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gr1.py
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gr1.py
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"""Algorithms for generalized Streett and Rabin games.
References
==========
Roderick Bloem, Barbara Jobstmann, Nir Piterman,
Amir Pnueli, Yaniv Sa'ar
"Synthesis of reactive(1) designs"
Journal of Computer and System Sciences
Vol.78, No.3, pp.911--938, 2012
Robert Konighofer
"Debugging formal specifications with
simplified counterstrategies"
Master's thesis
Inst. for Applied Information Processing and Communications,
Graz University of Technology, 2009
"""
import logging
import copy
from omega.symbolic import fixpoint as fx
from omega.symbolic import symbolic
logger = logging.getLogger(__name__)
# TODO:
# - switch to `dd.autoref`
# - expose both Mealy and Moore synthesizers
def solve_streett_game(aut, rank=1):
"""Return winning set and iterants for Streett(1) game.
@param aut: compiled game with <>[] | []<> winning
@type aut: `symbolic.Automaton`
"""
assert rank == 1, 'only rank 1 supported for now'
assert not aut.vars or aut.bdd.vars, (
'first call `Automaton.build`')
aut.assert_consistent(built=True)
assert len(aut.win['<>[]']) > 0
assert len(aut.win['[]<>']) > 0
bdd = aut.bdd
z = bdd.true
zold = None
while z != zold:
zold = z
xijk = list()
yij = list()
for goal in aut.win['[]<>']:
y, yj, xjk = _attractor_under_assumptions(z, goal, aut)
z = bdd.apply('and', z, y)
xijk.append(xjk)
yij.append(yj)
return z, yij, xijk
def _attractor_under_assumptions(z, goal, aut):
"""Targeting `goal`, under unconditional assumptions."""
bdd = aut.bdd
env_action = aut.action['env'][0]
sys_action = aut.action['sys'][0]
xjk = list()
yj = list()
y = bdd.false
yold = None
cox_z = fx.ue_preimage(env_action, sys_action, z, aut)
g = bdd.apply('and', goal, cox_z)
while y != yold:
yold = y
cox_y = fx.ue_preimage(env_action, sys_action, y, aut)
unless = bdd.apply('or', cox_y, g)
xk = list()
for safe in aut.win['<>[]']:
x = fx.trap(env_action, sys_action,
safe, aut, unless=unless)
xk.append(x)
y = bdd.apply('or', y, x)
yj.append(y)
xjk.append(xk)
return y, yj, xjk
def make_streett_transducer(z, yij, xijk, aut, bdd=None):
"""Return I/O `symbolic.Automaton` implementing strategy.
An auxiliary variable `_goal` is added,
to represent the counter of recurrence goals.
@param bdd: use this BDD manager
@type bdd: `BDD` in `dd.cudd` or `dd.bdd` or `dd.autoref`
"""
aut.assert_consistent(built=True)
assert z != aut.bdd.false, 'empty winning set'
# add goal counter var
c = '_goal'
dvars = copy.deepcopy(aut.vars)
n_goals = len(aut.win['[]<>'])
dvars[c] = dict(
type='saturating',
dom=(0, n_goals - 1),
owner='sys')
# compile transducer with refined shared BDD
t = symbolic.Automaton()
t.vars = dvars
t = t.build(bdd=bdd)
bdd = t.bdd
# copy functions of interest from solution BDD
r = [z, yij, xijk,
aut.init['env'][0], aut.init['sys'][0],
aut.action['env'][0], aut.action['sys'][0],
aut.win['<>[]'], aut.win['[]<>']]
r = _map_nested_lists(_copy_bdd, r, aut.bdd, t.bdd)
(z, yij, xijk, env_init, sys_init,
env_action, sys_action, holds, goals) = r
# compute strategy from iterates
# \rho_1: switch goals
rho_1 = bdd.false
for i, goal in enumerate(goals):
ip = (i + 1) % len(goals)
s = "({c} = {i}) & ({c}' = {ip})".format(c=c, i=i, ip=ip)
u = t.add_expr(s)
u = bdd.apply('and', u, goal)
rho_1 = bdd.apply('or', u, rho_1)
zp = bdd.rename(z, t.prime)
rho_1 = bdd.apply('and', rho_1, zp)
# \rho_2: descent in basin
rho_2 = bdd.false
for i, yj in enumerate(yij):
s = "({c} = {i}) & ({c}' = {i})".format(c=c, i=i)
