Advanced Artificial Intelligence Lecture 2: Logic

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Advanced Artificial Intelligence Lecture 2: Logic

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Title: Advanced Artificial Intelligence Lecture 2: Logic

1
Advanced Artificial IntelligenceLecture 2 Logic

Bob McKay
School of Computer Science and Engineering
College of Engineering
Seoul National University

2
Outline

Propositional Logic
Propositional Deduction
Predicate Logic
Predicae Deduction

3
References

Russell, S Norvig, P Artificial Intelligence
A Modern Approach, Prentice Hall
The library has edition 1, call number 006.3
R917a
To buy, edition 2 (1995), ISBN 0137903952
Either is fine
Nilsson, NJ Artificial Intelligence A New
Synthesis, Morgan Kaufmann, 1998, ISBN 1 55860
535 5
Library call no 006.3 N599a
More detail than youll ever need
Leitsch, A The Resolution Calculus, Springer
1997, ISBN 3 540 61882 1
Library call number 511.3 L537r

4
Logic in general

Logics are formal languages for representing
information such that conclusions can be drawn
Syntax defines the sentences in the language
Semantics define the "meaning" of sentences
i.e., define truth of a sentence in a world
E.g., the language of arithmetic
x2 y is a sentence x2y gt is not a
sentence
x2 y is true iff the number x2 is no less
than the number y
x2 y is true in a world where x 7, y 1
x2 y is false in a world where x 0, y 6

5
Entailment

Entailment means that one thing follows from
another
KB a
Knowledge base KB entails sentence a if and only
if a is true in all worlds where KB is true
E.g., the KB containing the Giants won and the
Reds won entails Either the Giants won or the
Reds won
E.g., xy 4 entails 4 xy
Entailment is a relationship between sentences
(i.e., syntax) that is based on semantics
(meaning)

6
Models

Logicians think a lot about models
Mathematical worlds in which we can check truth
We say m is a model of a sentence a if a is true
in m
M(a) is the set of all models of a
All the ways a can be true
So KB a means M(KB) ? M(a)
Ithat is, a is at least as true as KB is
E.g.
KB Giants won and Reds won
a Giants won

7
Entailment and Model Checking

Remember our definition
Knowledge base KB entails sentence a if and only
if a is true in all worlds where KB is true
One way to check this is to actually construct
all the possible words where KB is true, and
check whether a is also true in all of them

8
Whats Wrong with Model Checking?

Model checking is expensive
If there are N boolean variables, then there are
2N different models
Model checking cost is
Exponential in time
Linear in space
Can we do better?
Actually, well see later that the general answer
is probably not
But model checking is exponential for all
problems
Whether difficult or not
For many problems, we can do much better

9
Inference

Inference is the process of checking correctness
by procedures that preserve truth
KB i a means
sentence a can be derived from KB by procedure i
Soundness i is sound means
whenever KB i a, it is also true that KB a
Completeness i is complete means
whenever KB a, it is also true that KB i a

10
Inference the cheat sheet

We will define a logic expressive enough to say
many things of interest
Known as first-order logic
For which there is an inference method which is
Sound
Complete
That is, the method will answer any question
whose answer follows from what is known by the KB
Sounds too good to be true?
The fly in the soup
The method might go on forever
That is, it cant tell us for sure if the
question has a false answer
In fact, its possible to prove that no such
method could exist
But first, we define a weaker logic
Propositional Logic

11
Propositional logic Syntax

Propositional logic is the simplest practical
logic
The proposition symbols P1, P2 etc are sentences
If S is a sentence, (?S) is also (negation)
If S1 and S2 are sentences, (S1 ? S)) is also
(conjunction)
If S1 and S2 are sentences, (S1 ? S2) is also
(disjunction)
If S1 and S2 are sentences, S1 ? S2) is also
(implication)
If S1 and S2 are sentences, S1 ? S2) is also
(biconditional)
Since it often doesnt matter, in many cases we
will leave out the brackets
In digital logic, you might have seen for ?, .
for ?, for ?

