Download Quantifiers

Predicate Logic
From Propositional Logic to
Predicate Logic
• Last week, we dealt with propositional (or
truth-functional) logic: the logic of truthfunctional statements.
• Today, we are going to deal with predicate
(or quantificational) logic.
• Quantificational logic is an extension of,
and thus builds on truth-functional logic.
Recap: Formal Logic
• Step 1: Use certain symbols to express the
abstract form of certain statements
• Step 2: Use a certain procedure based on
these abstract symbolizations to figure out
certain logical properties of the original
statements.
Recap: Truth Tables
• Truth-Tables
–
–
–
–
Slow
Systematic
Reveals consequence as well as non-consequence
Only works for truth-functional logic
Recap: Formal Proofs
• Formal Proofs
–
–
–
–
Pretty fast (with practice!)
Not systematic
Can only reveal consequence
Can be made into systematic method (that can then also
check for non-consequence) but becomes inefficient
– Can be used for predicate logic
Recap: Truth Trees
• Truth Trees
–
–
–
–
Fast
Systematic
Can reveal consequence as well as non-consequence
Can be used for truth-functional as well as predicate
logic
Quantifiers
Individual Constants
• An individual constant is a name for an
object.
• Examples: john, marie, a, b
• Each name is assumed to refer to a unique
individual, i.e. we will not have two objects
with the same name.
• However, each individual object may have
more than one name.
Predicates
• Predicates are used to express properties of
objects or relations between objects.
• Examples: Tall, Cube, LeftOf, =
• Arity: the number of arguments of a
predicate (E.g. Tall: 1, LeftOf: 2)
Interpreted and Uninterpreted
Predicates
• Just as ‘P’ can be used to denote any
statement in propositional logic, a predicate
like ‘LeftOf’ is left ‘uninterpreted’ in
predicate logic. Thus, a statement like
LeftOf(a,a) can be true in predicate logic.
• The predicate ‘=‘ is an exception: it will
automatically be interpreted as the identity
predicate.
Quantification: ‘All’ and ‘Some’
• In quantificational logic, there are two
quantifiers: ‘all’ and ‘some’.
• Here are some examples:
– x Mortal(x) ‘All things are mortal’
– x Mortal(x) ‘Some things are mortal’
– x (Human(x)  Mortal(x)) ‘Every human is
mortal’
– x (Human(x)  Mortal(x)) ‘Some human is
not mortal’
The Four Aristotelian Forms
• “All P’s are Q’s”
– x (P(x)  Q(x))
• “Some P’s are Q’s”
– x (P(x)  Q(x))
• “No P’s are Q’s”
– x (P(x)  Q(x))
• “Some P’s are not Q’s”
– x (P(x)  Q(x))
Swapping Mixed Quantifiers:
Order Matters
x y Likes(x,y)
“Everything likes
something (possibly itself)”
y x Likes(x,y)
“Something is liked by
everything (including itself)”
Expressing Number of Objects
• How do we express that there are (at least) two
cubes?
• Note that x y (Cube(x)  Cube(y)) doesn’t
work: this will be true in a world with 1 object
(just pick that object for both x and y!)
• So, we have to make sure that x and y are different
objects: x y (xy  Cube(x)  Cube(y))
‘Exactly One’
• How can we say that “There is exactly one cube”?
• Saying that there is exactly one cube is saying two
things at once:
– There is at least one cube: xCube(x)
– There is at most one cube: xy(Cube(x)Cube(y)
xy)
– Thus: xCube(x)  xy(Cube(x)Cube(y)xy)
• Alternatively (and simpler):
– x(Cube(x)  y(Cube(y)  xy))
– x(Cube(x)  y(Cube(y)  x=y))
– x y(Cube(y)  x=y))
‘Exactly Two’
• How do we say “There are exactly two
cubes”?
• Similar set-up:
– x y(Cube(x)  Cube(y)  xy  z(Cube(z)  zx
 zy)) or:
– x y(Cube(x)  Cube(y)  xy  z(Cube(z)  (z=x
 z=y))) or:
– x y(xy  z(Cube(z)  (z=x  z=y)))
The Logic of Quantifiers
Quantifier Negation Equivalences
• x P(x)  x P(x)
• x P(x)  x P(x)
• Sometimes these are called the DeMorgan
Rules for Quantifiers, which makes sense:
• x P(x)  P(a)  P(b)  …
• x P(x)  P(a)  P(b)  …
Rewriting Example
If x (P(x)  Q(x)) (‘not all P’s are Q’s), then
x (P(x)  Q(x)) (some P’s are not Q’s), and
vice versa:
x (P(x)  Q(x))  (QN)
x (P(x)  Q(x))  (Impl)
x (P(x)  Q(x))
Other Quantifier Equivalences
•  over , and  over :
– x ((x)  (x))  x (x)  x (x)
– x ((x)  (x))  x (x)  x (x)
• Null Quantification:
– x P  P
– x P  P
• Replacing bound variables:
– x (x)  y (y)
– x (x)  y (y)
• Swapping quantifiers of same type:
– x y (x,y)  y x (x,y)
– x y (x,y)  y x (x,y)
The Assumption of Existential
Import
• The Assumption of Existential Import is the
assumption that the world in which we evaluate is
not empty, i.e. that at least one thing exists.
• Under this assumption, x P(x) is true if x P(x)
is true. Without the assumption, however, it’s not:
if the world in which we evaluate is empty, then
x P(x) is false, even though x P(x) is
(vacuously) true.
• In first-order logic, we usually make the
assumption of existential import. Thus, x P(x) is
considered a FO consequence of x P(x), even
though logically it is not.
Formal Proofs for Quantifiers
Quantifier Rules in F
• There are 4 quantifier rules in F:
– Universal Introduction and Elimination
– Existential Introduction and Elimination
• Universal Introduction and Existential Elimination
have important restrictions in that the rules cannot
be applied relative to just any individual constant.
The system F deals with those restrictions through
the use of subproofs. We’ll see later how that
works.
• Fortunately, Universal Elimination and Existential
Introduction do not have any restrictions, so we’ll
start with those.
Notation
• In describing the rules, the following
notation is useful:
– (x) is a wff with zero or more instances of x as
the only free variable.
– (a/x) is the statement that results when
substituting ‘a’ for all occurrences of ‘x’ that
are free in (x).
– If it is clear which variable we are subsituting,
we will simply write (a).
 Elim
• Universal Elimination ( Elim) allows one
to conclude that any thing has a certain
property if everything has that property:
x (x)

