LOGIC PROGRAMMING
(WEEK 1)
Eleni E. Mangina
Department of Computer Science
University College Dublin
Introduction to Logic
Distinction between Logic Programming and
Conventional Programming
Propositional & Predicate Calculi
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Lecture 1
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
General module information
• Module consists of 24 lectures and 5
practicals
• Final mark: 40% practical mark & 60%
exam mark
• All course material will be available on
the web
• Please note: this module starts slowly –
but it SPEEDS UP!!
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
LOGIC PROGRAMMING
Course Objectives
• To understand the placement of Logic Programming within
different Programming Paradigms
• To provide the student with an understanding of First Order Logic
• To examine in detail the theories and principles which underpins
automated deduction within First Order Logic
• To compare a particular Logic Programming language, namely
Prolog with the Pure Logical Foundations upon which it is based
• To understand the operation of Prolog (not to become proficient
programmers with the language)
• To review application of Logic Programming
• Review new developments in Logic Programming
eg. Parallelism
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
LOGIC PROGRAMMING
Lecture 1-2-3 Objectives
• To introduce the Importance of Logic Programming
• To outline a Taxonomy of Logics
• To distinguish between First Order Logics
& Higher Order Logics
• To review the fundamental ingredients of
Predicate Calculus
• To understand the basic inference rules &
propoties associated with First Order Logic
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Introduction to Logic
• ICOT (Institute for New Generation Computing) has identified
logic as the core programming formalism for its fifth generation
computers.
• The suitability of logic for the representation of real world
knowledge has been widly accepted.
• In the Fifth Generation Programme numerous relevant
techniques and technologies were investigated and this year
has seen the announcement of its successor the Real World
Computing (RWC) Programme. This 10 year strategic funding
initiative to the value of some $400 million will include work on
DAI, attention is given to the distribution and integration of
information, autonomous and cooperative control and
mechanisms for the integration of such functions as sensing,
perception, planning and action.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Advantages of Logic
• Well understood, ideas of logic have been
well refined.
• Concise notation.
• Mathematical deduction may be commissioned
to deduce new knowledge from old knowledge.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Disadvantages of Logic
• Lots of notation and terminology.
• Some people find programming in logic extremely
difficult. It differs significantly from conventional
programming in that the description of the logical
structure of problems must be given as opposed to
the description of how to solve them.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Taxonomy of Logics I
• Predicate Calculus, Predicate logic and first order logic are one
and the same.
• Predicate Calculus is an extension of Propositional Calculus.
• Three particular subsets of Predicate Calculus, which avoid the
full syntactic complexity of the aforementioned formalisms are
Conjunctive Normal Form (CNF), Clausal Form, The Horn Clause
Subset.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Taxonomy of Logics II
• Non-standard or higher order logics exist. Non-standard logic
refers to any other logic other than those of Propositional or
Predicate. These may be divided into:
a) Those that compete with these Classical logics
Multi-valued logic,
Fuzzy logic,
Intuitionistic logic
b) Those which extend these Classical logics
Modal logic,
Temporal logic.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Propositional Calculus
Consists of:
a) Propositions
b) Constants
The essential difference between Propositional Calculus
and Predicate Calculus are the latter:-
1) Incorporates Variables
2) Permits Quantification
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
NEWELL & SIMON
All sciences characterize the essential nature of the systems they study.
These characteristics are invariably qualitative in nature, for they set the
terms with which more detailed knowledge can be developed…
The study of logic and computers has revealed to us that intelligence resides
in physical symbol system. This is computer science’s most basic law of
qualitative structure. Symbol systems are collections of patterns and
processes, the latter being capable of producing, destroying and modifying
the former. The most important property of patterns is that they can designate
objects, processes or other patterns and that when they designate processes
they can be interpreted…
A second law of qualitative structure for artificial intelligence is that symbol
systems solve problems by generating potential solutions and testing them –
that is by searching. Solutions are usually sought by creating symbolic
expressions and modifying them sequentially until they satisfy the conditions
for a solution.
ACM Turing Award Lecture (1976)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Physical Symbol System Hypothesis
Intelligent activity, in either human or
machine, is achieved through the use of:
1. Symbol patterns to represent significant
aspects of a problem domain
2. Operations on these patterns to generate
potential solutions to problems
3. Search to select a solution from among
these possibilities
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicate Calculus
P: “it rained on Wednesday”
P:”weather (wednesday, rain)
Variable: X
Weather(X, rain)
Predicate Calculus Symbols:
The alphabet that makes up the symbols of the predicate
calculus consists of:
1. The set of letter, both upper- and lowercase of the English
alphabet
2. The set of digits 0,1,…,9
3. The underscore, _.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicate Calculus
Symbols and terms:
Predicate calculus symbols include:
1.
