap_rel), ap_rel))))
return syntax.DefinitionDecl(is_public_transition=True,
num_states=2,
name=decrease_name,
params=(),
mods=mods,
expr=syntax.And(*conjs))
