Logics for Computer Science

375 Pages·2003·1.464 MB·English

by  Arindama Singh

Logics For
Computer Science
Arindama Singh
Department of Mathematics
Indian Institute of Technology Madras
[email protected]
Logics For
Computer Science
Contents
Preface v
1 Propositional Logic 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Syntax of PL . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Semantics of PL . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6 Some Applications . . . . . . . . . . . . . . . . . . . . . . . 35
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 First Order Logic 51
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 Syntax of FL . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Preparing for Semantics . . . . . . . . . . . . . . . . . . . . 59
2.4 Semantics of FL . . . . . . . . . . . . . . . . . . . . . . . . 64
2.5 Some Useful Consequences . . . . . . . . . . . . . . . . . . . 77
2.6 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.7 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.8 Herbrand Interpretation . . . . . . . . . . . . . . . . . . . . 97
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3 Resolution 115
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.2 Resolution in PL . . . . . . . . . . . . . . . . . . . . . . . . 116
3.3 Unification of Clauses . . . . . . . . . . . . . . . . . . . . . 127
3.4 Extending Resolution . . . . . . . . . . . . . . . . . . . . . 134
3.5 Resolution for FL . . . . . . . . . . . . . . . . . . . . . . . . 139
3.6 Horn Clauses in FL . . . . . . . . . . . . . . . . . . . . . . . 142
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
iii
iv Contents
4 Proofs in PL and FL 152
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.2 Axiomatic System PC . . . . . . . . . . . . . . . . . . . . . 152
4.3 Axiomatic System FC . . . . . . . . . . . . . . . . . . . . . 165
4.4 Adequacy and Compactness . . . . . . . . . . . . . . . . . . 176
4.5 Natural Deduction . . . . . . . . . . . . . . . . . . . . . . . 184
4.6 Gentzen Systems . . . . . . . . . . . . . . . . . . . . . . . . 194
4.7 Analytic Tableau . . . . . . . . . . . . . . . . . . . . . . . . 204
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5 Program Verification 233
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.2 Issue of Correctness . . . . . . . . . . . . . . . . . . . . . . 235
5.3 The Core Language CL . . . . . . . . . . . . . . . . . . . . 238
5.4 Partial Correctness . . . . . . . . . . . . . . . . . . . . . . . 242
5.5 Hoare Proof and Proof Summary . . . . . . . . . . . . . . . 252
5.6 Total Correctness . . . . . . . . . . . . . . . . . . . . . . . . 259
5.7 The Predicate Transformer wp . . . . . . . . . . . . . . . . 263
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
6 Modal Logics 280
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
6.2 Syntax and Semantics of K . . . . . . . . . . . . . . . . . . 282
6.3 Axiomatic System KC . . . . . . . . . . . . . . . . . . . . . 294
6.4 Other Proof Systems for K . . . . . . . . . . . . . . . . . . 301
6.5 Other Modal Logics . . . . . . . . . . . . . . . . . . . . . . 312
6.6 Various Modalities . . . . . . . . . . . . . . . . . . . . . . . 320
6.7 Computation Tree Logic . . . . . . . . . . . . . . . . . . . . 325
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
7 Some Other Logics 336
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
7.2 Intuitionistic Logic . . . . . . . . . . . . . . . . . . . . . . . 337
7.3 L ukasiewicz Logics . . . . . . . . . . . . . . . . . . . . . . . 340
7.4 Probabilistic Logic . . . . . . . . . . . . . . . . . . . . . . . 345
7.5 Possibilistic and Fuzzy Logic . . . . . . . . . . . . . . . . . 346
7.6 Default Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 349
7.7 Autoepistemic Logic . . . . . . . . . . . . . . . . . . . . . . 352
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
References 356
Index 362
Preface
Each chapter in this book has an introduction, the first section. But there
is no introduction to the book itself. So, let me use the mask of the Preface
for this purpose. Of course, a plain introduction to the book is just to
place the book in your hand; but while you are reading it, you yourself
have already done that. Well done!
This is the way I will be talking to you throughout. I will ask you to
do exercises on the spot, often waiting for you up to some point, and then
give you a hint to proceed. It is something like the following commercial
for the book:
I would like to ask you three questions, would you answer them
with a plain ‘Yes’ or ‘No’?