count = t.add_expr(s)
rho_2j = bdd.false
basin = yj[0]
for y in yj[1:]:
next_basin = bdd.rename(basin, t.prime)
rim = bdd.apply('diff', y, basin)
u = bdd.apply('and', rim, next_basin)
rho_2j = bdd.apply('or', rho_2j, u)
basin = bdd.apply('or', basin, y)
u = bdd.apply('and', rho_2j, count)
rho_2 = bdd.apply('or', rho_2, u)
# \rho_3: persistence holds
rho_3 = bdd.false
for i, xjk in enumerate(xijk):
s = "({c} = {i}) & ({c}' = {i})".format(c=c, i=i)
count = t.add_expr(s)
rho_3j = bdd.false
used = bdd.false
for xk in xjk:
assert len(xk) == len(holds), xk
for x, hold in zip(xk, holds):
next_wait = bdd.rename(x, t.prime)
stay = bdd.apply('diff', x, used)
used = bdd.apply('or', used, x)
u = bdd.apply('and', stay, next_wait)
u = bdd.apply('and', u, hold)
rho_3j = bdd.apply('or', rho_3j, u)
u = bdd.apply('and', rho_3j, count)
rho_3 = bdd.apply('or', rho_3, u)
# \rho
u = bdd.apply('or', rho_1, rho_2)
u = bdd.apply('or', rho_3, u)
u = bdd.apply('and', sys_action, u)
# counter `c` limits
u = bdd.apply('and', t.action['sys'][0], u)
# include it only if `env_action` violation tracked
# with fresh bit in transducer memory
# u = bdd.apply('->', env_action, u)
action = u
t.action['sys'] = [action]
# initial condition
s = '{c} = 0'.format(c=c)
count = t.add_expr(s)
win_set = z
# counter initial limits (no closure taken for counter)
u = bdd.apply('and', sys_init, t.init['sys'][0])
u = bdd.apply('and', count, u)
u = bdd.apply('and', win_set, u)
init = bdd.apply('->', env_init, u)
t.init['sys'] = [init]
# \forall \exists init semantics
real = bdd.quantify(init, t.evars, forall=False)
real = bdd.quantify(real, t.uvars, forall=True)
assert real == t.bdd.true, 'cannot init in winning set'
return t
def solve_rabin_game(aut, rank=1):
"""Return winning set and iterants for Rabin(1) game.
@param aut: compiled game with <>[] & []<> winning
@type aut: `symbolic.Automaton`
"""
assert rank == 1, 'only rank 1 supported for now'
assert not aut.vars or aut.bdd.vars, (
'first call `Automaton.build`')
aut.assert_consistent(built=True)
# TODO: can these assertions be removed elegantly ?
assert len(aut.win['<>[]']) > 0
assert len(aut.win['[]<>']) > 0
bdd = aut.bdd
z = bdd.false
zold = None
zk = list()
yki = list()
xkijr = list()
while z != zold:
zold = z
xijr = list()
yi = list()
for hold in aut.win['<>[]']:
y, xjr = _cycle_inside(zold, hold, aut)
z = bdd.apply('or', z, y)
xijr.append(xjr)
yi.append(y)
zk.append(z)
yki.append(yi)
xkijr.append(xijr)
return zk, yki, xkijr
def _cycle_inside(z, hold, aut):
"""Cycling through goals, while staying in `hold`."""
bdd = aut.bdd
env_action = aut.action['env'][0]
sys_action = aut.action['sys'][0]
cox_z = fx.ue_preimage(env_action, sys_action,
z, aut, moore=True)
g = bdd.apply('or', cox_z, hold)
y = bdd.true
yold = None
while y != yold:
yold = y
cox_y = fx.ue_preimage(env_action, sys_action,
y, aut, moore=True)
inside = bdd.apply('and', cox_y, g)
xjr = list()
for goal in aut.win['[]<>']:
x, xr = _attractor_inside(inside, goal, aut, moore=True)
xjr.append(xr)
y = bdd.apply('and', y, x)
return y, xjr
def _attractor_inside(inside, goal, aut, moore=False):
bdd = aut.bdd
env_action = aut.action['env'][0]
sys_action = aut.action['sys'][0]
xr = list()
x = bdd.false
xold = None
while x != xold:
xold = x
cox_x = fx.ue_preimage(
env_action, sys_action, x, aut,
evars=aut.epvars, moore=moore)
x = bdd.apply('or', cox_x, goal)
x = bdd.apply('and', x, inside)
x = bdd.apply('or', x, xold)
xr.append(x)
return x, xr
def make_rabin_transducer(zk, yki, xkijr, aut):
"""Return O/I transducer for Rabin(1) game."""