Truth symbols, true or false
2.
Constant symbols are symbol expressions having the first
character lowercase
3.
Variable symbols are symbol expressions beginning with an
uppercase character
4.
Function symbols are symbol expressions having the first
character lowercase
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicate Calculus
Propositional Calculus Symbols:
•
Symbols of: P, Q, R, …
•
Truth symbols: true, false
•
Connectives: ^, , ¬, , 
Propositional Calculus Sentences:
•
Every propositional symbol and truth symbol: true, P, Q, …
•
The negation of a sentence: ¬P, ¬false
•
The conjunction, or and of two sentences: P^ ¬P
•
The disjunction, or or of two sentences: P  ¬P
•
The implication of one sentence for another: P Q
•
The equivalence of two sentences: PQR
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Truth Tables
Truth table lists all possible truth value assignments to the atomic proposition of an
expression and gives the truth value of the expression for each assignment. Truth
table enumerates all possible worlds of interpretation.
¬(¬P) = P
(P  Q) = (¬P  Q)
The contra positive law: (P  Q) = (¬Q  ¬P)
De Morgan’s law: ¬(P Q) = (¬P^¬Q)
Commutative law: (P^Q) = (Q^ P)
Distributive law: P^(Q R) = (P^Q)  (P^R)
P
Q
P^Q
T
T
T
T
F
F
F
T
F
F
F
F
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Examples
• John loves none but Mary
¬Y (loves(John, Mary) ^ loves(John,Y) ^(YMary)
¬Y (loves(John, Y)) ^ (¬ Y=Mary)
Y ¬loves(John,Y)  Y=Mary
Y ¬loves(John,Y)  Y=Mary
• Everybody loves exactly one person
X Y loves(X,Y) ^ (Z loves(X,Z)  Y=Z)
X Y  (Z loves(X,Z)  Y=Z)
• There are at least two (different) X such that A(X)
X Y (X Y ^ A(X) ^ A(Y))
• There is exactly one X such that A(X)
X A(X) ^ (A(Y)  Y=X)
X Y (A(Y)  Y=X
A(Y)  Y=X ^ Y=X  A(Y)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Examples
• He who is late is to be punished
X isLate(X)  punished(X)
• Boys who are late are to be punished
Y (isLate(Y) ^ isBoy(Y)  punished(Y)
• Everyone here is older than everyone there
X,Y (isHere(X) ^ isThere(Y))  olderThan(X,Y)
X isHere(X)  ¬ (Y isThere(Y) ^ olderThan(Y,X)
Y ¬ isThere(Y)  ¬olderThan(Y,X)
Y isThere(Y)  olderThan(Y,X)
• Everyone admires himself
X human(X)  admire(X,X)
• John loves Mary but Mary loves someone else
Loves(John,Mary) ^ Y loves(Mary,Y) ^ (YJohn)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Examples
Example of the use of predicate calculus to describe a simple
world. The domain of discourse is a set of family relationships in a
biblical genealogy:
mother (eve,abel) = Eve is the mother of Abel
mother(eve,cain) = …
father(adam, abel) = …
father(adam,cain) = …
In this example we use the predicates mother and father to
define a set of parent – child relationships. The implication give
general definitions of other relationships  inference rules!
parent(X,Y) ^ parent(X,Z)  sibling(Y,Z) YZ
parent(X,Y)  father(X,Y)  mother(X,Y)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Examples
If Jones envies Smith or vice versa but they do not envy each
other, then Jones envies Smith, if and only if Smith does not envy
Jones.
1. ((Jones envies Smith)  (Smith envies Jones)) ^
2. ¬ ((Jones envies Smith) ^ (Smith envies Jones)) 
3. (Jones envies Smith) 
4. ¬ (Smith envies Jones)
(AB) ^ (¬ (A^B) (A  ¬ B)
(A

B)
^
(¬
(A
^
B)

(A

¬ B)
T
T
T
F
F
T
T
T
T
T
F
F
T
T
F
T
T
T
F
F
T
T
T
T
F
T
T
T
T
F
F
T
T
F
T
F
F
F
F
F
T
F
F
F
T
F
F
T
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Lecture 2
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
A Logic-Based Financial Advisor
1.