(a)
Good and Bad Uses of  Elim
x SameSize(x,x)
Good

x SameSize(x,x)
Bad
SameSize(a,a)

SameSize(a,b)
 The same individual
constant should be used!
Bad
x SameSize(x,x)

SameSize(x,a)
 All free occurrences of x
should be replaced!
Bad
x (Tet(x)  x Large(x))

Tet(a)  x Large(a)
 Only free occurrences of x
should be replaced!
 Intro
• Existential Introduction ( Intro) allows one
to conclude that something has a certain
property if some thing has that property:
(a)

x (x)
Good and Bad Uses of  Intro
Good
SameSize(a,a)

x SameSize(x,x)
Good
SameSize(a,a)

x SameSize(a,x)
 Not all occurrences of a
have to be replaced!
Bad
SameSize(a,b)

x SameSize(x,x)
 The same individual
constant should be used!
x SameSize(a,x)
Bad

x x SameSize(x,x)
 Doesn’t follow the rule (no
free x’s in x SameSize(x,x))
Universal Proof
• A common proof in mathematics is a universal
proof.
• A universal proof proves something about
everything (of the Universe of Discourse) by
proving it to be true of some arbitrary thing.
• It usually starts with “Let ‘a’ be an arbitrary …”
• It then proves something about ‘a’
• Finally, since ‘a’ was just an arbitrary individual, it
must be true for all individuals.
 Intro
• Universal Introduction ( Elim) allows one
to conclude that everything has a certain
property if anything has that property:
a