Good, you have answered ‘Yes’, whatever be the reason. But
see, that was my first question. Would you answer the same to
the second as to the third?
Very good, you have answered ‘Yes’ again; of course, it does
not matter. If you have not bought a copy of this book, are you
going to buy it soon?
Look, if you have answered ‘Yes’ to the second question, you
are also answering ‘Yes’ to the third, and if you have answered
‘No’ to the second, you are not answering ‘No’ to the third, i.e.,
your answer to the third is undoubtedly, ‘Yes’. Excellent. That
is the end of the commercial.
You have participated well in the commercial; I hope you will be with
me throughout. It is easy to learn logic and easier to teach it; that is the
spirit of this book. I would not reveal what is logic; you will discover it
eventually. My aim is to equip you with the logical methods so that when
you take up computer science as your profession, you will feel like a fish in
water. A warning: though the book is ideal for self-learning, it would not
replace a teacher. At least on one count: you can ask a question to your
teacher and, hopefully, he1 will give you an answer, or you will be inspired
1Themasculinegenderisusedthroughoutthebooknotbecauseofsomechauvinistic
reasons, as it is easy to do that. It is easy to write ‘he’ rather than ‘she’ or ‘one’ or ‘a
person’ or ‘a human being’ etc. You need not be offended by it.
v
vi Preface
by him for finding your own answer; the book cannot do that always.
If you are a teacher, you may not need to learn all the topics, for
you had probably learnt them. But you can really teach logic better by
helping your students in the exercises. You can supplement the book with
more computer science applications which I had to omit, for it is already
a somewhat lengthy book. It is ideal for a two-semester course for not
so advanced students. But you can cover it in one semester2 by omitting
certain proof procedures.
In Chapter 1, you will find the logic of statements. The approach is
semantic. Thesemanticconsiderationsgiverisetothecalculationalmethod
of proof quite naturally. Calculations have been used very informally for
proving valid formulas just as you solve problems in school algebra. All
the metaresults have been discussed except compactness. That is done via
proof theory, again, for ease. In Chapter 2, a similar approach is taken for
thelogicofpredicatesorrelations, morecommonlyknownasthefirstorder
logic. Thefirstorderlogicdiscussedhereincludestheequalityrelation,and
nopreferenceisgiventoonewithoutequality,becauseapplicationsdemand
it. This chapter also deals with the Herbrand’s theorems without waiting
for the resolution method. The resolution method is the topic of Chapter
3. Chapter 4 is exclusively meant for various proof techniques. In a one-
semester course, you may go for only one proof method. Or, you may
do one of the proof methods formally in detail and present others in one
lecture each.
In Chapter 5, you come across a fundamental area of computer science,
theprogramverification. Itisalsooneofthemostusefulapplicationsoffirst
order logic. You should not stop just after that; you must hunt for some
more materials and pursue the topic a bit further. You must have a similar
approach to the model checking algorithms that have been introduced in
Chapter 6 as an application of modal logics. This chapter gives a very
comprehensive account of modal logics used in computer science. It does
cover the essential ingredients leaving one: decidability issues, for which
you may look through the References. Similarly, Chapter 7 introduces the
so-called nonstandard logics which are upcoming, but not entirely new.
We discuss only those nonstandard logics that have general frameworks
yet; the list is not exhaustive. You are led through the chapters towards
logic engineering, an activity that is becoming increasingly important in
current research in computer science.
The approach of the book is mathematical, in the sense that it involves
a continual interplay between the abstract and the concrete. The yearning
forunderstandingtheconcretephenomenagivesrisetoabstractionsandto
2It takes around 60 lectures to complete the book. In fact, it is developed from my
notes for a logic course offered to Engineering undergraduates at IIT Madras taking
theoretical computer science as their minor subject. Some of the postgraduate students
in Mathematics also credit the course as an elective.
Preface vii
understand the abstract, concrete phenomena are a necessity. The typical
feeling in any concrete problem is that when you abstract it, an existence
of a solution is an opaque mystery. The concrete problems do not come
with a label that such and such method would be applicable. It is you
who may think of applying an existing method to solve it. Often, patient
and continuous involvement with the problems help you to gradually gain
insight, leadingtoasolution. So, bepatientandpersistent, relyonyourself
as far as possible; start the journey now.
