def _process_rule(self, a, b):
# right part first
if isinstance(b, Logic):
# a -> b & c --> a -> b ; a -> c
# (?) FIXME this is only correct when b & c != null !
if b.op == '&':
for barg in b.args:
self.process_rule(a, barg)
# a -> b | c --> !b & !c -> !a
# --> a & !b -> c & !b
# --> a & !c -> b & !c
#
# NB: the last two rewrites add 1 term, so the rule *grows* in size.
# NB: without catching terminating conditions this could continue infinitely
elif b.op == '|':
# detect tautology first
if not isinstance(a, Logic): # Atom
# tautology: a -> a|c|...
if a in b.args:
raise TautologyDetected(a,b, 'a -> a|c|...')
self.process_rule(And(*[Not(barg) for barg in b.args]), Not(a))
for bidx in range(len(b.args)):
barg = b.args[bidx]
brest= b.args[:bidx] + b.args[bidx+1:]
self.process_rule(And(a, Not(barg)),
And(b.__class__(*brest), Not(barg)))
else:
raise ValueError('unknown b.op %r' % b.op)
# left part
elif isinstance(a, Logic):
# a & b -> c --> IRREDUCIBLE CASE -- WE STORE IT AS IS
# (this will be the basis of beta-network)
if a.op == '&':
assert not isinstance(b, Logic)
if b in a.args:
raise TautologyDetected(a,b, 'a & b -> a')
self.proved_rules.append((a,b))
# XXX NOTE at present we ignore !c -> !a | !b
elif a.op == '|':
if b in a.args:
raise TautologyDetected(a,b, 'a | b -> a')
for aarg in a.args:
self.process_rule(aarg, b)
else:
raise ValueError('unknown a.op %r' % a.op)
else:
# both `a` and `b` are atoms
na, nb = name_not(a), name_not(b)
self.proved_rules.append((a,b)) # a -> b
self.proved_rules.append((nb,na)) # !b -> !a
def _process_rule(self, a, b):
# right part first
if isinstance(b, Logic):
# a -> b & c --> a -> b ; a -> c
# (?) FIXME this is only correct when b & c != null !
if b.op == '&':
for barg in b.args:
self.process_rule(a, barg)
# a -> b | c --> !b & !c -> !a
# --> a & !b -> c & !b
# --> a & !c -> b & !c
#
# NB: the last two rewrites add 1 term, so the rule *grows* in size.
# NB: without catching terminating conditions this could continue infinitely
elif b.op == '|':
# detect tautology first
if not isinstance(a, Logic): # Atom
# tautology: a -> a|c|...
if a in b.args:
raise TautologyDetected(a, b, 'a -> a|c|...')
self.process_rule(And(*[Not(barg) for barg in b.args]), Not(a))
for bidx in range(len(b.args)):
barg = b.args[bidx]
brest = b.args[:bidx] + b.args[bidx + 1:]
self.process_rule(And(a, Not(barg)),
And(b.__class__(*brest), Not(barg)))
else:
raise ValueError('unknown b.op %r' % b.op)
# left part
elif isinstance(a, Logic):
# a & b -> c --> IRREDUCIBLE CASE -- WE STORE IT AS IS
# (this will be the basis of beta-network)
if a.op == '&':
assert not isinstance(b, Logic)
if b in a.args:
raise TautologyDetected(a, b, 'a & b -> a')
self.proved_rules.append((a, b))
# XXX NOTE at present we ignore !c -> !a | !b
elif a.op == '|':
if b in a.args:
raise TautologyDetected(a, b, 'a | b -> a')
for aarg in a.args:
self.process_rule(aarg, b)
else:
raise ValueError('unknown a.op %r' % a.op)
else:
# both `a` and `b` are atoms
na, nb = name_not(a), name_not(b)
self.proved_rules.append((a, b)) # a -> b
self.proved_rules.append((nb, na)) # !b -> !a
def deduce_alpha_implications(implications):
"""deduce all implications
Description by example
----------------------
given set of logic rules:
a -> b
b -> c
we deduce all possible rules:
a -> b, c
b -> c
implications: [] of (a,b)
return: {} of a -> [b, c, ...]
