1. First-order Logic

2. Assertions; t/f FOPC Prob FOPC Ontological commitment Prob prop logic
Facts
Objects
relations
FOPC
Prob
FOPC
Ontological
commitment
Prob
prop
logic
Prop
logic
facts
t/f/u
Deg
belief
Epistemological
commitment
Assertions;
t/f

3. AtomicPropositionalRelationalFirst order
Expressiveness of Representations
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 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

4. 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!"

6. Connection to propositional logic:
general object referent
Can’t have predicates of predicates..
thus first-order
Connection to propositional logic:
Think of “atomic sentences” as propositions…

10. 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

11. 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, x depends on y
as it is in the scope of
the quantification on y
(foreshadowing Skolemization)
English is Expressive but Ambiguous.

12. Caveat: Order of quantifiers matters
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)
“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”

13. 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

14. 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

15. 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

16. 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 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
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
The “Semantic Web” Connection

19. 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

20. Connection to propositional logic:
Think of “atomic sentences” as propositions…

21. Herbrand Interpretations
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

23. 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

24. 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

25. 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

27. Want mgu (maximal general unifiers)

28. How about knows(x,f(x)) knows(u,u)?
x/u; u/f(u)leads to infinite regress (“occurs check”)

29. 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.

30. 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?

32. Your Project 4!

33. 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.

34. Forward (bottom-up) vs. Backward (top-down) chaining
Suppose we have
P => Q
Q & R =>S
S => Z
Z & Q => W
Q => 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 Q=>J, we can subgoal on Q
Using P=>Q, we can subgoal on P
P holds. We are done.

35. Datalog and Deductive Databases
Connection to Progression becoming goal directed w.r.t.
P.G. reachability heuristics 
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.
?R(z)
R(c); R(b)..
Rules
P(x,y),Q(y)=>R(y)
Base facts
P(a,b),Q(b)
R(c)..
RDBMS

36. Similar to “Integer Programming” or “Constraint Programming”

37. Generate compilable
matchers for each
pattern, and use them

42. 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?
z/SK(rao);x’/rao
y/z;x/Rao
~loves(z,Rao)

43. 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

44. 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

45. 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).

46. 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”