(a)
x (x)
a may not occur before the subproof,
unless all subproofs in which it occurs
have been closed. a may not occur in
(x) either.
Good and Bad Uses of  Intro
Good
a

SameSize(a,a)
Bad
x SameSize(x,x)
Still
Good
 a occurs before subproof!
a

Tet(a)
x Tet(x)
 a occurs outside subproof,
but only in a subproof that has
been closed.
Tet(a)
a

SameSize(a,a)
x SameSize(x,x)
Bad
a

SameSize(a,a)
x SameSize(a,x)
 a occurs in SameSize(a,x)!
Existential Proof
• Sometimes, we know that something has a certain
property, but we don’t know who or what this
something is.
• In order to perform some reasoning, we will give
this something a name, and whatever we can infer
from that point on, we can infer from the original
statement.
• Like the universal proof, the name should be an
arbitrary name, but in this case it denotes a
specific individual: that individual that had the
relevant property.
 Elim
• Existential Elimination ( Elim) allows one
to conclude anything that follows from
some thing having a certain property, given
that something has that property.
x (x)
a (a)

Q
Q
a may not occur before the subproof,
unless all subproofs in which it occurs
have been closed. a may not occur in
Q either.
Good and Bad Uses of  Elim
Good
Still
Good
x SameSize(x,x)
a SameSize(a,a)

x Cube(x)
Bad
x Cube(x)
a Cube(a)

Tet(b)
Tet(b)
 a occurs before subproof,
but only in a subproof
which has been closed.
Tet(a)

x SameSize(x,x)
a SameSize(a,a)

x Large(x)
x Large(x)
 a occurs before subproof!
Bad
a SameSize(a,a)

Large(a)
Large(a)
 a occurs in Large(a)!
= Intro
• At any point, you can assert any statement of the
form a=a
• = Intro does not require any statements as part of
its justification, and reflects the reflexivity of
identity.

a=a
= Intro
= Elim
• = Elim: If you have a statement of the form
a=b, and a statement in which a occurs
(written as P(a)), then you may infer P(b),
which is the statement that results when
replacing any number of occurences of a by
b in the statement P(a):
n
P(a)

