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## First-order Logic

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Objects

relations

FOPC

Prob

FOPC

Ontological

commitment

Prob

prop

logic

Prop

logic

facts

t/f/u

Deg

belief

Epistemological

commitment

Assertions;

t/f

Expressiveness of Representations

AtomicPropositionalRelationalFirst order- Atomic representations: States as blackboxes..
- Propositional representations: States as made up of state variables
- Relational representations: States made up of objects and relations between them
- First-order: there are functions which “produce” objects.. (so essentially an infinite set of objects

- Propositional can be compiled to atomic (with exponential blow-up)
- Relational can be compiled to propositional (with exponential blo-up) if there are no functions
- With functions, we cannot compile relational representations into any finite propositional representation

“higher-order” representations

can (sometimes) be compiled to lower order

Can’t have predicates of predicates..

thus first-order

Connection to propositional logic:

Think of “atomic sentences” as propositions…

Important facts about quantifiers

- Forall and There-exists are related through negation..
- ~[forall x P(x)] = Exists x ~P(x)
- ~[exists x P(x)] = forall x ~P(x)
- Quantification is allowed only on variables
- can’t quantify on predicates; can’t say
- [Forall P Reflexive(P) forall x,y P(x,y) => P(y,x)—you have to write it once per relation)
- Order of quantifiers matters

Family Values:Falwell vs. Mahabharata

- According to a recent CTC study,

“….90% of the men surveyed said they will marry the same woman..”

“…Jessica Alba.”

Intuitively, xdepends on y

as it is in the scope of

the quantification on y

(foreshadowing Skolemization)

English is Expressive but Ambiguous.

Caveat: Order of quantifiers matters

Loves(x,y) means

x loves y

Intuitively, xdepends on y

as it is in the scope of

the quantification on y

(foreshadowing Skolemization)

“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”

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 too—The real

World assertion

corresponding to a

proposition

Connection to propositional logic:

Think of “atomic sentences” as propositions…

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)))….
- Herbrand Base
- All ground atomic sentences made with terms in Herbrand universe
- Friend(Rao,Pat);Friend(Pat,Rao);Friend(Pat,Pat);Friend(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 = 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)
- 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

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

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?

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”

~loves(z,Rao)

z/SK(rao);x’/rao

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?

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?

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

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”

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