"""
res = {}
for a, b in implications:
if a == b:
continue # skip a->a cyclic input
I = res.setdefault(a, [])
list_populate(I, b)
# UC: -------------------------
# | |
# v |
# a='rat' -> b='real' ==> (a_='int') -> 'real'
for a_ in res:
ra_ = res[a_]
if a in ra_:
list_populate(ra_, b, skipif=a_)
# UC:
# a='pos' -> b='real' && (already have b='real' -> 'complex')
# ||
# vv
# a='pos' -> 'complex'
if b in res:
ra = res[a]
for b_ in res[b]:
list_populate(ra, b_, skipif=a)
# let's see if the result is consistent
for a, impl in res.iteritems():
na = name_not(a)
if na in impl:
raise ValueError('implications are inconsistent: %s -> %s %s' %
(a, na, impl))
return res
def deduce_alpha_implications(implications):
"""deduce all implications
Description by example
----------------------
given set of logic rules:
a -> b
b -> c
we deduce all possible rules:
a -> b, c
b -> c
implications: [] of (a,b)
return: {} of a -> [b, c, ...]
"""
res = {}
for a,b in implications:
if a == b:
continue # skip a->a cyclic input
I = res.setdefault(a,[])
list_populate(I,b)
# UC: -------------------------
# | |
# v |
# a='rat' -> b='real' ==> (a_='int') -> 'real'
for a_ in res:
ra_ = res[a_]
if a in ra_:
list_populate(ra_, b, skipif=a_)
# UC:
# a='pos' -> b='real' && (already have b='real' -> 'complex')
# ||
# vv
# a='pos' -> 'complex'
if b in res:
ra = res[a]
for b_ in res[b]:
list_populate(ra, b_, skipif=a)
# let's see if the result is consistent
for a, impl in res.iteritems():
na = name_not(a)
if na in impl:
raise ValueError('implications are inconsistent: %s -> %s %s' % (a, na, impl))
return res
def deduce_alpha_implications(implications):
"""deduce all implications
Description by example
----------------------
given set of logic rules:
a -> b
b -> c
we deduce all possible rules:
a -> b, c
b -> c
implications: [] of (a,b)
return: {} of a -> set([b, c, ...])
"""
res = defaultdict(set)
for a, b in implications:
if a == b:
continue # skip a->a cyclic input
res[a].add(b)
# (x >> a) & (a >> b) => x >> b
for fact in res:
implied = res[fact]
if a in implied:
implied.add(b)
# (a >> b) & (b >> x) => a >> x
if b in res:
res[a] |= res[b]
# Clean up tautologies and check consistency
for a, impl in res.iteritems():
impl.discard(a)
na = name_not(a)
if na in impl:
raise ValueError('implications are inconsistent: %s -> %s %s' %
(a, na, impl))
return res
def deduce_alpha_implications(implications):
"""deduce all implications
Description by example
----------------------
given set of logic rules:
a -> b
b -> c
we deduce all possible rules:
a -> b, c
b -> c
implications: [] of (a,b)
return: {} of a -> set([b, c, ...])
"""
res = defaultdict(set)
for a, b in implications:
if a == b:
continue # skip a->a cyclic input
res[a].add(b)
# (x >> a) & (a >> b) => x >> b
for fact in res:
implied = res[fact]
if a in implied:
implied.add(b)
# (a >> b) & (b >> x) => a >> x
if b in res:
res[a] |= res[b]
# Clean up tautologies and check consistency
for a, impl in res.iteritems():
impl.discard(a)
na = name_not(a)
if na in impl:
raise ValueError('implications are inconsistent: %s -> %s %s' % (a, na, impl))
return res
def deduce_all_facts(self, facts, base=None):
"""Deduce all facts from known facts ({} or [] of (k,v))
*********************************************
* This is the workhorse, so keep it *fast*. *
*********************************************
base -- previously known facts (must be: fully deduced set)
attention: base is modified *in place* /optional/
providing `base` could be needed for performance reasons -- we don't
want to spend most of the time just re-deducing base from base
(e.g. #base=50, #facts=2)
"""
# keep frequently used attributes locally, so we'll avoid extra
# attribute access overhead
rels = self.rels
beta_rules = self.beta_rules
if base is not None:
new_facts = base
else:
new_facts = {}
# XXX better name ?
def x_new_facts(keys, v):
for k in keys:
if k in new_facts and new_facts[k] is not None:
assert new_facts[k] == v, \
('inconsistency between facts', new_facts, k, v)
continue
else:
new_facts[k] = v
if type(facts) is dict:
fseq = facts.iteritems()
else:
fseq = facts
while True:
beta_maytrigger = set()
# --- alpha chains ---
#print '**'
for k,v in fseq:
#print '--'
# first, convert name to be not a not-name
if k[:1] == '!':
k = name_not(k)
v = fuzzy_not(v)
#new_fact(k, v)
if k in new_facts:
assert new_facts[k] is None or new_facts[k] == v, \
('inconsistency between facts', new_facts, k, v)
# performance-wise it is important not to fire implied rules
# for already-seen fact -- we already did them all.
continue
else:
new_facts[k] = v
# some known fact -- let's follow its implications
if v is not None:
# lookup routing tables
try:
tt, tf, tbeta, ft, ff, fbeta = rels[k]
except KeyError:
pass
else:
# Now we have routing tables with *all* the needed
# implications for this k. This means we do not have to
# process each implications recursively!
# XXX this ^^^ is true only for alpha chains
# k=T
if v:
x_new_facts(tt, True) # k -> i
x_new_facts(tf, False) # k -> !i
beta_maytrigger.update(tbeta)
# k=F
else:
x_new_facts(ft, True) # !k -> i
x_new_facts(ff, False) # !k -> !i
beta_maytrigger.update(fbeta)
# --- beta chains ---
# if no beta-rules may trigger -- it's an end-of-story
if not beta_maytrigger:
break
#print '(β) MayTrigger: %s' % beta_maytrigger
fseq = []
# XXX this is dumb proof-of-concept trigger -- we'll need to optimize it
# let's see which beta-rules to trigger
for bidx in beta_maytrigger:
bcond,bimpl = beta_rules[bidx]
# let's see whether bcond is satisfied
for bk in bcond.args:
try:
if bk[:1] == '!':
bv = fuzzy_not(new_facts[bk[1:]])
else:
bv = new_facts[bk]
except KeyError:
break # fact not found -- bcond not satisfied
# one of bcond's condition does not hold
if not bv:
break
else:
# all of bcond's condition hold -- let's fire this beta rule
#print '(β) Trigger #%i (%s)' % (bidx, bimpl)
if bimpl[:1] == '!':
bimpl = bimpl[1:]
v = False
else:
v = True
fseq.append( (bimpl,v) )
return new_facts
def __init__(self, rules):
"""Compile rules into internal lookup tables"""
if isinstance(rules, basestring):
rules = rules.splitlines()
# --- parse and process rules ---
P = Prover()
for rule in rules:
# XXX `a` is hardcoded to be always atom
a, op, b = rule.split(None, 2)
a = Logic.fromstring(a)
b = Logic.fromstring(b)
if op == '->':
P.process_rule(a, b)
elif op == '==':
P.process_rule(a, b)
P.process_rule(b, a)
else:
raise ValueError('unknown op %r' % op)
#P.print_proved('RULES')
#P.print_beta('BETA-RULES')
# --- build deduction networks ---
# deduce alpha implications
impl_a = deduce_alpha_implications(P.rules_alpha)
# now:
# - apply beta rules to alpha chains (static extension), and
# - further associate beta rules to alpha chain (for inference at runtime)
impl_ab = apply_beta_to_alpha_route(impl_a, P.rules_beta)
if 0:
print '\n --- ALPHA-CHAINS (I) ---'
for a,b in impl_a.iteritems():
print '%s\t-> α(%2i):%s' % (a,len(b),b)
print ' - - - - - '
print
if 0:
print '\n --- ALPHA-CHAINS (II) ---'
for a,(b,bb) in impl_ab.iteritems():
print '%s\t-> α(%2i):%s β(%s)' % (a,len(b),b, ' '.join(str(x) for x in bb))
print ' - - - - - '
print
# extract defined fact names
self.defined_facts = set()
for k in impl_ab.keys():
if k[:1] == '!':
k = k[1:]
self.defined_facts.add(k)
#print 'defined facts: (%2i) %s' % (len(self.defined_facts), self.defined_facts)