2.
3.
The function of the advisor is to help a user decide whether to invest in a
savings account or the stock market. Some investors may want to to split
their money between the two. The investment that will be recommended
for individual investors depends on their income and the current amount
they have saved according to the following criteria:
Individuals with an inadequate savings account should always make
increasing the amount saved their first priority, regardless of their
income.
Individuals with an adequate savings account and an adequate income
should consider a riskier but potentially more profitable investment in the
stock market.
Individuals with a lower income who already have an adequate savings
account may want to consider splitting their surplus income between
savings and stocks, to increase the cushion in savings while attempting
to increase their income through stocks
The adequacy of both savings and income is determined by the number
of dependents an individual must support. Our rule is to have at least
$5000 in the bank for each dependent. An adequate income must be a
steady income and supply at least $15000 per year plus an additional
$4000 for each dependent.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Solution
• First task: Determine the major features
savings_account and income (adequate and inadequate arguments):
savings_account(adequate)
savings_account(inadequate)
income(adequate)
income(inadequate)
investment (stock, savings and combination arguments)
• Second task: Determine the rules
First rule:
(1) savings_account(inadequate)  investment(savings)
Second rule:
(2) savings_account(adequate) ^ income(adequate)  investment(stocks)
Third rule:
(3) savings_account(adequate) ^ income(inadequate) 
investment(combination)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Solution (cont.)
•
To determine the minimum adequate savings the function minsavings is defined:
(4) X amountSaved(X) ^ Y (dependents(Y) ^greater(X,minsavings(Y)) 
savings_account(adequate)
(5) X amountSaved(X) ^ Y (dependents(Y) ^greater(X,minsavings(Y)) 
savings_account(inadequate)
Where:
minsavings(X) = 5000*X and
minincome(X) = 15000 + (4000*X)
(6) X earnings(X,steady) ^ Y (dependents(Y) ^greater(X,minincome(Y)) 
income(adequate)
(7) X earnings(X,steady) ^ Y (dependents(Y) ^  greater(X,minincome(Y)) 
income(inadequate)
(8) X earnings(X,unsteady)  income(inadequate)
In order to perform a consultation, a description of a particular investor is added
to this set of predicate calculus sentences using the predicates amountSaved,
earnings and dependents. Thus an individual with 3 dependents, $22000 in
savings and a steady income of $25000 would be described by:
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Solution (cont.)
•
•
•
•
•
•
amount_saved(22000)
earnings(25000, steady)
dependents(3)
minsavings(3) = 15000
minincome(3) = 27000
Earnings(25000,steady) ^ dependents(3) ^
greater(25000,minincome(3))  income(inadequate)
• Amount_saved(22000) ^ dependents(3) ^
greater(22000,minsavings(3))  savings_account(adequate)
• Savings_account(adequate) ^ income (inadequate) 
investment(combination)
This is the suggested investment for this individual!
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicate Calculus
Predicate Calculus is comprised of the following
basic building blocks:
a) Predicates
b) Functions
c) Constants
d) Variables
e) Connectives
f) Quantifiers
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicates, Constants, Variables
& Functions I
A logic program is a collection of sentences. Sentences are constructed
from predicates, constants, variables, functions, formulae, connectives
and quantifiers. Predicates are chosen from a particular vocabularly.
Some examples are...
hates (greg, computer_science)
Predicates
Constant arguments
loves (greg, cakes)
The argument greg must denote a particular object. Thus we assume
that it denotes a particular person called greg. The object denoted by
a term does not have to be a person it could be a class, an event, a thing
depending on the domain.
Once, however, the context of a term has been decided it remains fixed.
If we decided greg refers to greg o’hare then it cannot be concluded that
greg howard hates computer science or that all gregs love cake.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicates, Constants, Variables
& Functions II
The arguments of a predicate can be either
a) Constants
b) Variable
c) Function
is_a (greg, lecturer)
greg and lecturer are constants, they refer to a particular object
or class of object.
is_a (X, lecturer)
is a predicate which has as terms a variable X and a constant
lecturer.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicates, Constants, Variables
& Functions III
At some stage a variable may take a particular value, say greg.
The variable would then be said to have been instantiated.