I have to pause here for a second. I would like to thank all those who
have made this happy journey of yours possible. Though the book has
taken shape now, there has been a continual effort throughout. There
must be many good reasons for this, the most basic of all these being my
interest in higher studies. I thank my parents and my school teacher Mr.
Rupakar Sarangi who nourished me physically and mentally by keeping me
fit for higher studies. I thank my wife Archana for continuing in a simi-
lar vein after our marriage. She, along with my children, Anindya Ambuj
and Ananya Asmita, did suffer a lot due to my post-marriage engagement
with this book. I thank them for their patience and encouragement. I
am indebted to my friend and logic tutor Professor Chinmoy Goswami
of the University of Hyderabad. My students helped me considerably to
learn logic; in a way, this book is a reward for that. I thank my colleague
Dr.M.Thamban Nair of IIT Madras for encouraging me to bring out the
book from some scratches on a note book. I also thank the Publishers,
Prentice Hall of India, particularly, Mr.K.C.Devasia for his expert guid-
ance and the production team for their assistance.
Any constructive suggestions for improving the contents would be most
welcome. These may be emailed to me at [email protected].
Arindama Singh
1
Propositional Logic
1.1 Introduction
Logic is all about how an object follows from other objects. In Proposi-
tional logic, our objects are propositions such as
Bapuji was indeed a Mahatma.
The title of a book on logic could be mis-spelt.
Five men cannot have eleven eyes.
Buddha’s original name was Arindama.
Alexander the great did not set foot in India.
The woman who committed the crime did not have three legs.
No bachelor is married.
Some unmarried men get married.
Propositions are declarative sentences which may be asserted to be true
or false. It is quite possible that you may not be able to say for certain
whetheragivenpropositionistrue(orfalse), withoutgoingtoitsmeanings
or external factors. But that does not really matter. All that we need is
that it is either true or false. The so-called conjectures or open problems
are propositions, the truth of which you do not know yet. For example,
Goldbach’s conjecture: “Every even number bigger than 2 is a sum of two
prime numbers”, or the P 6= NP: “There is at least one nondeterministic
polynomial time solvable problem which is not deterministic polynomial
time solvable”. As of now, we do not have any way of showing the truth or
falsity of these propositions. However, each of them is either true or false.
Note that we are not defining here what a proposition is. In fact, we
will not even attempt to do so. We are only getting familiarized with the
kind of objects in question.
The sentences which are not propositions include questions, orders, ex-
clamations, etc., for which we may not like to associate a truth value. We
do not know how to say whether “Are you OK?” is true or false. Similarly,
we may not assert that “How beautiful is the morning sky!” is true.
Our building blocks here are propositions; we will not try to go beyond
them. Itisnotourconcerntodeterminewhether‘eachbachelorismarried’,
1
2 Propositional Logic
for we do not know the meanings of the words uttered in the proposition.
This is so because our units here are propositions and nothing less (or
more). However, we seem to know that two propositions such as “I know
logic” and “You know logic” can be composed to get another proposition
such as “I and you know logic”. We are only interested in propositions
and how they are composed to yield other propositions. This is what we
mean when we say that propositions are our building blocks. Thus we are
interested in the forms rather than the meanings of propositions. Since
propositions can be true or false, we must know how to assign truth values
to the compound propositions.
If indeed I like logic and you like logic, then we must agree that the
proposition“Iandyoulikelogic”istrue. Butwhatabouttheproposition“I
likelogicandyoulikelogicoryoudonotlikelogic”? Thisisproblematic,for
wedonotknowexactlyhowthiscompoundpropositionhasbeencomposed
of or formed. We do not know which way to parse it:
(I like logic and you like logic) or (you do not like logic)
OR
(I like logic) and (you like logic or you do not like logic)
We must use some such device, say, use of parentheses for disambiguat-
ing compound propositions. Further, we must know the connectives like
‘and’, ‘or’, as used above. It is enough to start with some commonly used
connectives. If need arises, we will enrich our formalization later by adding
new ones. Furthermore, we have the responsibility to say what ‘follows
from’ means formally.