12
Propositional logic Semantics

A model specifies true/false for each proposition
symbol
E.g. P1,2 P2,2 P3,1
false true false
With three symbols, 8 possible models, can be
enumerated automatically.
Rules for evaluating truth with respect to a
model m
?S is true iff S is false
S1 ? S2 is true iff S1 is true and S2 is
true
S1 ? S2 is true iff S1is true or S2 is true
S1 ? S2 is true iff S1 is false or S2 is true
i.e., is false iff S1 is true and S2 is
false
S1 ? S2 is true iff S1?S2 is true andS2?S1 is
true
A simple recursive process evaluates any
sentence, e.g.
?P1,2 ? (P2,2 ? P3,1) true ? (true ? false)
true ? true true

13
Truth tables for connectives
14
Truth tables for inference
15
Logical equivalence

Two sentences are logically equivalen means
They are true in the same models
a ß means
a ß and ß a

Some well-known equivalences
16
Validity

A sentence is valid means
it is true in all models,
True, A ??A, A ? A, (A ? (A ? B)) ? B
Validity is connected with inference in a very
useful way
KB a if and only if (KB ? a) is valid
The Deduction Theorem

17
Satisfiability

A sentence is satisfiable means
it is true in some model
A? B, C
A sentence is unsatisfiable means
it is true in no models
A??A
Satisfiability is also connected to inference
KB a if and only if (KB ??a) is unsatisfiable

18
Inference Rules

Inference rules allow us to deduce new true
sentences from previous ones. There are many
possible sets, the set below are taken under fair
dealing use from http//en.wikipedia.org/wiki/Prop
ositional_logic
Double negative elimination
From the sentence f, we may infer f
Conjunction introduction
From any sentences f and ?, we may infer ( f ? ?
).
Conjunction elimination
From any sentence ( f ? ? ), we may infer f and ?
Disjunction introduction
From any sentence f, we may infer (f ? ?) and (?
? f), where ? is any sentence.

19
Inference Rules

Disjunction elimination
From sentences ( f ? ? ), ( f ? ? ), and ( ? ? ?
), we may infer ?.
Biconditional introduction
From sentences ( f ? ? ) and ( ? ? f ), we may
infer ( f ? ? ).
Biconditional elimination
From the sentence ( f ? ? ), we may infer ( f ? ?
) and ( ? ? f ).
Modus ponens
From sentences f and ( f ? ? ), we may infer ?.
Conditional proof
If ? can be derived by assuming f, we can infer (
f ? ? ).
Reductio ad absurdum
If we can derive both ? and ? by assuming f, we
may infer f.

20
Example Proof

Prove (A ? (A ? B)) ? B
1) Assume (A ? (A ? B))
2) Deduce A
Conjunction Elimination from 1)
3) Deduce A ? B
Conjunction Elimination from 1)
4) Deduce B
Modus Ponens from 2) and 3)
5) Deduce (A ? (A ? B)) ? B
Conditional Proof from 1) and 4)

21
Properties of our Inference Rules

Good
Sound
We cant derive invalid sentences
Complete
We will always be able to derive valid sentences
Readily used by people
People can readily use these rules to derive
proofs
People are good at choosing the right rules to
use
Not so good
Not so effective for automated use
Its not easy to get machines to choose the right
rules to make a proof

22
Proof methods

Proof methods divide (roughly) into two kinds
Application of inference rules
Legitimate (sound) generation of new sentences
from old
Like our method from last lecture
Proof a sequence of inference rule applications
We could use inference rules as steps in a
standard search algorithm
Effective computer searches usually transform
sentences into a normal form which is simpler for
automated searching
Model checking
truth table enumeration (always exponential in n)
improved backtracking, e.g., Davis--Putnam-Logeman
n-Loveland (DPLL)
heuristic search in model space (sound but
incomplete)
e.g., min-conflicts-like hill-climbing
algorithms

23
Proof methods for automated search

Our proof method from last lecture isnt very
good for automated search
In essence, guessing the right combination of
rules to use is a hard problem (even if humans do
it well)
Its even harder for the predicate calculus (next
lecture)
Well look at an alternative, resolution, which
works well for automated search in both the
propositional and predicate calculus
It requires sentences to be converted to a normal
form
Conjunctive Normal Form (CNF)

