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Ramprasad Joshi, September 12, 2013 Header - p. 1
Logic in Computer Science
CS/IS F214 Handout, Slides, and Notes
Ramprasad Joshi
BITS, Pilani - K. K. Birla Goa Campus
http://www.bits-goa.ac.in/CSIS/CSIS.htm
mailto:[email protected]://www.bits-goa.ac.in/CSIS/CSIS.htmhttp://www.bits-goa.ac.in/CSIS/CSIS.htmmailto:[email protected]
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Handout Part II
Handout
Introduction
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 2
Handout Part II
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Handout Part II
Handout
Introduction
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 3
Handout-II
In addition to Part -I (General Handout for all courses in
theBulletin) this portion gives further details pertaining to
thecourse.
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Handout Part II
Introduction
Puzzles?
Puzzles?
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 4
Introduction
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Handout Part II
Introduction
Puzzles?
Puzzles?
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 5
Can you make sense of this?-I
All men are mortal. Socrates is a man. So Socrates ismortal.
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Handout Part II
Introduction
Puzzles?
Puzzles?
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 5
Can you make sense of this?-I
All men are mortal. Socrates is a man. So Socrates ismortal.
A mortal must die some time. So must Socrates.
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Handout Part II
Introduction
Puzzles?
Puzzles?
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 5
Can you make sense of this?-I
All men are mortal. Socrates is a man. So Socrates ismortal.
A mortal must die some time. So must Socrates.
All men are mortal. Can we say anything about womenbeing mortal
or not from this?
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Handout Part II
Introduction
Puzzles?
Puzzles?
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 5
Can you make sense of this?-I
All men are mortal. Socrates is a man. So Socrates ismortal.
A mortal must die some time. So must Socrates.
All men are mortal. Can we say anything about womenbeing mortal
or not from this?
Belgaumskis are slard after they are teahed. If they arelooced,
they become hoft. Tawer psees from them if you puttheiwgh on them
while they are hoft. Chikkodiovs areBelgaumskis. Tuppappa is a
Chikkodiov having tonten onhim while being looced. If tonten is a
theiwgh, will Tuppappapsee tawer?
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Handout Part II
Introduction
Puzzles?
Puzzles?
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 6
Can you make sense of this?-II
Naistib must take his cat, hen, and grain-sack across theriver
in a boat that can carry at most one of the three alongwith him.
The cat is looking for the first opprtunity when
Naistib is not around to eat the hen. The hen is
constantlytrying to peck the sack and he is constantly shooing it
away.Naistib does not want to lose his grain or poultry. Can
hecross the river with them and the cat?
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Handout Part II
Introduction
Puzzles?
Puzzles?
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 6
Can you make sense of this?-II
Naistib must take his cat, hen, and grain-sack across theriver
in a boat that can carry at most one of the three alongwith him.
The cat is looking for the first opprtunity when
Naistib is not around to eat the hen. The hen is
constantlytrying to peck the sack and he is constantly shooing it
away.Naistib does not want to lose his grain or poultry. Can
hecross the river with them and the cat?
... what the king fundamentally insisted upon was that
hisauthority should be respected. He tolerated nodisobedience. He
was an absolute monarch. But, becausehe was a very good man, he
made his orders reasonable.
If I ordered a general, he would say, by way of example, if
Iordered a general to change himself into a sea bird, and ifthe
general did not obey me, that would not be the fault ofthe general.
It would be my fault.
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Handout Part II
Introduction
Puzzles?
Puzzles?
Logic and Common Sense
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 6
Can you make sense of this?-II
Naistib must take his cat, hen, and grain-sack across theriver
in a boat that can carry at most one of the three alongwith him.
The cat is looking for the first opprtunity when
Naistib is not around to eat the hen. The hen is
constantlytrying to peck the sack and he is constantly shooing it
away.Naistib does not want to lose his grain or poultry. Can
hecross the river with them and the cat?
... what the king fundamentally insisted upon was that
hisauthority should be respected. He tolerated nodisobedience. He
was an absolute monarch. But, becausehe was a very good man, he
made his orders reasonable.
If I ordered a general, he would say, by way of example, if
Iordered a general to change himself into a sea bird, and ifthe
general did not obey me, that would not be the fault ofthe general.
It would be my fault.
What is common between all these quizzical paragraphs?
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 7
Logic and Common Sense
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 8
The Law of the Excluded Middle
In symbolic logic, it is assumed that there are two
distinctvalues, and no more.
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 8
The Law of the Excluded Middle
In symbolic logic, it is assumed that there are two
distinctvalues, and no more.
Each symbol takes either one or the other of the two values.
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 8
The Law of the Excluded Middle
In symbolic logic, it is assumed that there are two
distinctvalues, and no more.