# now split each rule into four logic chains
# (removing betaidxs from impl_ab view) (XXX is this needed?)
impl_ab_ = dict( (k,impl) for k, (impl,betaidxs) in impl_ab.iteritems())
rel_tt, rel_tf, rel_ft, rel_ff = split_rules_tt_tf_ft_ff(impl_ab_)
# XXX merge me with split_rules_tt_tf_ft_ff ?
rel_tbeta = {}
rel_fbeta = {}
for k, (impl,betaidxs) in impl_ab.iteritems():
if k[:1] == '!':
rel_xbeta = rel_fbeta
k = name_not(k)
else:
rel_xbeta = rel_tbeta
rel_xbeta[k] = betaidxs
self.rel_tt = rel_tt
self.rel_tf = rel_tf
self.rel_tbeta = rel_tbeta
self.rel_ff = rel_ff
self.rel_ft = rel_ft
self.rel_fbeta = rel_fbeta
self.beta_rules = P.rules_beta
# build rels (forward chains)
K = set (rel_tt.keys())
K.update(rel_tf.keys())
K.update(rel_ff.keys())
K.update(rel_ft.keys())
rels = {}
empty= ()
for k in K:
tt = rel_tt.get(k,empty)
tf = rel_tf.get(k,empty)
ft = rel_ft.get(k,empty)
ff = rel_ff.get(k,empty)
tbeta = rel_tbeta.get(k,empty)
fbeta = rel_fbeta.get(k,empty)
rels[k] = tt, tf, tbeta, ft, ff, fbeta
self.rels = rels
# build prereq (backward chains)
prereq = {}
for rel in [rel_tt, rel_tf, rel_ff, rel_ft]:
rel_prereq = rules_2prereq(rel)
for k,pitems in rel_prereq.iteritems():
kp = prereq.setdefault(k,[])
for p in pitems:
list_populate(kp, p)
self.prereq = prereq
def deduce_alpha_implications(implications):
"""deduce all implications
Description by example
----------------------
given set of logic rules:
a -> b
b -> c
we deduce all possible rules:
a -> b, c
b -> c
implications: [] of (a,b)
return: {} of a -> [b, c, ...]
"""
res = {} # a -> [] of implications(a)
# NOTE: this could be optimized with e.g. toposort (?), but we need this
# only at FactRules creation time (i.e. only once), so there is no demand
# in further optimising this.
for a,b in implications:
if a==b:
continue # skip a->a cyclic input
I = res.setdefault(a,[])
list_populate(I,b)
# UC: -------------------------
# | |
# v |
# a='rat' -> b='real' ==> (a_='int') -> 'real'
for a_ in res:
ra_ = res[a_]
if a in ra_:
list_populate(ra_, b, skipif=a_)
# UC:
# a='pos' -> b='real' && (already have b='real' -> 'complex')
# ||
# vv
# a='pos' -> 'complex'
if b in res:
ra = res[a]
for b_ in res[b]:
list_populate(ra, b_, skipif=a)
#print 'D:', res
# let's see if the result is consistent
for a, impl in res.iteritems():
na = name_not(a)
if na in impl:
raise ValueError('implications are inconsistent: %s -> %s %s' % (a, na, impl))
return res
def deduce_all_facts(self, facts, base=None):
"""Deduce all facts from known facts ({} or [] of (k,v))
*********************************************
* This is the workhorse, so keep it *fast*. *
*********************************************
base -- previously known facts (must be: fully deduced set)
attention: base is modified *in place* /optional/
providing `base` could be needed for performance reasons -- we don't
want to spend most of the time just re-deducing base from base
(e.g. #base=50, #facts=2)
"""
# keep frequently used attributes locally, so we'll avoid extra
# attribute access overhead
rels = self.rels
beta_rules = self.beta_rules
if base is None:
new_facts = {}
else:
new_facts = base
def x_new_facts(keys, v):
for k in keys:
if k in new_facts and new_facts[k] is not None:
assert new_facts[k] == v, \
('inconsistency between facts', new_facts, k, v)
continue
else:
new_facts[k] = v
if type(facts) is dict:
fseq = facts.iteritems()
else:
fseq = facts
while True:
beta_maytrigger = set()
# --- alpha chains ---
for k, v in fseq:
# first, convert name to be not a not-name
if k[:1] == '!':
k = name_not(k)
v = fuzzy_not(v)
#new_fact(k, v)
if k in new_facts:
assert new_facts[k] is None or new_facts[k] == v, \
('inconsistency between facts', new_facts, k, v)
# performance-wise it is important not to fire implied rules
# for already-seen fact -- we already did them all.
continue
else:
new_facts[k] = v
# some known fact -- let's follow its implications
if v is not None:
# lookup routing tables
try:
tt, tf, tbeta, ft, ff, fbeta = rels[k]
except KeyError:
pass
else:
# Now we have routing tables with *all* the needed
# implications for this k. This means we do not have to
# process each implications recursively!
# XXX this ^^^ is true only for alpha chains
# k=T
if v:
x_new_facts(tt, True) # k -> i
x_new_facts(tf, False) # k -> !i
beta_maytrigger.update(tbeta)
# k=F
else:
x_new_facts(ft, True) # !k -> i
x_new_facts(ff, False) # !k -> !i
beta_maytrigger.update(fbeta)
# --- beta chains ---
# if no beta-rules may trigger -- it's an end-of-story
if not beta_maytrigger:
break
fseq = []
# let's see which beta-rules to trigger
for bidx in beta_maytrigger:
bcond, bimpl = beta_rules[bidx]
# let's see whether bcond is satisfied
for bk in bcond.args:
try:
if bk[:1] == '!':
bv = fuzzy_not(new_facts[bk[1:]])
else:
bv = new_facts[bk]
except KeyError:
break # fact not found -- bcond not satisfied
# one of bcond's condition does not hold
if not bv:
break
else:
# all of bcond's condition hold -- let's fire this beta rule
if bimpl[:1] == '!':
bimpl = bimpl[1:]
v = False
else:
v = True
fseq.append((bimpl, v))
return new_facts
def __init__(self, rules):
"""Compile rules into internal lookup tables"""
if isinstance(rules, basestring):
rules = rules.splitlines()
# --- parse and process rules ---
P = Prover()
for rule in rules:
# XXX `a` is hardcoded to be always atom
a, op, b = rule.split(None, 2)
a = Logic.fromstring(a)
b = Logic.fromstring(b)
if op == '->':
P.process_rule(a, b)
elif op == '==':
P.process_rule(a, b)
P.process_rule(b, a)
else:
raise ValueError('unknown op %r' % op)
# --- build deduction networks ---
# deduce alpha implications
impl_a = deduce_alpha_implications(P.rules_alpha)
# now:
# - apply beta rules to alpha chains (static extension), and
# - further associate beta rules to alpha chain (for inference at runtime)
impl_ab = apply_beta_to_alpha_route(impl_a, P.rules_beta)
# extract defined fact names
self.defined_facts = set()
for k in impl_ab.keys():
if k[:1] == '!':
k = k[1:]
self.defined_facts.add(k)
# now split each rule into four logic chains
# (removing betaidxs from impl_ab view) (XXX is this needed?)
impl_ab_ = dict(
(k, impl) for k, (impl, betaidxs) in impl_ab.iteritems())
rel_tt, rel_tf, rel_ft, rel_ff = split_rules_tt_tf_ft_ff(impl_ab_)
rel_tbeta = {}
rel_fbeta = {}
for k, (impl, betaidxs) in impl_ab.iteritems():
if k[:1] == '!':
rel_xbeta = rel_fbeta
k = name_not(k)
else:
rel_xbeta = rel_tbeta
rel_xbeta[k] = betaidxs
self.rel_tt = rel_tt
self.rel_tf = rel_tf
self.rel_tbeta = rel_tbeta
self.rel_ff = rel_ff
self.rel_ft = rel_ft
self.rel_fbeta = rel_fbeta
self.beta_rules = P.rules_beta
# build rels (forward chains)
K = set(rel_tt.keys())
K.update(rel_tf.keys())
K.update(rel_ff.keys())
K.update(rel_ft.keys())
rels = {}
empty = ()
for k in K:
tt = rel_tt.get(k, empty)
tf = rel_tf.get(k, empty)
ft = rel_ft.get(k, empty)
ff = rel_ff.get(k, empty)
tbeta = rel_tbeta.get(k, empty)
fbeta = rel_fbeta.get(k, empty)
rels[k] = tt, tf, tbeta, ft, ff, fbeta
self.rels = rels
# build prereq (backward chains)
prereq = {}
for rel in [rel_tt, rel_tf, rel_ff, rel_ft]:
rel_prereq = rules_2prereq(rel)
for k, pitems in rel_prereq.iteritems():
kp = prereq.setdefault(k, [])
for p in pitems:
list_populate(kp, p)
self.prereq = prereq
def deduce_alpha_implications(implications):
"""deduce all implications
Description by example
----------------------
given set of logic rules:
a -> b
b -> c
we deduce all possible rules:
a -> b, c
b -> c
implications: [] of (a,b)
return: {} of a -> [b, c, ...]