Consider the following predicates...
loves (greg, eating_icecream, hot)
loves (greg, home)
loves (Y, sleeping, morning)
An incompatibility exists, loves takes differing number of arguments,
with the second argument taking a distinct meaning in each case.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicates, Constants, Variables
& Functions IV
If we wished to use both forms we would have to introduce unique predicates.
loves (greg, sister (mandy))
sister (mandy) is a function which refers to an object that is the sister of mandy.
loves (greg, sweet_and_fattening (X))
greg loves any object that is sweet and fattening.
supports (chris, topoftable (X))
chris supports an object (team) that is at top of the table.
loves_action (T, eating_icecream, hot)
loves_object (greg, home)
loves_action (Y, sleeping, morning)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Predicates, Constants, Variables
& Functions V
Note
• Number of arguments can vary
• Once number, order and interpretation decided at the outset
it must be maintained throughout.
In general because the meaning or denotation of predicate calculus
is dependent upon the meaning of its parts, it is said to exhibit
compositional semantics. Consequently when introducing a function
or predicate the number and meaning and indeed order of each of
its predicates must be stated exactly.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Connectives & Quantifiers I
We combine formuals by use of connectives.
Within Predicate Calculus these are as follows...
AND

OR

NOT
~
IF

IF AND ONLY IF (IFF)

Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Connectives & Quantifiers II
We will concentrate on the first four of these, suffice to note
that the distinction between IF and IF and only IF is illustrated
in the following truth table...
p
T
T
F
F
q
T
F
T
F
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
p -> q
T
F
T
T
p <-> q
T
F
F
T
DEPT. OF COMPUTER SCIENCE
UCD
Connectives & Quantifiers III
• It should be noted that
p -> q is equivalent to ~p v q
This can be verified.
• Furthermore such compound predicates are
Commutative
p,q=q,p
Distributive
p , (q v r) = (p , q) v (p , r)
p v (q , r) = (p v q) , (p v r)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Connectives & Quantifiers IV
• Associative
p , (q , r) = (p , q) , r
p v (q v r) = (p v q) v r
• Obey De Morgan’s Laws
~(p , q) = ~p v ~q
~(p v q) = ~p , ~q
• Annihilation
~(~p) = p
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Example Formula
shy (paula) , pretty (paula) -> attractive(paula)
equal_sides (Shape) , equals (angles , 4) , equal_angles (Shape)
-> square(Shape)
wears (X , tanktop) v ~(trendy (X))
-> computer_scientist (X)
thief (X) , valuable (Y) -> steal (X , Y)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Connectives & Quantifiers V
• Quantifiers in Predicate Calculus indicate the number of
instantiations of a variable that need to be true in order
that the entire proposition is true.
They come in two forms...
a) Universal quantifier
b) Existential quantifier
 (X) - all instantiations must be true in order
for entire proposition to hold true.
 (X) - only some instantiations (at least one) need be true.
(X) (mortal (X) -> will_die (X))
 (Y) (rectangle (Y) , square (Y))
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Quantifier Order
• If both universal and existential quantifiers are commissioned
in the same proposition the ordering may prove significant.
According to Alty and Coombs a typical such example may be
(X) (Y) (employee (X) -> manager (Y, X))
- for every X there exists a Y which manages them.
 (Y) (X) (employee (X) -> manager (Y , X))
- there exists a manager who manages every employee.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
What exactly are Formulas?
Well formed formulas (wff) are formed recursively by applying
one or more of the following rules.
a) If P is a 0-ary predicate, P is a wff
b) If P is an n-ary predicate and term1...termN are terms then
P(t1...tn) is a wff.
c) Wffs connected by the connectives V , |, ->, <-> are
well formed formulas.
d) If P is a wff then ~(P) is a wff.
e) Any wff surrounded by either a universal or
existential quantifiers are wff.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Some Terminology
• A sentence is a wff in which all variables
are bound. That is none are free.
• A clause is a wff consisting of
a disjunction of terms.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Things to Do !
• Experiment expressing relationships between objects
in Predicate Calculus.
Describe the world of your bedroom as a series of
sentences in Predicate Calculus
• Find out clearly what is meant by Second Order Logics
• Become confident with the distinction between
Propositional and Predicate Logic
•
Expand your notes on Monotonic and NonMonotonic
Logic
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Lecture 3
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Scope of Variables I
Consider the following proposition
 (P) (X) (train_to (X, london)
-> ( (Y) (first_class_carriage (Y) v
second_class_carriage (Y))
,  (Z) (buffer_car (Z))
, will_become (X , manchester_train)))
Some of quantifiers generally follows laws observed in say Pascal.
