10/29

Plan(s) for make-up Class

1. Extend four classes until 12:15pm

2. Have a separate make-up class on a Friday morning

Prop logic

First order predicate logic(FOPC)

Prob. Prop. logic

Objects,relations

Degree ofbelief

First order Prob. logic

Objects,relations

Degree ofbelief

Degree oftruth

Fuzzy Logic

Time

First order Temporal logic(FOPC)

Assertions;t/f

Epistemological commitment

Ontological commitment

t/f/u Degbelief

facts

FactsObjectsrelations

Proplogic

Probproplogic

FOPC ProbFOPC

Cannot say that “happy people smile” except by writing one sentence for each person in your KB

Roughly speaking, the “atomic sentences” take the place of proposition symbols

Terms correspond to generalized object referents

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. Can’t say “A symmetric predicate is one which holds even if the arguments are reversed” in a single sentenceE.g2. Can’t write Mathematical Induction schema as a single FOPC sentence (Goedel’s incompleteness theorem won’t hold for FOPC )

Why roughly? Because if you have a function symbol, then the conjunction (or disjunction) can be infinitely large

....))((),((

))(,(

))(,(

TomoffatheroffatherTomoffatherloves

TomoffatherTomloves

xoffatherxlovesx

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.”

),( xyMarryyx),( xyMarryxy

Caveat: Order of quantifiers matters

),( yxlovesyx),( yxlovesxy

)],(),([

)],(),([

)],(),([),(

)],(),([

)],(),([

)],(),([),(

TweetyTweetylovesTweetyFidoloves

FidoTweetylovesFidoFidoloves

yTweetylovesyFidolovesyyxlovesxy

TweetyTweetylovesFidoTweetyloves

TweetyFidolovesFidoFidoloves

TweetyxlovesFidoxlovesxyxlovesyx

TweetyandFidowithworldaConsider

“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”

Loves(x,y) means x loves y

More on writing sentences

• Forall usually goes with implications (rarely with conjunctive sentences)

• There-exists usually goes with conjunctions—rarely with implications

Everyone at ASU is smart

Someone at UA is smart

)(),(

)(),(

xSmartASUxxAt

xSmartASUxAtx

)(),(

)(),(

xSmartUAxAtx

xSmartUAxAtx

Will hold if there exists a single person who doesn’t go to UA (in which case At(x,UA) will be false, and so the implication will be true, making the entire disjunction true..

Notes on encoding English statements to FOPC

• 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 doesn’t hold and the English one holds and vice versa

• 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

The “Semantic Web” Connection

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 realWorld assertion corresponding to a proposition

10/31

Midterm returned

Make-up class on Friday 11/9 (morning—usual class time)

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

))(,(

),(

),(),(,

RaoofsonPatFriend

PatRaoFriend

yxLikesyxFriendyx

If there are n constants; andp 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

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

But what about Godel?

• 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

)(),()(),( aQbaPxQyxyPx

)(),();(),( bqbaPxQyxyPx

)1()( SKPxxP

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

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?

)(?

)(.11

)(.10

)(.9

)(.8

)()(.7

)()(.6

)()(.5

)()(.4

)()(.3

)()(.2

)()()(.1

xaptPet

clydeelephant

garfieldcat

louieskunk

fidodog

xsmallxdog

xsmallxcat

xsmallxskunk

xpetxelephant

xpetxcat

xpetxdog

xaptPetxpetxsmall

)(?

)(.11

)(.10

)(.9

)(.8

)()(.7

)()(.6

)()(.5

)()(.4

)()(.3

)()(.2

)()()(.1

xaptPet

clydeelephant

garfieldcat

louieskunk

fidodog

xsmallxdog

xsmallxcat

xsmallxskunk

xpetxelephant

xpetxcat

xpetxdog

xaptPetxpetxsmall

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

If x loves y, x will give a valentine card to y

Will anyone give Rao a valentine card?

)'),'((),( xxSKlovesxylovesyx

),(),(),(),( xyGVxylovesxyGVxylovesyx

),(),(),(),( RaozGVRaozGVzRaozzGVRaozzGV

y/z;x/Rao

~loves(z,Rao)

z/SK(rao);x’/raoAnyone who gets a valentines card will not go on a rampage on 2/14 Prove: Rao won’t go on rampage..

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/homeYou 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?

22

22

2

Ration

al

2qp

222

qp

Irrational

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..