"""
res = {} # a -> [] of implications(a)
# NOTE: this could be optimized with e.g. toposort (?), but we need this
# only at FactRules creation time (i.e. only once), so there is no demand
# in further optimising this.
for a,b in implications:
if a==b:
continue # skip a->a cyclic input
I = res.setdefault(a,[])
list_populate(I,b)
# UC: -------------------------
# | |
# v |
# a='rat' -> b='real' ==> (a_='int') -> 'real'
for a_ in res:
ra_ = res[a_]
if a in ra_:
list_populate(ra_, b, skipif=a_)
# UC:
# a='pos' -> b='real' && (already have b='real' -> 'complex')
# ||
# vv
# a='pos' -> 'complex'
if b in res:
ra = res[a]
for b_ in res[b]:
list_populate(ra, b_, skipif=a)
#print 'D:', res
# let's see if the result is consistent
for a, impl in res.iteritems():
na = name_not(a)
if na in impl:
raise ValueError('implications are inconsistent: %s -> %s %s' % (a, na, impl))
return res
def __init__(self, rules):
"""Compile rules into internal lookup tables"""
if isinstance(rules, basestring):
rules = rules.splitlines()
# --- parse and process rules ---
P = Prover()
for rule in rules:
# XXX `a` is hardcoded to be always atom
a, op, b = rule.split(None, 2)
a = Logic.fromstring(a)
b = Logic.fromstring(b)
if op == '->':
P.process_rule(a, b)
elif op == '==':
P.process_rule(a, b)
P.process_rule(b, a)
else:
raise ValueError('unknown op %r' % op)
# --- build deduction networks ---
# deduce alpha implications
impl_a = deduce_alpha_implications(P.rules_alpha)
# now:
# - apply beta rules to alpha chains (static extension), and
# - further associate beta rules to alpha chain (for inference at runtime)
impl_ab = apply_beta_to_alpha_route(impl_a, P.rules_beta)
# extract defined fact names
self.defined_facts = set()
for k in impl_ab.keys():
if k[:1] == '!':
k = k[1:]
self.defined_facts.add(k)
# now split each rule into four logic chains
# (removing betaidxs from impl_ab view) (XXX is this needed?)
impl_ab_ = dict( (k,impl) for k, (impl,betaidxs) in impl_ab.iteritems())
rel_tt, rel_tf, rel_ft, rel_ff = split_rules_tt_tf_ft_ff(impl_ab_)
rel_tbeta = {}
rel_fbeta = {}
for k, (impl,betaidxs) in impl_ab.iteritems():
if k[:1] == '!':
rel_xbeta = rel_fbeta
k = name_not(k)
else:
rel_xbeta = rel_tbeta
rel_xbeta[k] = betaidxs
self.rel_tt = rel_tt
self.rel_tf = rel_tf
self.rel_tbeta = rel_tbeta
self.rel_ff = rel_ff
self.rel_ft = rel_ft
self.rel_fbeta = rel_fbeta
self.beta_rules = P.rules_beta
# build rels (forward chains)
K = set (rel_tt.keys())
K.update(rel_tf.keys())
K.update(rel_ff.keys())
K.update(rel_ft.keys())
rels = {}
empty= ()
for k in K:
tt = rel_tt[k]
tf = rel_tf[k]
ft = rel_ft[k]
ff = rel_ff[k]
tbeta = rel_tbeta.get(k,empty)
fbeta = rel_fbeta.get(k,empty)
rels[k] = tt, tf, tbeta, ft, ff, fbeta
self.rels = rels
# build prereq (backward chains)
prereq = defaultdict(set)
for rel in [rel_tt, rel_tf, rel_ff, rel_ft]:
rel_prereq = rules_2prereq(rel)
for k, pitems in rel_prereq.iteritems():
prereq[k] |= pitems
self.prereq = prereq