In the above...
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Scope of Variables III
Scope X - A and any proposition nested within, namely B and C
Scope Y - B and any propositions nested within
Scope Z - C and any propositions nested within
Scope P - free variable unbounded
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Automated Deduction I
To process knowledge represented in predicate logic
we must be able to take a given set of assertions and implications
(or facts and rules as they are often referred to) and
be able to infer new facts and rules based on this
corpus of knowledge.
In addition such an automated technique should perform it’s
inferences in such a manner that it’s conclusions can be
viewed with confidence.
There exists many inference rules which include:
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Inference Rules I
1) Modus Ponendo Ponens (MPP)
A -> B
If we know A we can conclude B
2) Modus Tollendo Tollens (MTT)
A -> B
If we know ~B then we can conclude
~A
3) Double Negation (DN)
A
We know ~(~A)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Inference Rules II
4) And Introduction (&I)
A
B
We can conclude A , B
5) Reductio Ad Absurdum (RAA)
A -> B
A ->~B
We conclude ~A
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Inference Rules III
6) Universal Specialisation (US)
(X) W(X) , A
We conclude W(A)
eg. Unreliable (Computers) Vax Computer
Unreliable(Vax)
Where A is a specific instance of X
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
The Choice of Inference Rule
Often a single inference rule or a group of inference rules
may be commissioned.
Two problems prevail:
1) Choice of inference rule
Which rule to employ at a particular instance.
2) Generation of numerous uninteresting propositions
How to direct or focus deduction?
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Resolution I
Having encoded real world knowledge in the form of
predicate calculus the next stage is that of utilising
the advantages the formalism affords.
The precise, unambiguous and formal nature of the logic
make it easy to automate the deductive process.
Many Computer Scientists argue high-level languages
and notations sholud be natural and easy to use this is
hardly the case with logic.
One method of overcoming these is to use a more general technique.
One such technique is that of Resolution Theorem Proving,
and it is this very technique which is employed within PROLOG.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Resolution II
Resolution reduces the proof procedure to the
application of a single rule.
Resolution is a procedure that carries out the variety
of processes involved when reasoning with a series
of predicate calculus statements.
Resolution produces proofs by Refutation.
That is to say in order to prove a statement,
it attempts to show that the negation of the statement
produces a contradiction, with the existing facts and rules.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Resolution - The Algorithm
In generalist form it involves...(assume we want to prove A)
1) Assume ~A True.
2) Show ~A and axioms lead to us concluding something is
True that cannot be True.
3) Result ~A cannot be true
since given rise to a contradiction.
4) Conclude A since ~A cannot be True.
In order to execute efficiently Resolution operates upon statements
which have been converted to a very convenient
standard form, namely Clausal form.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
A Problem
Yellow
Green
D
B
C
A
H
E
F
Blue
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
The Conversion Process
A Worked Example
Consider the following well formed formula...
(X) [P(X) -> ( (Y) [Q(X,Y), R(Y)]
, ~ (Y) [Q(X,Y), Q(Y,X)]
, (Y) [~P(Y) -> ~S(X,Y)])]
We will now work through the nine stages involved in
converting this Predicate Calculus sentence to Clausal Form.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
The Conversion Process
Converting Predicate Calculus sentences to the normalised
form of Clausal form involves a sequence of nine steps.
These steps are:1. Eliminate Implications
2. Move Negations Down to Atomic Formula
3. Eliminate Existential Quantifiers
4. Rename Variables
5. Move Universal Quantifiers to the Left
6. Convert Matrix into a Conjunction of Disjuncts
7. Eliminate Conjunctions
8. Rename Variables
9. Purge Universal Quantifiers
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Eliminate Implications I
Remove implications by replacing
a -> b by ~a v b
Two nested implications exist here. Where more than one
implication exists we always remove innermost first.
Work from the inside out. Removing first the innermost implication
gives....
(X) [P(X) -> ( (Y) [Q(X,Y), R(Y)]
, ~ (Y) [Q(X,Y), Q(Y,X)]
, (Y) [~(~P(Y)) v ~S(X,Y)])]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Eliminate Implications II
Then removing the outermost...