1.2 Syntax of PL
For any simple proposition, called a propositional variable, we will use
any of the symbols p ,p ,... For connectives ‘not’, ‘and’, ‘or’, ‘if ... then
0 1
...’, ‘if and only if’, we use the symbols ¬,∧,∨,→,↔, respectively; their
names are negation, conjunction, disjunction, conditional, biconditional.
We use the parentheses ‘)’ and ‘(’ as punctuation marks. We also have
the special propositions > and ⊥, called propositional constants, which
standforpropositionswhichare‘true’and‘false’,respectively. Bothpropo-
sitional variables and propositional constants are commonly called atomic
propositions. So, the alphabet of Propositional Logic, PL, is the set
{),(,¬,∧,∨,→,↔,>,⊥,p ,p ,p ,...}
0 1 2
consisting of all these symbols. Any expression over this alphabet is a
string of symbols such as
(¬p → ()∧p ∨, ¬p )(→ ∨, (¬p → p )
0 1 100 0 1
Only the last one of these is a propositional formula or a proposition. In
Syntax of PL 3
fact, we are interested only in such expressions. Now how to define them?
Here are some of the ways of defining the set of all propositions, PROP.
Definition 1.1 The set of all propositions, PROP is the smallest set
of expressions such that {>,⊥,p ,p ,...} ⊆ PROP, and whenever x,y ∈
0 1
PROP, the expressions ¬x,(x ∧ y),(x ∨ y),(x → y),(x ↔ y) are also in
PROP. The ‘smallest’ is in the sense of the relation ⊆, i.e., A is smaller
than B iff (if and only if) A ⊆ B.
The set PROP can also be defined inductively (or recursively).
Definition 1.2
PROP(0) = {p ,p ,...}∪{>,⊥}
0 1
PROP(i+1) = PROP(i)∪{¬x,(x∧y),(x∨y),(x → y),
(x ↔ y) : x,y ∈ PROP(i)}
PROP = ∪i∈N PROP(i)
Can you see that both the definitions above define the same set PROP?
What are the sets PROP(i)?
PROP(0)containsallatomicpropositionswhichdonotcontainanycon-
nective. PROP(1) contains all propositions having a single occurrence of a
connective, along with those having no connectives. In general, PROP(i)
contains all propositions having i or less number of occurrences of connec-
tives. It is of course clear that any proposition has only a finite number
of occurrences of connectives,and thus, it must be in some PROP(i). Now,
can you prove it? What is there to prove? Well, if PROP(i) has been
defined by Definition 1.2, PROP has been defined by Definition 1.1, and
if PROP0 = ∪i∈N PROP(i), then you must show that PROP ⊆ PROP0 and
PROP0 ⊆ PROP. Do the following exercise before reading further.
Exercise 1.1 Show that PROP = PROP0.
We see that {>,⊥,p ,p ,...} ⊆ PROP(0) ⊆ PROP0. For x,y ∈ PROP0,
0 1
we also see that x ∈ PROP(i) and y ∈ PROP(j) for some i,j ∈ N. Then,
bothx,y ∈PROP(k),wherek = max(i,j). Thisshowsthattheexpressions
¬x,(x∧y),(x∨y),(x → y),(x ↔ y) are in PROP(k+1). Since PROP(k+1)
⊆PROP0,wheneverx,y ∈PROP0,allof¬x,(x∧y),(x∨y),(x → y),(x ↔ y)
are also in PROP0. Since PROP is the smallest set of expressions with this
property, we have PROP ⊆ PROP0.
To see the converse, it is enough to see that PROP(i) ⊆ PROP for
each i ∈ N. This is proved by induction on i. To start with, we have al-
readyseenthatPROP(0) ⊆PROP.Assumethat(theinductionhypothesis)
PROP(i) ⊆ PROP. Now let w ∈ PROP(i+1). By Definition 1.2, w is one
of ¬x,(x∧y),(x∨y),(x → y),(x ↔ y), for some x,y ∈ PROP(i). Due to
the induction hypothesis, x,y ∈ PROP. By Definition 1.1, the expressions
¬x,(x ∧ y),(x ∨ y),(x → y),(x ↔ y) are in PROP. That is, w ∈PROP,
proving that PROP(i+1) ⊆ PROP. This completes the induction proof.
We summarize the above discussion in the following statement.