aut.assert_consistent(built=True)
win_set = zk[-1]
assert win_set != aut.bdd.false, 'empty winning set'
dvars = dict(aut.vars)
n_holds = len(aut.win['<>[]'])
n_goals = len(aut.win['[]<>'])
# add transducer memory as two indices:
# - `w`: persistence hold index
# - `c`: recurrence goal index
w = '_hold'
c = '_goal'
n_w = n_holds - 1 + 1 # last value used as "none"
n_c = n_goals - 1
dvars[w] = dict(type='saturating', dom=(0, n_w), owner='sys')
dvars[c] = dict(type='saturating', dom=(0, n_c), owner='sys')
# compile
t = symbolic.Automaton()
t.vars = dvars
t = t.build()
# copy functions of interest from solution BDD
r = [zk, yki, xkijr,
aut.init['env'][0], aut.init['sys'][0],
aut.action['env'][0], aut.action['sys'][0],
aut.win['<>[]'], aut.win['[]<>']]
r = _map_nested_lists(_copy_bdd, r, aut.bdd, t.bdd)
(zk, yki, xkijr, env_init, sys_init,
env_action, sys_action, holds, goals) = r
t.action['env'] = [env_action]
# compute strategy from iterates
bdd = t.bdd
# \rho_1: descent in persistence basin
s = "({c}' = {c}) & ({w}' = {none})".format(
c=c, w=w, none=n_holds)
count = t.add_expr(s)
rho_1 = bdd.false
basin = zk[0]
for z in zk[1:]:
trans = _moore_trans(basin, t)
rim = bdd.apply('diff', z, basin)
u = bdd.apply('and', rim, trans)
u = bdd.apply('and', u, count)
rho_1 = bdd.apply('or', rho_1, u)
basin = z
rho_2 = bdd.false
rho_3 = bdd.false
rho_4 = bdd.false
basin = bdd.false
for z, yi, xijr in zip(zk, yki, xkijr):
cox_basin = fx.ue_preimage(env_action, sys_action,
basin, t, moore=True)
rim = bdd.apply('diff', z, basin)
rim = bdd.apply('and', rim, -cox_basin)
# rho_2: pick persistence set
s = "({c}' = {c}) & ({w} = {none})".format(
c=c, w=w, none=n_holds)
count = t.add_expr(s)
u = bdd.apply('and', rim, count)
v = bdd.false
for i, y in enumerate(yi):
s = "{w}' = {i}".format(w=w, i=i)
count = t.add_expr(s)
trans = _moore_trans(y, t)
q = bdd.apply('and', count, trans)
v = bdd.apply('or', v, q)
u = bdd.apply('and', u, v)
rho_2 = bdd.apply('or', rho_2, u)
# rho_3: descent in recurrence basin
s = (
"({c}' = {c}) &"
"({w} != {none}) &"
"({w}' = {w})").format(
c=c, w=w, none=n_holds)
count = t.add_expr(s)
u = bdd.apply('and', rim, count)
v = bdd.false
for i, xjr in enumerate(xijr):
for j, (xr, goal) in enumerate(zip(xjr, goals)):
s = (
"({c} = {j}) &"
" ({w} = {i})").format(c=c, w=w, i=i, j=j)
count = t.add_expr(s)
x_basin = xr[0]
p = bdd.false
for x in xr[1:]:
trans = _moore_trans(x_basin, t)
q = bdd.apply('and', trans, -x_basin)
q = bdd.apply('and', x, q)
p = bdd.apply('or', p, q)
x_basin = x
p = bdd.apply('and', p, count)
p = bdd.apply('and', p, -goal)
v = bdd.apply('or', v, p)
u = bdd.apply('and', u, v)
rho_3 = bdd.apply('or', rho_3, u)
# rho_4: advance to next recurrence goal
u = bdd.false
for j, goal in enumerate(goals):
jp = (j + 1) % len(goals)
s = "({c} = {j}) & ({c}' = {jp})".format(
c=c, j=j, jp=jp)
count = t.add_expr(s)
p = bdd.apply('and', count, goal)
u = bdd.apply('or', u, p)
s = (
"({w} != {none}) &"
"({w}' = {w})").format(
c=c, w=w, none=n_holds)
count = t.add_expr(s)
u = bdd.apply('and', u, count)
u = bdd.apply('and', u, rim)
v = bdd.false
for i, y in enumerate(yi):