m a=b

P(b)
= Elim n,m
Rules for other Predicates
• Of course, one could define inference rules for
predicates other than ‘=‘. For example, given the
reflexivity of the SameSize relationship, one could
make it a rule that SameSize(a,a) can be inferred
at any time.
• However, ‘=‘ is the only predicate for which F has
defined inference rules as it is the only interpreted
predicate.
• We’ll see later how we can deal with logical truths
about other predicates.
Truth Trees for
Predicate Logic
Running Examples
Valid Argument
x (Cube(x)  Small(x))
x Cube(x)  x Small(x)
Invalid Argument
x Cube(x)  x Small(x)
x (Cube(x)  Small(x))
Truth-Functional Expansions
• Suppose that our Universe of Discourse (UD)
contains only the objects a and b.
• Given this UD, the claim x Cube(x) is true iff
Cube(a)  Cube(b) is true.
• Similarly, the claim x Cube(x) is true iff Cube(a)
 Cube(b) is true.
• The truth-functional interpretation of the FO
statements given a fixed UD is called the truthfunctional expansion of the original FO statement
with regard to that UD.
Truth-Functional Expansions and
Proving FO Invalidity
• Truth-Functional expansions can be used to
prove FO invalidity. Example:
x Cube(x)  x Small(x)
x (Cube(x)  Small(x))
UD = {a,b}
T
T
F
T
F
T T
(Cube(a)  Cube(b))  (Small(a)  Small(b))
(Cube(a)  Small(a))  (Cube(b)  Small(b))
T
F F
F
F
F T
This shows that there is a world in which the premise is
true and the conclusion false. Hence, the original argument
is FO invalid.
Truth-Functional Expansions and
Proving FO Validity
• If the truth-functional expansion of an FO argument in
some UD is truth-functionally invalid, then the original
argument is FO invalid, but if it is truth-functionally valid,
then that does not mean that the original argument is FO
valid.
• For example, with UD = {a}, the expansion of the
argument would be truth-functionally valid. In general, it is
always possible that adding one more object to the UD
makes the expansion invalid.
• Thus, we can’t prove validity using the expansion method,
as we would have to show the expansion to be valid in
every possible UD, and there are infinitely many UD’s.
• The expansion method is therefore only good for proving
invalidity. Indeed, it searches for countermodels.
The Expansion Method as a
Systematic Procedure
• Still, what is nice about the expansion
method is that it can be made into a
systematic procedure:
– Step 1: Expand FO argument (which can be
done systematically) in UD = {a}.
– Step 2: Use some systematic procedure (e.g.
truth-table method or truth-tree method) to test
whether the expansion is TF invalid. If it is TF
invalid, then stop: the FO argument is FO
invalid. Otherwise, expand FO argument in UD
= {a,b}, and repeat step 2.
Incompleteness of
the Expansion Method
• We saw that the expansion method is not a test for
FO validity, but only for invalidity.
• However, even as such it is an incomplete test!
• Proof: Consider the following argument:
xy(xy  ((x>y  y>x) 
(x>y  y>x)))
xyz((x>y  y>z)  x>z)
xy(xy  x>y)
For any UD with an arbitrarily
large yet finite number of objects,
the expansion of this argument
will be truth-functionally valid.
However, the argument is FO
invalid (consider the natural
numbers)!
A More Focused Search
• A further drawback of the expansion method is
that the search for a counterexample is very
inefficient.
• A focused search for a counterexample is more
efficient:
– (for the invalid argument) I want there to be at least one
cube, and at least one small object, but no small cubes.
So, if we have a cube, a, then a cannot be small, so I
need a second object, b, which is small, but not a cube.
Counterexample, so the argument is invalid.
Advantage of a Focused Search
• The focused search method is like the indirect
truth-table method.
• Indeed, like the indirect truth-table method, the
focused search method can prove validity:
– (for the valid argument) I want there to be at least one
small cube. Let us call this small cube a. How, I don’t
want it to be true that there is at least one cube and at
least one small object. However, a is both a cube and
small. Contradiction, so I can’t generate a
counterexample.
Truth-Trees for Predicate Logic
• Like the direct method, the focused search
method needs to be systematized, especially
since the search often involves making
choices.
• Fortunately, the truth-tree method, which
systematized the indirect truth-table method
in truth-functional logic, can be extended
for predicate logic.
Truth-Tree Rules for Quantifiers
x (x)

x (x)
x (x)
x (x) 
x (x)

(c)
with ‘c’ a
new constant
in that branch
x (x)
(c)
with ‘c’ any
constant
Truth-Tree Rules for Identity
(c)
c = d (or d = c)
(d)
(where (d) is the result
of replacing any number
of c’s with d’s in (c))
aa
×
Truth-Tree Example I
x Cube(x)  x Small(x) 
x (Cube(x)  Small(x)) 
x Cube(x)

x Small(x)

x (Cube(x)  Small(x))
Cube(a)
Small(b)
(Cube(a)  Small(a)) 
(Cube(b)  Small(b)) 
Cube(a)
Small(a)
×
Cube(b)
Small(b)
Open branch,
×
so it’s invalid
Truth-Tree Example II
x (Cube(x)  Small(x))