24
Conjunctive Normal Form (CNF)

An atom means one of the variables of our
language
(A, B, C,.)
A literal means either an atom or its negation
(A, B, C,.)
A clause means a disjunction (or) of literals
(A ? B ? C,.)
A CNF sentence means a conjunction (and) of
clauses
(A ? ?B) ? (B ? ?C ? ?D)

25
Converting to CNF

Every propositional sentence can be
(systematically) converted to CNF
First, replace all equivalences using
biconditional elimination
(A ? B) (A ? B) ? (B ? A)
Next, replace all implications using implication
elimination
(A ? B) (?A ? B)
Next, move all negations inwards towards atoms
using de Morgans laws
?(A ? B) (?A ? ?B)
?(A ? B) (?A ? ?B)
And remove all multiple negations
? ?A A
Now, always move conjunctions to the right using
commutativity
(A ? B) (B ? A)
But at the same time, move disjunctions inside
conjunctions when possible, using distributivity
(A ? (B ? C)) (A ? B) ? (A ? C)

26
Converting to CNF

The algorithm for converting to CNF finishes in a
finite amount of time
If you wish, you can calculate the maximum time
it will take to convert a formula of n symbols
The sentence it generates is logically equivalent
to the original sentence
Ie it has the same models
In particular
If one of the sentences is valid, so is the other
If one of the sentences is unsatisfiable, so is
the other

27
Representing in Clausal Form

A sentence in CNF consists of a conjunction of
disjunctions of literals
Remember that we dont care about the order of
conjunctions or disjunctions, or about
repetitions
So we can just represent CNF as a set of sets
A, ?B,B, ?C, ?D
Instead of
(A ? ?B) ? (B ? ?C ? ?D)
We just have to remember that the outer set means
conjunction, the inner one disjunction
This is known as clausal form
Its easier to explain the resolution method in
clausal form

28
The Resolution Method

Resolution is the basic process used in the
resolution method
There are a number of different (equivalent) ways
of explaining it
Well use a different (slightly more general) one
from the book
Suppose we want to prove A is valid
One way of doing this is to prove that ?A is
unsatisfiable
So we start by converting ?A into clausal form
Lets take last lectures
(A ? (A ? B)) ? B
As an example

29
Clausal Form Conversion

?((A ? (A ? B)) ? B)
?(?(A ? (? A ? B)) ? B)
(? ?(A ? (? A ? B)) ? ? B)
(A ? (? A ? B)) ? ? B
A, ?A, B, ?B

30
The Resolution Method

The resolution method uses an operator called
resolution
Resolution takes two clauses which have
complementary literals
The same literal with opposite signs
X1, X2, Xn, Y and Z1, Z2, Zm, ?Y
And creates their resolvent by joining them
together, deleting the complementary literals
X1, X2, Xn, Z1, Z2, Zm
You can add the resolvent to the original clause
set without changing its validity

31
The Resolution Method (Cont)

Sometimes, the resolvent is bigger than the
original clauses
But sometimes, its also smaller
Notice that as clauses get smaller, they are more
difficult to satisfy
A is harder to satisfy than (A ? B ? C)
In the limit, it is consistent to regard the
empty clause as unsatisfiable ( false)
(this requires proof, but its not hard)
So a clause set which contains the empty clause
is a conjunction of clauses, one of which is
unsatisfiable
So the clause set is also unsatisfiable
The resolution method uses repeated resolutions
to try to derive the empty set

32
Resolution Example

Take our example
?((A ? (A ? B)) ? B)
Which we converted to
A, ?A, B, ?B
We can resolve the first two clauses
A resolved with ?A, B gives B
Adding this to our clause set gives
A, ?A, B, ?B, B
And resolving the last two clauses
?B and B gives

33
Resolution Example (continued)

So our new clause set is
A, ?A, B, ?B, B
Since is unsatisfiable, so is the whole clause
set
So ?((A ? (A ? B)) ? B) is unsatisfiable
Which means ((A ? (A ? B)) ? B) is valid
Notice that there were fewer choices to make than
in our previous proof
Theres only one rule to choose (resolution)
Our only choice is which (of possibly many)
resolutions to choose