 (X) [~P(X) v (  (Y) [Q(X,Y), R(Y)]
, ~ (Y) [Q(X,Y), Q(Y,X)]
,
(Y) [~(~P(Y)) v ~S(X,Y)])]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Move Negations Down to the
Atomic Formula I
Reduce scope of negation by employing the following rules...
~(~P) = P
(1)
~(a , b) = ~a v ~b
(2)
~(a v b) = ~a , ~b
(3)
and
~ (X) P(X) =
(X) ~ P(X) (4)
~ (X) P(X) = (X) ~ P(X) (5)
Rules (2) and (3) are those of De Morgans laws
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Move Negations Down to Atomic
Formula II
Consequently we get...
(X) [~P(X) v ( (Y) [Q(X,Y), R(Y)]
, (Y) [~Q(X,Y) v ~Q(Y,X)]*
, (Y) [P(Y) v ~S(X,Y)]) **
]
* From (2) and (5)
** From (1)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Eliminate Existential Quantifiers I
If a variable contains an existentially quantified variable
it means there is a certain value of the variable which
will make the formula true.
We may eliminate the quantifier by substituting a reference
to a function that will produce the required value.
We do not need to know how the function works or the form it will
take. All we need to know is that such a function will exist.
Consider the following simple example:
(Y) Lecturer (Y)
could be replaced by
Lecturer (f1)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Skolem Constants
Lecturer (f1)
f1 is a function with no arguments, sometimes called a
Skolem Constants that produces a value that satisfies lecturers.
More particularly if existential quantifiers exist within
the scope of universal quantifiers the value that satisfies
the predicate may depend on the value of the universal quantifier.
eg  (X) (Y) father_of (Y, X)
Thus the value y depends on the value x thus substitute
function f2 which returns the appropriate value given the
dependence on x (i.e. accepts x as an argument to the function)
(X) father_of( f2(X), X)
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Skolem Functions
These functions are known as Skolem Functions.
In general the process of removing existential quantifiers
is known as Skolemization after Thoraf Skolem the logician.
In our particular wff we only have one occurence of an existential
quantifier namely Y.
The scope of Y lies within that of the universal quantification of X,
and consequently we replace Y by a Skolem function lets call it f1.
The value f1 returns is dependent upon the value of X.
Thus we introduce the skolem function f1(X).
This function returns the appropriate value of Y dependent
upon the value of X.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Examples of Skolemisation
Consider the following WFF’s
(X) (Q) (Z) related_to (X,Z,Q)
=> (X) (Q) related_to (X, g(X,Q),Q)
What if the WFF looked like...
(Z) (X) (Q) related_to (X,Z,Q)
=> (X) (Q) related_to (X,g,Q)
Where g is a skolem constant in the latter case and a
skolem function in the former.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Skolemising
Hence returing to our wff skolemising results
in...
(X) [ ~P(X) v ([Q(X, f1(X), R(f1(X))]
, (Y) [~Q(X,Y) v ~Q(Y,X)]
, (Y) [P(Y) v ~S(X,Y)])
]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Rename Variables
Rename variables so that each quantifier within each expression
has a unique name.
For example the formula:
(X) rich(X) v (X) tall(X)
Note : Different X’s used, not the same variable (cf scope).
Could become
(X) rich(X) v (Y) tall(Y)
Thus each quantifier binds to a unique variable.
Since variables only dummy names this won’t affect true
value of formula.
The logical equivalence of the formula will be preserved.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Rename Variables II
Having performed Stage 4 our wff looks like...
 (X)[~P(X) v ([Q(X, f1(X)), R(f1(X))]
, (Y) [~Q(X,Y) v ~Q(Y,X)]
, (Z) [ P(Z) v ~S(X,Z)]
)
]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Move Universal Quantifiers
to the Left I
This may be permitted since each quantifier now has a unique
name and consequently the leftward migration of the universal
quantifiers will not cause any associated confusion.
Our wff now simply becomes ......
 (X), (Y), (Z)
[ ~ P(X) v ( [ Q(X,f1(X)), R(f1(X)) ]
, [ ~ Q(X,Y) v ~ Q(Y,X)]
, [ P(Z) v ~ S(X,Z)]
)
]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Move Universal Quantifiers to
the Left II
At this stage the formula is said to be in PRENEX NORMAL FORM.