Each symbol takes either one or the other of the two values.
No symbol can take more than one value, nor can it take anyother
value than the designated two values.
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 8
The Law of the Excluded Middle
In symbolic logic, it is assumed that there are two
distinctvalues, and no more.
Each symbol takes either one or the other of the two values.
No symbol can take more than one value, nor can it take anyother
value than the designated two values.
Each symbol must take one of the two values; it cannot
beindeterminate.
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 8
The Law of the Excluded Middle
In symbolic logic, it is assumed that there are two
distinctvalues, and no more.
Each symbol takes either one or the other of the two values.
No symbol can take more than one value, nor can it take anyother
value than the designated two values.
Each symbol must take one of the two values; it cannot
beindeterminate.
In sum: each statement is either true or false; it cannot
beboth; it cannot be none.
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 8
The Law of the Excluded Middle
In symbolic logic, it is assumed that there are two
distinctvalues, and no more.
Each symbol takes either one or the other of the two values.
No symbol can take more than one value, nor can it take anyother
value than the designated two values.
Each symbol must take one of the two values; it cannot
beindeterminate.
In sum: each statement is either true or false; it cannot
beboth; it cannot be none.
It is important to understand the implication: we cannot,
forinstance, talk about a photon being a particle and a wave
atonce. They must be two difference entities, or they mustreside in
two different worlds.
C S
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 9
Common Sense
Wrote Tom Paine: One of the strongest NATURAL proofs ofthe folly
of hereditary right in kings, is, that naturedisapproves it,
otherwise she would not so frequently turn it
into ridicule by giving mankind an ASS FOR A LION.
C S
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 9
Common Sense
Wrote Tom Paine: One of the strongest NATURAL proofs ofthe folly
of hereditary right in kings, is, that naturedisapproves it,
otherwise she would not so frequently turn it
into ridicule by giving mankind an ASS FOR A LION. Here, words
like natural proof, disapproves, frequently,
etc. are telltale signs that this is a sentence in commonsense
(indeed, in a pamphlet entitled Common Sense), but
not in formal sense.
C S
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Handout Part II
Introduction
Logic and Common Sense
Aristotle
Tom Paine
Formal Languages
Ramprasad Joshi, September 12, 2013 Header - p. 9
Common Sense
Wrote Tom Paine: One of the strongest NATURAL proofs ofthe folly
of hereditary right in kings, is, that naturedisapproves it,
otherwise she would not so frequently turn it
into ridicule by giving mankind an ASS FOR A LION. Here, words
like natural proof, disapproves, frequently,
etc. are telltale signs that this is a sentence in commonsense
(indeed, in a pamphlet entitled Common Sense), but
not in formal sense. Here is an answer to the biggest problem of
Computer
Science of today ;-) P = NP iff N = 1 P = 0
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Handout Part II
Introduction
Logic and Common Sense
Formal LanguagesSymbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proofThe first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 10
Formal Languages
The Players
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Handout Part II
Introduction
Logic and Common Sense
Formal LanguagesSymbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proofThe first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 11
The Players
Symbols are letters A . . . Z , a . . . z and their
extensionsA1, A2, . . . , B1, B2, . . . , . . . , Z 1, Z2, . . . ,
. . . , . . .
The Players
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Handout Part II
Introduction
Logic and Common Sense
Formal LanguagesSymbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proofThe first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 11
The Players
Symbols are letters A . . . Z , a . . . z and their
extensionsA1, A2, . . . , B1, B2, . . . , . . . , Z 1, Z2, . . . ,
. . . , . . .
T and F are special symbols, in that all other symbols are
equivalent to either one of them.
The Players
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Handout Part II
Introduction
Logic and Common Sense
Formal LanguagesSymbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proofThe first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 11
The Players
Symbols are letters A . . . Z , a . . . z and their
extensionsA1, A2, . . . , B1, B2, . . . , . . . , Z 1, Z2, . . . ,
. . . , . . .
T and F are special symbols, in that all other symbols are
equivalent to either one of them. Operators are or,and,not.
The Players
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Handout Part II
Introduction
Logic and Common Sense
Formal LanguagesSymbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 11
The Players
Symbols are letters A . . . Z , a . . . z and their
extensionsA1, A2, . . . , B1, B2, . . . , . . . , Z 1, Z2, . . . ,
. . . , . . .
T and F are special symbols, in that all other symbols are
equivalent to either one of them. Operators are or,and,not.
Other interesting operators (called syntactic sugar) are:implies
, impliedby , equivalent , nor , nand ,xor.
The Rules of the Game
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 12
The Rules of the Game
Buchstaben (Alphabet)by Hermann Hessetranslation by Richard S.