(x Cube(x)  x Small(x)) 
Cube(a)  Small(a) 
Cube(a)
Small(a)
x Cube(x) 
x Cube(x)
x Small(x) 
x Small(x)
Small(a)
Cube(a)
×
×
All branches close,
so it’s valid
Completeness and Incompleteness
FO
•  is a FO consequence of  = {1, …, n}
iff  is a logical consequence of  in virtue
of truth-functional, quantificational, and
identity properties.
• Let us use the symbol FO to indicate FO
consequence:
–  FO  iff  is a FO consequence of .
FO Provability
• Let us define FO provability with regard to
some formal deductive logic system S (e.g
F) as follows:  is FO provable from a set
of premises {1, …, n} in the system S iff
there exists a formal proof in S with 1, …,
n as premises and  as the conclusion
using the FO rules of S.
FO(S)
• Let us use the symbol FO(S) to indicate FO
provability in S:
–  FO(S)  iff  is FO provable from  in the
system S.
• The subscript FO(S) indicates that we
restrict our proofs to the FO rules of S.
Two Important Properties
• For every deductive system of formal logic
S we can define the following 2 properties:
– 1. FO Deductive Soundness: A system S is FO
deductively sound iff for any  and  :
• if  FO(S)  then  FO 
– 2. FO Deductive Completeness: A system S is
FO deductively complete iff for any  and  :
• if  FO  then  FO(S) 
F is FO Sound and Complete
•
•
•
•
F is both FO sound and FO complete!
Soundness is pretty tricky to prove.
Completeness is very hard to prove.
The first proof of completeness was given
by Kurt Gödel in 1929. Hence it’s called
Gödel’s Completeness Result.
• If you want to see the proofs, take
Computability and Logic
Completeness of
the Tree Method
• The tree method is sound with regard to
both FO validity and FO invalidity (i.e. it
will never claim something to be FO valid
or FO invalid when in fact it is not).
• Moreover, it can be shown that the tree
method is complete with regard to FO
validity!
Infinite Trees
x y Likes(x,y)
y Likes(a,y) 
Likes(a,b)
y Likes(b,y)

Likes(b,c)
y Likes(c,y)

Likes(c,d)
y Likes(d,y)

Likes(d,e)