34
Properties of the Resolution Method

For the Propositional calculus, the resolution
method is
Sound
It will never give incorrect answers
Complete
For any unsatisfiable clause set, the empty
clause can be derived in finite time
In fact, in exponential time
Algorithmic
There is an algorithm to guarantee this
Relatively efficient
Proofs can be quite effective
Highly efficient for some sorts of formulas

35
Horn Clauses

A Horn Clause is a clause which has at most one
positive literal
There are three cases
1) No positive literal (generally not used much)
? X1, ? X2, ? Xn (? X1 ? ? X2 ? ? ? Xn)
2) No negative literal
X (X)
A fact
3) General Case
? X1, ? X2, ? Xn, Y (? X1 ? ? X2 ? ? ?
Xn ? Y)
(X1 ? X2 ? ? Xn) ? Y
A rule

36
Horn Clauses and Resolution

Lets look at the sub-cases
1) and 1) - Cant resolve (no positive literals)
1) and 2) - only one way to do it (not very
useful)
(? X1 ? ? X2 ? ? ? Xn), X1
_________________________
(? X2 ? ? ? Xn)
1) and 3) - again only one choice (not very
useful)
(? X1 ? ? X2 ? ? ? Xn), (Y1 ? Y2 ? ? Yn) ?
X1
_________________________
(? X2 ? ? ? Xn ? ? Y1 ? ? Y2 ? ? ? Yn)

37
Horn Clauses and Resolution

2) and 2) - Cant resolve (no negative literals)
2) and 3) - only one way to do it
(Generalised Modus Ponens)
X1, (X1 ? X2 ? ? Xn) ? Y
_________________________
(X2 ? ? Xn) ? Y
3) and 3) - Now there are two choices, lets take
one
(X1 ? X2 ? ? Xn) ? Y1, (Y1 ? Y2 ? ? Yn) ?Z
_________________________
(X1 ? X2 ? ? Xn ? Y2 ? ? Yn) ?Z
In fact, we only need generalised modus ponens
for Horn clause reasoning
Runs in Linear Time

38
First-order logic

Whereas propositional logic assumes the world
contains facts,
first-order logic (like natural language) assumes
the world contains
Objects people, houses, numbers, colors,
baseball games, wars,
Relations red, round, prime, brother of, bigger
than, part of, comes between,
Functions father of, best friend, one more than,
plus,

39
Syntax of FOL Basic elements

Constants KingJohn, 2, SNU,...
Predicates Brother, sibling, gt,...
(sibling means either brother or sister)
Functions Sqrt, LeftLegOf,...
Variables x, y, a, b,...
Connectives ?, ?, ?, ?, ?
Equality
(can be treated as just a special predicate)
Quantifiers ?, ?
Note we follow standard mathematical
terminology, so constants have capitals,
variables are lower case
This is the opposite convention to prolog

40
Atomic sentences

Atomic sentence predicate (term1,...,termn)
or term1 term2
Term function (term1,...,termn)
or constant or variable
E.g.,
Brother(KingJohn,RichardTheLionheart)
gt (Length(LeftLegOf(Richard)), Length(LeftLegOf(Ki
ngJohn)))

41
Complex sentences

Complex sentences are made from atomic sentences
using the same connectives as in the propositional
calculus
?S, S1 ? S2, S1 ? S2, S1 ? S2, S1 ? S2,
E.g.
Sibling(KingJohn,Richard)?
Sibling(Richard,KingJohn)
gt(1,2) ? (1,2)
gt(1,2) ? ? gt(1,2)
(just as in propositional calculus, they dont
have to be true!)

42
Models in first-order logic

Sentences are true with respect to a model and an
interpretation
Model contains objects (domain elements) and
relations among them
Interpretation specifies meanings for
constant symbols ? objects
predicate symbols ? relations
function symbols ? functional relationships
An atomic sentence predicate(term1,...,termn) is
true
iff the objects referred to by term1,...,termn
are in the relation referred to by predicate

43
Universal quantification

?ltvariablesgt ltsentencegt
Says that ltsentencegt is true for all values of
ltvariablesgt
Everyone at SNU is smart
?x (At(x,SNU) ? Smart(x))
Formally, ?x P is true in a model m iff for every
possible value of x as an object in the model, P
is true
Roughly speaking, equivalent to the conjunction
of instantiations of P
(since the model might be infinite, it might be
an infinite conjunction)
At(KingJohn,SNU) ? Smart(KingJohn)
? At(Richard,SNU) ? Smart(Richard)
? At(SNU,SNU) ? Smart(SNU)
? ...