Such formula consists of a set of prefixing quantifiers followed
by a quantifier free Matrix.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Convert Matrix to a Conjunction
of Disjuncts I
In simpler terms this merely involves moving
OR’s inside AND’s
This involves the commissioning of the
Distributive (Dist) and Associative (Ass) laws
thus:-
Dist
Ass
p v (q,r)
 (p v q), (p v r)
p v (q v r)  (p v q) v r
pvqvr
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Convert Matrix into a Conjunction
of Disjuncts II
Firstly, distributing P(X) we get
 (X),  (Y),  (Z)
[(~ P(X) v ( [Q(X) f1(X)), R(f1(X)) ] )
, ( ~ P(X) v [ ~ Q(X,Y) v ~ Q(Y,X) ]
, ( ~ P(X) v [ P(Z) v ~ S(X, Z) ] )
)
]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Convert Matrix into a Conjunction
of Disjuncts III
Employing Distributive and Associative laws.....
 (X),  (Y),  (Z)
[ ~ (P(X) v Q(X, f1(X)) , ( ~ P(X) v R(f1(X))
, ( ~ P(X) v ~ Q(X,Y) v ~ Q(Y,X))
, ( ~ P(X) v P(Z) v ~ S(X,Z))
]
The wff is said to be in Conjunctive Normal Form (CNF).
This is a particular normalised form of Predicate Calculus.
CNF represents the original wff as a conjunction of
disjunctions of literals.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Eliminate the Conjunctions
Remove the explicit conjunctions, writing each as if it were a
separate axiom. Each separate conjunct is called a Clause.
We can do this since in a conjunction all parts must be true in
order for the entire wff to be true.
Hence each component can be stated as a separate axiom.
This produces .....
 (X) [ ~ P(X) v Q(X, f1(X)) ]
 (X) [ ~ P(X) v R(f1(X)) ]
 (X),  (Y) [ ~ P(X) v ~ Q(X,Y) v ~ Q(Y,X) ]
 (X),  (Z) [ ~ P(X) v P(Z) v ~ S(X,Z) ]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Rename Variables
We rename variables so that each clause references
different variables.
Giving: (X) [ ~ P(X) v Q(X, f1(X)) ]
 (W) [ ~ P(W) v R(f1(W))]
 (U),  (V) [ ~ P(U) v ~ Q(U,V) v ~ Q(V,U)]
 (A),  (B) [ ~ P(A) v P(B) v ~ S(A,B) ]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Purge Universal Quantifiers
Remove universal quantifiers so that these become implicit.
[ ~ P(X) v Q(X, f1(X)) ]
[ ~ P(W) v R(f1(W)) ]
[ ~ P(U) v ~ Q(U,V) v ~ Q(V,U) ]
[ ~ P(A) v P(B) v ~ S(A,B) ]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Example Conversions
Convert the Following to Clausal Form....
1.  (X),  (Y) L(X, Y) , Q(X, Y) -> Z(X, Y)
2. ~  (X),  (Q), M(X,Q) -> S(X, Q)
3. ( ~  (X) P(X)) -> (  (X), P(X))
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Conversion Exercises II
Convert the following well formed formula into Conjunctive Normal
Form.
 (X) [ P(X) -> ( (Y) [ Q(X, Y) , R(Y) ]
, ~ (Y) [ Q(X, Y) , Q(Y, X) ]
)
]
Ensure that your answer clearly highlights the various discrete
stages involved in the conversion.
[14 Marks]
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Has Logical Equivalence
Been Preserved ?
All Steps with the exception of Step 3 that of Skolemisation
preserves the logical form of the original sentence.
If no skolemisation is demanded within the conversion process
then we can say that the conjunction of all the resulting clauses (RC)
are logically equivalent to the original sentence (OS).
OS  RC
If skolemisation is necessitated then we can not say that
such an equivalence holds. We can only state that :• RC are satisfiable iff OS is satisfiable and
• RC implies OS.
Is this set of weaker properties sufficient for our purposes?
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
Things to Do List
1. Review the worked example.
2. Rework the example without reference to your notes.
3. Attempt the Skolemisation Examples Presented
4. Supplement your notes on Skolemisation
5. Consult past examination papers and attempt
conversion related questions.
Remember that past papers together with model answers
are held in the library behind the issue desk.
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD
1st assignment! Due within
next 10 days!
Dr Eleni Mangina – COURSE: LOGIC PROGRAMMING
(during a joint degree with Fudan University in Software Engineering)
DEPT. OF COMPUTER SCIENCE
UCD