This tree will
never be finished,
so the tree method
will not give us
any answer!
Decision Procedures and
Decidability
• A decision procedure is a systematic procedure that
correctly decides whether something is or is not the case
for all relevant cases.
• The truth-table method is a decision procedure for truthfunctional consequence. That is, for any  and , the truthtable will systematically and correctly decide whether 
TF  or not.
• Because a decision procedure for truth-functional
consequence exists, we say that truth-functional
consequence is decidable.
• Question: is FO consequence decidable? In other words,
could there be a systematic test that correctly decides
whether something is a FO consequence of something or
not? (maybe FO Con is such a test?)
A Common Response
• Well, given that we have a sound and complete test for FO
validity, we should be able to make this into a test for FO
invalidity as follows: Have the procedure test for validity.
If it is valid, then eventually the procedure will say it is
valid (e.g. it says “Yes, it’s valid”), and hence we will
know (because the procedure is sound) that it is not
invalid. If it is invalid, then the procedure will not say so
(e.g. it outputs “Bananas on Mars”), but we can simply
interpret anything other than “Yes, it’s valid” as the claim
that it is invalid and, given that the procedure is complete,
it should indeed be invalid, for otherwise it would say
“Yes, it’s valid”. So, I would have a decision procedure for
FO validity!
The Mistake in the Reasoning
• There are two ways in which a positive test may
not say that some thing has some property:
– The test finishes but does not say that the certain
something has that property (“Bananas on Mars”)
– The test never finishes
• In the first case we know that the thing does not
have the property. But, in the second case, we may
not know this, as we may not know whether the
test is going to finish or not!
• The moral: positive tests do not guarantee
negative tests and vice versa.
Undecidability of FO validity
• It can be proven that no such decision
procedure can exist. This proof was found
by Alonzo Church in 1936. This year is no
accident: it’s the year of Turing’s famous
paper in which he lays out the TuringMachine, Turing’s Thesis, The Universal
Machine, and the Halting Problem. Indeed,
the undecidability of FOL follows from the
uncomputability of the Halting Problem.
For a full proof, take Computability and
Logic.
Extending Our Reasoning
• Since FO validity is undecidable, we know
that for any complete test for validity there
exists at least one case of FO invalidity for
which the test will never finish.
• For, if it would always finish, then we in
fact could make the positive test into a
negative one, and hence FO validity would
be decidable after all.
Incompleteness of FO Con
• Since it is unacceptable for FO Con to never
finish, we can’t make FO Con into a positive test.
• We thus know that FO Con is incomplete with
regard to FO validity as well as FO invalidity.
• Still, FO Con is sound, and will classify most
cases of FO validity as FO valid.
• Moreover, FO Con will also correctly classify
many cases of FO invalidity as FO invalid.
• What I say here about FO Con holds for ATP’s in
general of course.
Axiomatization
Limits to Predicate Logic
• Since it is not the case that Cube(a) FO Tet(a),
it is not the case either that Cube(a) FO(F)
Tet(a), even though Cube(a)  Tet(a)
• Thus, even though predicate logic is very
powerful, and more powerful than propositional
logic, it still doesn’t capture logical consequence!
• So, while we have that:
– if  FO  then  FO(S)  for some system S,
• what we really want is:
– if    then  FO(S)  for some system S.
Axioms
• An axiom regarding one or more predicates is a
statement that expresses a (usually, very basic)
truth regarding those predicates.
• Example: An axiom expressing a basic truth
regarding the predicate Adjoins is:
xy(Adjoins(x,y)  Adjoins(y,x))
• By adding axioms to the premises, we can prove
things we couldn’t before. For example, if we add
the axiom x(Cube(x)  Tet(x)) to our
premises, then we can infer Tet(a) from Cube(a).
Bridging the Gap
• An interesting question is now: can axioms be
used to bridge the gap between provability and
logical consequence?
• That is, focusing on a certain set of predicates R,
can we find a set of axioms A regarding R such
that for any  and  :    iff A FO  and
hence (since F is sound and complete)    iff
A FO(F) ?
• If we can, then all truths regarding R are said to be
axiomatizable or systematizable, and the axiom set
A is called complete with regard to the body of
truths involving R.
Axiomatizing Mathematics
• Around 1900, shortly after the formulation
of first-order logic was completed,
mathematicians started to wonder if all of