44
Existential quantification

?ltvariablesgt ltsentencegt
(there is a way of making sentence true by
substituting values for variables)
Someone at SNU is smart
?x At(x,SNU) ? Smart(x)
Formally, ?x P is true in a model m iff P is true
with x being some possible object in the model.
Roughly speaking, it can be seen as equivalent to
the disjunction of instantiations of P
At(KingJohn,SNU) ? Smart(KingJohn)
? At(Richard,SNU) ? Smart(Richard)
? At(SNU,SNU) ? Smart(SNU)
? ...

45
Properties of quantifiers

?x ?y is the same as ?y ?x
?x ?y is the same as ?y ?x
?x ?y is not the same as ?y ?x
?x ?t can_fool_at(x,t)
Some people can always be fooled (same people
all the time)
?t ?x can_fool_at(x,t)
Always, there is someone who can be fooled
(maybe different people)
?x ?t can_fool_at(x,t)
Sometimes you can fool everyone
?x ?t can_fool_at(x,t)
Everyone has a time when they can be fooled
??x ?t can_fool_at(x,t)
You cant fall all of the people all of the
time
Quantifier duality each can be expressed using
the other
?x Likes(x,IceCream)?? ??x ?Likes(x,IceCream)
?x Likes(x,Broccoli) ? ??x ?Likes(x,Broccoli)
? x ? t ?can_fool_at(x,t)
Theres a person and a time where the person
cant be fooled

46
Equality

term1 term2 is true under a given
interpretation if and only if term1 and term2
refer to the same object
E.g., definition of Sibling in terms of Parent
?x,y Sibling(x,y) ?
?(x y) ?
?m,f ? (m f) ? Parent(m,x) ? Parent(f,x) ?
Parent(m,y) ? Parent(f,y)

47
Universal instantiation (UI)

Every instantiation of a universally quantified
sentence is entailed by it
?v aSubst(v/g, a)
for any variable v and ground term g
E.g., ?x King(x) ? Greedy(x) ? Evil(x) yields
King(John) ? Greedy(John) ? Evil(John)
King(Richard) ? Greedy(Richard) ? Evil(Richard)
King(Father(John)) ? Greedy(Father(John)) ?
Evil(Father(John))
.
.

48
Existential instantiation (EI)

For any sentence a, variable v, and constant
symbol k that does not appear elsewhere in the
knowledge base
?v a
Subst(v/k, a)
E.g., ?x Crown(x) ? OnHead(x,John) yields
Crown(C1) ? OnHead(C1,John)
provided C1 is a new constant symbol, called a
Skolem constant
(Easiest to think of this in terms of adding a
new individual to the World it can have any
relationships you like)
(There is a slight complication here
what about ? x (xJohn)?
Generates a new Skolem constant C2 which is
not John, but is in every respect identical to
John.
Even worse, you can make the statement that
there is only one thing equal to John, and still
have two things satisfy it!

49
Reduction to propositional inference

Suppose the KB contains just the following
?x King(x) ? Greedy(x) ? Evil(x)
King(John)
Greedy(John)
Brother(Richard,John)
Instantiating the universal sentence in all
possible ways, we have
King(John) ? Greedy(John) ? Evil(John)
King(Richard) ? Greedy(Richard) ? Evil(Richard)
King(John)
Greedy(John)
Brother(Richard,John)
The new KB is propositionalised proposition
symbols are
King(John), Greedy(John), Evil(John),
King(Richard), etc.