s = "{w} = {i}".format(w=w, i=i)
count = t.add_expr(s)
trans = _moore_trans(y, t)
q = bdd.apply('and', count, trans)
v = bdd.apply('or', v, q)
u = bdd.apply('and', u, v)
rho_4 = bdd.apply('or', rho_4, u)
# update
basin = z
# \rho
u = bdd.apply('or', rho_1, rho_2)
u = bdd.apply('or', rho_3, u)
u = bdd.apply('or', rho_4, u)
u = bdd.apply('and', sys_action, u)
# counter limits
u = bdd.apply('and', t.action['sys'][0], u)
# record whether env lost
# env_blocked = - bdd.quantify(env_action, t.upvars, forall=False)
# s = "{env_blocked} -> {won}".format(
# env_blocked=env_blocked, won=won)
# action = t.add_expr(s)
# action = bdd.apply('and', action, u)
action = u
t.action['sys'] = [action]
# initial condition
s = '({c} = 0) & ({w} = {none})'.format(
c=c, w=w, none=n_holds)
count = t.add_expr(s)
win_set_ = zk[-1]
u = bdd.apply('and', sys_init, t.init['sys'][0])
u = bdd.apply('and', count, u)
v = bdd.apply('->', env_init, win_set_)
u = bdd.apply('and', u, v)
init = u
assert init != t.bdd.false, 'no init in winning set'
t.init['sys'] = [init]
# \exists \forall init semantics
real = bdd.quantify(init, t.uvars, forall=True)
real = bdd.quantify(real, t.evars, forall=False)
assert real == t.bdd.true, 'cannot init in winning set'
return t
def _moore_trans(target, aut):
"""Return controllable transitions for progress."""
bdd = aut.bdd
env_action = aut.action['env'][0]
sys_action = aut.action['sys'][0]
uvars = aut.upvars
u = bdd.rename(target, aut.prime)
u = bdd.apply('->', env_action, u)
u = bdd.apply('and', sys_action, u)
u = bdd.quantify(u, uvars, forall=True)
return u
def trivial_winning_set(aut_streett, bdd=None):
"""Return set of trivially winning nodes for Streett(1).
@param bdd: use this BDD manager
@type bdd: `BDD` in `dd.cudd` or `dd.bdd` or `dd.autoref`
@return: `(trivial, aut_streett)` where:
- `trivial`: node in `aut_streett.bdd`
- `aut_streett`: `symbolic.Automaton`
"""
aut_rabin = symbolic.Automaton()
for var, d in aut_streett.vars.iteritems():
d = d.copy()
owner = d['owner']
owner = 'env' if owner == 'sys' else 'sys'
d['owner'] = owner
aut_rabin.vars[var] = d
aut_rabin.action['env'] = aut_streett.action['sys']
aut_rabin.action['sys'] = aut_streett.action['env']
win = ['!({w})'.format(w=w) for w in aut_streett.win['<>[]']]
aut_rabin.win['[]<>'] = win
symbolic.fill_blanks(aut_rabin, rabin=True)
aut_streett = aut_streett.build(bdd=bdd)
aut_rabin = aut_rabin.build(bdd=bdd)
# solve
win_set_streett, _, _ = solve_streett_game(aut_streett)
zk, _, _ = solve_rabin_game(aut_rabin)
win_set_rabin = zk[-1]
# find trivial win set
win_set_rabin_ = _copy_bdd(win_set_rabin,
aut_rabin.bdd, aut_streett.bdd)
trivial = aut_streett.bdd.apply('diff', win_set_streett, win_set_rabin_)
return trivial, aut_streett
def _map_nested_lists(f, x, *arg, **kw):
"""Recursively map lists, with non-lists at the bottom.
Useful for applying `dd.bdd.copy_bdd` to several lists.
"""
if isinstance(x, list):
return [_map_nested_lists(f, y, *arg, **kw) for y in x]
else:
return f(x, *arg, **kw)
def _copy_bdd(u, a, b):
"""Copy bdd `u` from manager `a` to `b`.
No effect if `a is b`.
"""
if a is b:
return u
return a.copy(u, b)