mathematics could be axiomatized. That is,
is it possible to find a finite set of axioms
expressing basic truths regarding
mathematics (e.g. xy (x + y = y + x))
such that every mathematical theorem is a
logical consequence of these axioms?
Peano Axioms for Natural Number
Arithmetic
• Where s(x) stands for the successor of x:
–
–
–
–
–
–
–
x y ((s(x) = s(y))  x = y)
x s(x) = 0
x (x  0  y s(y) = x))
x x + 0 = x
x y x + s(y) = s(x + y)
x x * 0 = x
x y x * s(y) = x * y + x
Gödel’s Incompleteness Result
• In 1931, the bomb dropped: Kurt Gödel proved
that not all of mathematics is axiomatizable. In
fact, hardly anything of mathematics is
axiomatizable, as Gödel proved that you can’t
even axiomatize all arithmetical truths involving
only the addition and multiplication of natural
numbers.
• Gödel’s Incompleteness Theorem is regarded as
one of the most important theorems of the 20th
century, as it shows fundamental limitations to
formal logic and, as such, to symbolic information
processing (i.e. computation) in general.
Resolution
The Rule of Resolution
• The Rule of Resolution is defined over
disjunctions of one or more literals:
P1  …  Pi-1  X  Pi+1  …  Pm
Q1  …  Qi-1  X  Qi+1  …  Qn
P1  …  Pi-1  Pi+1  …  Pm 
Q1  …  Qi-1  Qi+1  …  Qn
(each of Pi and Qi are literals; X is atomic)
Clauses
• A clause is a set of literals.
• Assuming a clause to represent a disjunction of all
literals that are in that clause, we can resolve two
clauses as follows:
{P1 , … , Pi-1 , X , Pi+1 , … , Pm}
{Q1 , … , Qi-1 , X , Qi+1 , … , Qn}
{P1 , … , Pi-1 , Pi+1 , … , Pm ,
Q1 , … , Qi-1 , Qi+1 , … , Qn}
(each of Pi and Qi are literals; X is atomic)
The Method of Resolution
• The method of resolution checks whether
some set of statements S is consistent. It
does this as follows:
– 1. Make a set T of clauses representing all
conjuncts of the CNF of each statement in S.
– 2. Resolve any two clauses from T that can be
resolved, and add the result to T.
– 3. If two clauses resolve to the empty set, stop:
the original set of statements was inconsistent.
Resolution Example
(Elusive(a)  Dangerous(a))  (Elusive(a)  Rare(a))
(Rare(a)  Dangerous(a))  Horned(a)
Horned(a)  Magical(a)
Negate Conclusion
and put into CNF
Magical(a)
(E  D)  (E  R)
(R  D)  H
HM
M
(E  D)  (E  R)
(R  D)  H
H  M
(R  D)  H
(R  H)  (D  H)
Resolution Example (Cont’d)
(E  D)  (E  R)
{E, D}
{E, R}
{E, H}
From CNF to
clauses and resolve
(R  H)  (D  H)
H  M
M
{R, H}
{H, M}
{M}
{D, H}
{H}
{E, H}
{H}
{}
Inconsistent, so valid!
Soundness and Completeness of
Resolution
• The method of Resolution is sound and
complete with regard to truth-functional
consistency in the sense that:
– If the method finds a set of statements to be
inconsistent, then that set of statements is
indeed inconsistent (soundness).
– If a set of statements is inconsistent, then the
method can find that set of statements to be
inconsistent by deriving the empty clause
(completeness).
Algorithms for Resolution and ATP’s
• Algorithms for resolution will differ in the
order in which clauses get resolved.
• Many ATP’s are based on resolution:
– Con mechanisms in Fitch
– Vampire (winner of world-wide ATP
competition last few years)
Prolog
Prolog
• The programming language Prolog is based on
Horn clauses.
• A Prolog program consists of 2 types of lines:
– Facts: Statements of the form P.
– Rules: Statements of the form (P1  …  Pn)  Q.
• A Prolog program is run by asking whether some
atomic statement Q follows from the facts and
rules. In Prolog: Q?
• The Prolog program will answer ‘Yes’ or ‘No’.
The Prolog Algorithm
• Prolog checks whether Q follows from the facts or
rules as follows:
–
–
–
–
1. Make a set of goals G, starting with Q.
2. If G is empty, stop with answer ‘Yes’.
3. If a statement P is in G that is a fact, remove P from G.
4. If P is in G and there is a rule P :- P1 , … , Pn, then
remove P from G, and add each Pi to G.
– 5. If you get stuck, try a different rule P :- P1 , … , Pn.
– 6. If all options fail, stop with answer ‘No’.
Prolog Example
Putting into Prolog:
Query: R?
H  H
{R}
H  E  E :- H.
H  D  D :- H.
(E  M)  R  R :- E, M.
{E, M}
{D, E}
{H, M}
{H, E}
{M}
{E}
(D  E)  R  R :- D, E.
{H}
{} ‘Yes’!
Power and Limitations of Prolog
• Prolog can only handle arguments whose
premises and conclusion are of the type as
discussed.
• So, many logic arguments cannot be
verified.
• However, because of the restriction, Prolog
becomes more efficient than generalpurpose provers.
Prolog and Production Rules
• Prolog’s rules are reminiscent of production
systems (bunch of if … then … statements).
• However, one big difference is:
– production systems are forward chaining
systems (start with given facts, apply rules to
proceed and get new stuff)
– Prolog is a backward chaining system (start
with the goal, and try and satisfy it, working
backwards)