Plan(s) for make-up Class

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10/29

• Plan(s) for make-up Class
• Extend four classes until 1215pm
• Have a separate make-up class on a Friday morning

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Facts Objects relations
FOPC
Prob FOPC
Ontological commitment
Prob prop logic
Prop logic
facts
t/f/u
Deg belief
Epistemological commitment
Assertions t/f
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Cannot say that happy people smile except by
writing one sentence for each person in your KB
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Why FOPC

• If your thesis is utter vacuous
• Use first-order predicate calculus.
• With sufficient formality
• The sheerest banality
• Will be hailed by the critics "Miraculous!"

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Roughly speaking, the atomic sentences take the
place of proposition symbols Terms
correspond to generalized object referents
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Note that quantification is over objects. This
is what makes it First-order If you can
quantify over predicate symbols, then it will be
Second-order E.g. Cant say A symmetric
predicate is one which holds even if the
arguments are reversed in a single
sentence E.g2. Cant write Mathematical Induction
schema as a single FOPC sentence
(Goedels incompleteness theorem wont hold for
FOPC ? )
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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 and cats are small.
• Some dogs are cute
• Each dog hates some cat
• Fido is a dog

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Caveat Decide whether a symbol is predicate,
constant or function

• Make sure you decide what are your constants,
  what are your predicates and what are your
  functions
• Once you decide something is a predicate, you
  cannot use it in a place where a predicate is not
  expected! In the previous example, you cannot say

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Family ValuesFalwell vs. Mahabharata

• According to a recent CTC study,
• .90 of the men surveyed said they
  will marry the same woman..
• Jessica Alba.

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Caveat Order of quantifiers matters
Loves(x,y) means x loves y
either Fido loves both Fido and Tweety or
Tweety loves both Fido and Tweety
Fido or Tweety loves Fido and Fido or Tweety
loves Tweety
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More on writing sentences
Everyone at ASU is smart Someone at UA is
smart

• Forall usually goes with implications (rarely
  with conjunctive sentences)
• There-exists usually goes with conjunctionsrarely
  with implications

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Notes on encoding English statements to FOPC

• Since you are allowed to make your own predicate
  and function names, it is quite possible that two
  people FOPCizing the same KB may wind up writing
  two syntactically different KBs
• If each of the DBs is used in isolation, there is
  no problem. However, if the knowledge written in
  one DB is supposed to be used in conjunction with
  that in another DB, you will need Mapping
  axioms which relate the vocabulary in one DB
  to the vocabulary in the other DB.
• This problem is PRETTY important in the context
  of Semantic Web

• You get to decide what your predicates,
  functions, constants etc. are. All you are
  required to do it be consistent in their usage.
• When you write an English sentence into FOPC
  sentence, you can double check by asking
  yourself if there are worlds where FOPC sentence
  doesnt hold and the English one holds and vice
  versa

The Semantic Web Connection
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Two different Tarskian Interpretations
This is the same as the one on The left except
we have green guy for Richard
Problem There are too darned many Tarskian
interpretations. Given one, you can change it
by just substituting new real-world objects
? Substitution-equivalent Tarskian
interpretations give same valuations to the
FOPC statements (and thus do not change
entailment) ? Think in terms of equivalent
classes of Tarskian Interpretations
(Herbrand Interpretations)
We had this in prop logic tooThe real World
assertion corresponding to a proposition
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10/31

• ?Midterm returned
• ?Make-up class on Friday 11/9 (morningusual
  class time)

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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?

• 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

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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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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11/7
Why is the set that is the set of all sets
jumping up and down excitedly?
..it couldnt contain itself

• Are there irrational numbers p and q such that pq
  is a rational number?
• Hint Suppose pq

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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?
Anyone who gets a valentines card will not go
on a rampage on 2/14 Prove Rao wont go on
rampage..
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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