First-order Logic - PowerPoint Presentation

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First-order LogicSlide2

Assertions;

t/f

Epistemological

commitment

Ontological

commitment

t/f/u

Deg

belief

facts

Facts

Objects

relations

Prop

logic

Probproplogic

FOPC

ProbFOPCSlide3

Atomic

PropositionalRelationalFirst 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 themFirst-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 functionsWith functions, we cannot compile relational representations into any finite propositional representation

“higher-order” representations

can (sometimes) be compiled to lower order

Expressiveness of RepresentationsSlide4

Why FOPC

If your thesis is utter vacuous

Use first-order predicate calculus.

With sufficient formality

The sheerest banalityWill be hailed by the critics: "Miraculous!"Slide5
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Connection to propositional logic:

Think of “atomic sentences” as propositions…

general object referent

Can’t have predicates of predicates..

thus first-orderSlide7
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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 mattersSlide11

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.”

English is Expressive but Ambiguous.

Intuitively,

x

depends on y

as it is in the scope of the quantification on y (foreshadowing Skolemization)Slide12

Caveat: Order of quantifiers matters

“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

Intuitively,

x

depends on y

as it is in the scope of the quantification on

y (foreshadowing Skolemization)Slide13

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 saySlide14

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 smartSlide15

Apt-pet

An apartment pet is a pet that is small

Dog is a pet

Cat is a petElephant is a petDogs and cats are small. Some dogs are cute

Each dog hates some catFido is a dogSlide16

Notes on encoding English statements to FOPC

You get to decide what your predicates, functions, constants etc. are. All you are required to do is 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 KBs is used in isolation, there is no problem. However, if the knowledge written in one KB is supposed to be used in

conjunction

with that in another KB, you will need

“Mapping axioms” which relate the “vocabulary” in one KB to the vocabulary in the other KB. This problem is PRETTY important in the context of Semantic Web

The “Semantic Web” ConnectionSlide17
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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 too—The real

World assertion

corresponding to a

propositionSlide20

Connection to propositional logic:

Think of “atomic sentences” as propositions…Slide21

Herbrand Interpretations

Herbrand Universe

All constants

Rao,Pat

All “ground” functional terms Son-of(Rao);Son-of(Pat);Son-of(Son-of(…(Rao)))….

Herbrand BaseAll ground atomic sentences made with terms in Herbrand universeFriend(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*nkBut 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 logicSlide22
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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 statementsIf 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 numbersYou 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 theoremsSlide24

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 UNIFICATIONSlide25

UI can be applied several

times to add new sentences

--The resulting KB is

equivalent

to the old oneEI can only applied once --The resulting DB is

not equivalent to the old one BUT will be satisfiable only when the old one isSlide26
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Want

mgu

(maximal general unifiers)Slide28

How about knows(x,f(x)) knows(u,u)?

x/u; u/f(u)

leads to infinite regress (“occurs check”)Slide29

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.Slide30

Apt-pet

An apartment pet is a pet that is small

Dog is a pet

Cat is a pet

Elephant is a petDogs, cats and skunks are small. Fido is a dogLouie is a skunk

Garfield is a catClyde is an elephantIs there an apartment pet?Slide31
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Your Project 4!Slide33

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. Slide34

Forward (bottom-up)

vs. Backward (top-down) chaining

Forward chaining fires rules starting from facts

Using P, derive Q

Using Q & R, derive S Using S, derive Z Using Z, Q, derive WUsing Q, derive J

No more inferences. Check if J holds. It does. So proved

Backward chaining starts from the theorem to be provedWe want to prove J. Using Q=>J, we can subgoal on Q

Using P=>Q, we can subgoal on PP holds. We are done.

Suppose we have P => Q Q & R =>S S => Z Z & Q => W Q => J

P RWe 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…Slide35

Datalog and Deductive Databases

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.

Base facts

P(a,b),Q(b)R(c)..

RulesP(x,y),Q(y)=>R(y)

?R(z)

RDBMS

R(c); R(b)..Connection to Progression becoming goal directed w.r.t.

P.G. reachability heuristics Slide36

Similar to “Integer Programming” or “Constraint Programming”Slide37

Generate compilable

matchers for each

pattern, and use themSlide38
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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?

y/z;x/Rao

~loves(z,Rao)

z/SK(rao);x’/raoSlide43

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) 3Resolve 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 officeSlide44

Existential proofs..

Are there irrational numbers p and q such that p

q

is rational?

Rational

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..Slide45

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). Slide46

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” Slide47
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