50
Reduction

Every first order KB can be propositionalised so
as to preserve entailment
( ground sentences are entailed by new KB iff
entailed by original KB)
A ground sentence is one with no variables
Idea
propositionalise the KB
ask a query
apply resolution
return the result
Problem
with function symbols, there are infinitely many
ground terms,
Father(Father(Father(John)))

51
Reduction contd.

Theorem Herbrand (1930) For first order logic
If a sentence a is entailed by a knowledge base
Then it is entailed by a finite subset of
propositionalised KB
Idea For n 0 to 8 do
create a propositional KB by instantiating
with depth-n terms
see if a is entailed by this KB
Problem works if a is entailed, loops if a is
not entailed
Theorem Turing (1936), Church (1936) Entailment
for FOL is
semi-decidable
Algorithms exist that say yes to every entailed
sentence
But no algorithm exists that also says no to
every non-entailed sentence

52
Problems with propositionalisation

Propositionalisation generates many
irrelevant-seeming sentences.
E.g., from
?x King(x) ? Greedy(x) ? Evil(x)
King(John)
?y Greedy(y)
Brother(Richard,John)
it seems obvious that Evil(John), but
propositionalisation produces lots of facts such
as Greedy(Richard) that seem irrelevant
With p k-ary predicates and n constants, there
are pnk instantiations.

53
Unification

if we can find a substitution ? such that King(x)
and Greedy(x) match King(John) and Greedy(y)
We dont need to generate all these irrelevant
instances
We can generate the inference immediately
? x/John,y/John does this
In general,
Unify(a,ß) ? if a? ß?
p q ?
Knows(John,x) Knows(John,Jane)
Knows(John,x) Knows(y,OJ)
Knows(John,x) Knows(y,Mother(y))
Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of
variables, e.g., Knows(z17,OJ)

54
Unification

if we can find a substitution ? such that King(x)
and Greedy(x) match King(John) and Greedy(y)
We dont need to generate all these irrelevant
instances
We can generate the inference immediately
? x/John,y/John does this
In general,
Unify(a,ß) ? if a? ß?
p q ?
Knows(John,x) Knows(John,Jane) x/Jane
Knows(John,x) Knows(y,OJ)
Knows(John,x) Knows(y,Mother(y))
Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of
variables, e.g., Knows(z17,OJ)

55
Unification

if we can find a substitution ? such that King(x)
and Greedy(x) match King(John) and Greedy(y)
We dont need to generate all these irrelevant
instances
We can generate the inference immediately
? x/John,y/John does this
In general,
Unify(a,ß) ? if a? ß?
p q ?
Knows(John,x) Knows(John,Jane) x/Jane
Knows(John,x) Knows(y,OJ) x/OJ, y/John
Knows(John,x) Knows(y,Mother(y))
Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of
variables, e.g., Knows(z17,OJ)

56
Unification

if we can find a substitution ? such that King(x)
and Greedy(x) match King(John) and Greedy(y)
We dont need to generate all these irrelevant
instances
We can generate the inference immediately
? x/John,y/John does this
In general,
Unify(a,ß) ? if a? ß?
p q ?
Knows(John,x) Knows(John,Jane) x/Jane
Knows(John,x) Knows(y,OJ) x/OJ, y/John
Knows(John,x) Knows(y,Mother(y))
x/Mother(John), y/John
Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of
variables, e.g., Knows(z17,OJ)

57
Unification

if we can find a substitution ? such that King(x)
and Greedy(x) match King(John) and Greedy(y)
We dont need to generate all these irrelevant
instances
We can generate the inference immediately
? x/John,y/John does this
In general,
Unify(a,ß) ? if a? ß?
p q ?
Knows(John,x) Knows(John,Jane) x/Jane
Knows(John,x) Knows(y,OJ) x/OJ, y/John
Knows(John,x) Knows(y,Mother(y))
x/Mother(John), y/John
Knows(John,x) Knows(x,OJ) fail
Standardizing apart eliminates overlap of
variables, e.g., Knows(z17,OJ)

58
Unification

To unify Knows(John,x) and Knows(y,z),
? y/John, x/z or ? y/John, x/John,
z/John
The first unifier is more general than the
second.
A unifier is a most general unifier (MGU) if no
other unifier is more general
Theorem any two MGUs are equivalent in the sense
that we can get one from the other just by
renaming variables
We say the MGU is unique up to renaming of
variables
Note that we dont guarantee an MGU exists
MGU y/John, x/z