Download Week 11: Unification, Generalized Modus

Let us think of interpretations for FOPC that are more
like interpretations for prop logic
Herbrand Interpretations
•
Herbrand Universe
– All constants
• Rao,Pat
– All “ground” functional terms
• Son-of(Rao);Son-of(Pat);
• Son-of(Son-of(…(Rao)))….
•
x, y Friend ( x, y )  Likes( x, y )
Friend ( Rao , Pat )
Friend ( Pat , son  of ( Rao ))
Herbrand Base
– All ground atomic sentences made with
terms in Herbrand universe
• Friend(Rao,Pat);Friend(Pat,Rao);Friend(P
at,Pat);Friend(Rao,Rao)
• Friend(Rao,Son-of(Rao));
• Friend(son-of(son-of(Rao),son-of(sonof(son-of(Pat))
– We can think of elements of HB as
propositions; interpretations give T/F
values to these. Given the interpretation,
we can compute the value of the FOPC
database sentences
If there are n constants; and
p k-ary predicates, then
--Size of HU = n
--Size of HB = p*nk
But if there is even one function,
then |HU| is infinity and so is |HB|.
--So, when there are no function
symbols, FOPC is really just
syntactic sugaring for a (possibly
much larger) propositional database
But what about Godel?
• In First Order Logic
– We have finite set of constants
– Quantification allowed only over variables…
• Godel’s incompleteness theorem holds only in a system that includes
“mathematical induction”—which is an axiom schema that requires
infinitely many FOPC statements
– If a property P is true for 0, and whenever it is true for number n, it is also
true for number n+1, then the property P is true for all natural numbers
– You can’t write this in first order logic without writing it once for each P
(so, you will have to write infinite number of FOPC statements)
• So, a finite FOPC database is still semi-decidable in that we can prove
all provably true theorems
Proof-theoretic
Inference in first order logic
•
For “ground” sentences (i.e., sentences without any quantification), all the old
rules work directly (think of ground atomic sentences as propositions)
– P(a,b)=> Q(a); P(a,b) |= Q(a)
– ~P(a,b) V Q(a) resolved with P(a,b) gives Q(a)
•
What about quantified sentences?
– May be infer ground sentences from them….
– Universal Instantiation (a universally quantified statement entails every
instantiation of it)
xyP( x, y )  Q( x)entailsP (a, b)  Q(a)
– Existential instantiation (an existentially quantified statement holds for some term
(not currently appearing in the KB).
•
xP( x) entails P( SK1) sk1 is a new const.
Can we combine these (so we can avoid unnecessary instantiations?) Yes.
Generalized modus ponens
xyP( x, y )  Q( x); P(a, b) entails q(b)
•
Needs UNIFICATION
UI can be applied several
times to add new sentences
--The resulting KB is
equivalent to the old one
EI can only applied once
--The resulting DB is
not equivalent to the
old one
BUT will be satisfiable
only when the old one is
Want mgu (maximal general unifiers)
How about knows(x,f(x)) knows(u,u)?
x/u; u/f(u)leads to infinite regress (“occurs check”)
GMP can be used in the “forward” (aka “bottom-up”) fashion
where we start from antecedents, and assert the consequent
or in the “backward” (aka “top-down”) fashion where we start
from consequent, and subgoal on proving the antecedents.
Apt-pet
• An apartment pet is a pet
that is small
• Dog is a pet
• Cat is a pet
• Elephant is a pet
• Dogs, cats and skunks are
small.
• Fido is a dog
• Louie is a skunk
• Garfield is a cat
• Clyde is an elephant
• Is there an apartment pet?
1.small ( x)  pet ( x)  aptPet( x)
2.dog ( x)  pet ( x)
3.cat ( x)  pet ( x)
4.elephant ( x)  pet ( x)
5.skunk ( x)  small ( x)
6.cat ( x)  small ( x)
7.dog ( x)  small ( x)
8.dog ( fido)
9.skunk (louie )
10.cat ( garfield )
11.elephant (clyde )
? aptPet( x)
1.small ( x)  pet ( x)  aptPet( x)
2.dog ( x)  pet ( x)
3.cat ( x)  pet ( x)
4.elephant ( x)  pet ( x)
5.skunk ( x)  small ( x)
6.cat ( x)  small ( x)
7.dog ( x)  small ( x)
8.dog ( fido)
9.skunk (louie )
10.cat ( garfield )
11.elephant (clyde )
? aptPet( x)
Efficiency can be improved by re-ordering subgoals adaptively
e.g., try to prove Pet before Small in Lilliput Island; and
Small before Pet in pet-store.
Similar to “Integer Programming” or “Constraint Programming”
Generate compilable
matchers for each
pattern, and use them
Example of FOPC Resolution..
Everyone is loved by someone
xy loves( y, x)  loves( SK ( x' ), x' )
If x loves y, x will give a valentine card to y
xy loves( y, x)  GV ( y, x)  loves( y, x)  GV ( y, x)
Will anyone give Rao a valentine card?
zGV ( z, Rao )  zGV ( z, Rao )  zGV ( z, Rao )  GV ( z, Rao )
y/z;x/Rao
~loves(z,Rao)
z/SK(rao);x’/rao
Finding where you left your key..
Atkey(Home) V Atkey(Office) 1
Where is the key?
Ex Atkey(x)
Negate
Forall x ~Atkey(x)
CNF ~Atkey(x) 2
Resolve 2 and 1 with x/home
You get Atkey(office) 3
Resolve 3 and 2 with x/office
You get empty clause
So resolution refutation “found”
that there does exist a place
where the key is…
Where is it?
what is x bound to?
x is bound to office once
and home once.
so x is either home or office
Existential proofs..
• Are there irrational numbers p and q such
that pq is rational?
2
2
2
pq 2
2 

 2 

2
p 2 q 2
This and the previous examples
show that resolution refutation is
powerful enough to model
existential proofs.
In contrast, generalized modus
ponens is only able to model
constructive proofs..
Existential proofs..
• The previous example shows that resolution
refutation is powerful enough to model existential
proofs. In contrast, generalized modus ponens is
only able to model constructive proofs..
• (We also discussed a cute example of existential
proof—is it possible for an irrational number
power another irrational number to be a rational
number—we proved it is possible, without
actually giving an example).
GMP vs. Resolution Refutation
• While resolution refutation is a complete inference for
FOPC, it is computationally semi-decidable, which is a far
cry from polynomial property of GMP inferences.
• So, most common uses of FOPC involve doing GMP-style
reasoning rather than the full theorem-proving..
• There is a controversy in the community as to whether the
right way to handle the computational complexity is to
– a. Develop “tractable subclasses” of languages and require the
expert to write all their knowlede in the procrustean beds of those
sub-classes (so we can claim “complete and tractable inference”
for that class) OR
– Let users write their knowledge in the fully expressive FOPC, but
just do incomplete (but sound) inference.
– See Doyle & Patil’s “Two Theses of Knowledge Representation”