Herbrand Interpretations

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Herbrand Interpretations
Let us think of interpretations for FOPC that are
more like interpretations for prop logic

• Herbrand Universe
• All constants
• Rao,Pat
• All ground functional terms
• Son-of(Rao)Son-of(Pat)
• Son-of(Son-of((Rao))).
• Herbrand Base
• All ground atomic sentences made with terms in
  Herbrand universe
• Friend(Rao,Pat)Friend(Pat,Rao)Friend(Pat,Pat)Fr
  iend(Rao,Rao)
• Friend(Rao,Son-of(Rao))
• Friend(son-of(son-of(Rao),son-of(son-of(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
pnk 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
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But what about Godel?

• In First Order Logic
• We have finite set of constants
• Quantification allowed only over variables
• Godels incompleteness theorem holds only in a
  system that includes mathematical
  inductionwhich 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
  n1, then the property P is true for all natural
  numbers
• You cant 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

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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)gt 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)
• Existential instantiation (an existentially
  quantified statement holds for some term (not
  currently appearing in the KB).
• Can we combine these (so we can avoid unnecessary
  instantiations?) Yes. Generalized modus ponens
• Needs UNIFICATION

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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
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Want mgu (maximal general unifiers)
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How about knows(x,f(x)) knows(u,u)?
x/u u/f(u)?leads to infinite regress (occurs
check)
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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.
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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?

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Your Project 4!
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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.
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Forward (bottom-up) vs. Backward (top-down)
chaining
Suppose we have P gt Q Q R gtS S gt Z
Z Q gt W Q gt J P R We want to prove J
Forward chaining allows parallel derivation of
many facts together but it may derive facts
that are not relevant for the theorem. Backward
chaining concentrates on proving subgoals that
are relevant to the theorem. However, it
proves theorems one at a time.
Some similarity with progression vs. regression

• Forward chaining fires rules starting from facts
• Using P, derive Q
• Using Q R, derive S
• Using S, derive Z
• Using Z, Q, derive W
• Using Q, derive J
• No more inferences. Check if J holds. It does. So
  proved

• Backward chaining starts from the theorem to be
  proved
• We want to prove J.
• Using QgtJ, we can subgoal on Q
• Using PgtQ, we can subgoal on P
• P holds. We are done.

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Datalog and Deductive Databases
Connection to Progression becoming goal directed
w.r.t. P.G. reachability heuristics ?

• A deductive database is a generalization of
  relational database, where in addition to the
  relational store, we also have a set of rules.
• The rules are in definite clause form
  (universally quantified implications, with one
  non-negated head, and a conjunction of
  non-negated tails)
• When a query is asked, the answers are retrieved
  both from the relational store, and by deriving
  new facts using the rules.
• The inference in deductive databases thus
  involves using GMP rule.
• Since deductive databases have to derived all
  answers for a query, top-down evaluation winds
  up being too inefficient.
• So, bottom-up (forward chaining) evaluation is
  used (which tends to derive non-relevant facts ?
• A neat idea called magic-sets allows us to
  temporarily change the rules (given a specific
  query), such that forward chaining on the
  modified rules will avoid deriving some of the
  irrelevant facts.

?R(z)
R(c) R(b)..
Rules P(x,y),Q(y)gtR(y)
Base facts P(a,b),Q(b) R(c)..
RDBMS
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Similar to Integer Programming or Constraint
Programming
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Generate compilable matchers for each
pattern, and use them
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Example of FOPC Resolution..
Everyone is loved by someone If x loves y, x
will give a valentine card to y Will anyone
give Rao a valentine card?
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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
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Existential proofs..

• Are there irrational numbers p and q such that pq
  is rational?

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..
Rational
Irrational
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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
  proofis it possible for an irrational number
  power another irrational number to be a rational
  numberwe proved it is possible, without actually
  giving an example).

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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 Patils Two Theses of Knowledge
  Representation