Ellis
We now and then take pen in handAnd make some marks on empty
paper.Just what they say, all understand.It is a game with rules
that matter.
The Rules of the Game
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 12
The Rules of the Game
Buchstaben (Alphabet)by Hermann Hessetranslation by Richard S.
Ellis
We now and then take pen in handAnd make some marks on empty
paper.Just what they say, all understand.It is a game with rules
that matter.
formula ::= atomformula ::= formulaformula ::= (formula)formula
::= formula op formulaatom ::= a|b| . . . |a
1|a2
| . . . |b1
| . . . |p for any p Pop ::= | | | | | | |
Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 13
Structural Induction
P is all atoms. F is all formulas, including atoms. Anassignment
is a mapping : P {T, F}. An interpretationis an extension of such
an assignment to : F {T, F}
Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 13
Structural Induction
P is all atoms. F is all formulas, including atoms. Anassignment
is a mapping : P {T, F}. An interpretationis an extension of such
an assignment to : F {T, F}
Each formula is either an atom, a negation of a formula, or
acomposite of two formulas joined by an operator.
Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proofThe first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 13
Structural Induction
P is all atoms. F is all formulas, including atoms. Anassignment
is a mapping : P {T, F}. An interpretationis an extension of such
an assignment to : F {T, F}
Each formula is either an atom, a negation of a formula, or
acomposite of two formulas joined by an operator.
To show a property(A) for any A U F, it is sufficient toshow
that:
Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 13
Structural Induction
P is all atoms. F is all formulas, including atoms. Anassignment
is a mapping : P {T, F}. An interpretationis an extension of such
an assignment to : F {T, F}
Each formula is either an atom, a negation of a formula, or
acomposite of two formulas joined by an operator.
To show a property(A) for any A U F, it is sufficient toshow
that:
1. For any atom p U, property(p) holds.
Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 13
Structural Induction
P is all atoms. F is all formulas, including atoms. Anassignment
is a mapping : P {T, F}. An interpretationis an extension of such
an assignment to : F {T, F}
Each formula is either an atom, a negation of a formula, or
acomposite of two formulas joined by an operator.
To show a property(A) for any A U F, it is sufficient toshow
that:
1. For any atom p U, property(p) holds. 2. If for any A U,
property(A) holds and A U, then,
necessarily, property(A) can be shown.
Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 13
P is all atoms. F is all formulas, including atoms. Anassignment
is a mapping : P {T, F}. An interpretationis an extension of such
an assignment to : F {T, F}
Each formula is either an atom, a negation of a formula, or
acomposite of two formulas joined by an operator.
To show a property(A) for any A U F, it is sufficient toshow
that:
1. For any atom p U, property(p) holds. 2. If for any A U,
property(A) holds and A U, then,
necessarily, property(A) can be shown.
3. If for any A, B U, property(A) and property(B) hold,then,
necessarily, whenever A B U for each {, , , , , , , }, property(A
B) can be shown.
Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 13
P is all atoms. F is all formulas, including atoms. Anassignment
is a mapping : P {T, F}. An interpretationis an extension of such
an assignment to : F {T, F}
Each formula is either an atom, a negation of a formula, or
acomposite of two formulas joined by an operator.
To show a property(A) for any A U F, it is sufficient toshow
that:
1. For any atom p U, property(p) holds. 2. If for any A U,
property(A) holds and A U, then,
necessarily, property(A) can be shown.
3. If for any A, B U, property(A) and property(B) hold,
then, necessarily, whenever A B U for each {, , , , , , , },
property(A B) can be shown.
Principal Operator: The operator at the root of the parse treeof
a formula.
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Satisfiability, Validity, Consequence
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 15
y y q
A propositional formula A is satisfiable iff there is
aninterpretation such that (A) = T. Such a is called amodel for A.
A is valid, or a tautology, denoted |= A, iff
(A) = T for all interpretations .
Satisfiability, Validity, Consequence
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 15
A propositional formula A is satisfiable iff there is
aninterpretation such that (A) = T. Such a is called amodel for A.
A is valid, or a tautology, denoted |= A, iff
(A) = T for all interpretations . A propositional formula A is
unsatisfiableor contradictory or
invalid, iff it is not satisfiable. A is not-valid or
falsifiable,denoted |= A iff it is not valid.
The Universe of Formulas
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 16
Each formula can be negated by just putting the negationoperator
before it. Thus each valid formula has a negationthat is a
contradiction, and vice versa. All formulas that are
satisfiable as well as falsifiable will form a middle space,each
such formula has its negation also in this middle space.We can show
the space of formulas divided up like thesymmetric figure in Figure
1:
Valid
Satisfia
bleas
well
asfalsifi
able
Satisfiable
= Not Invalid
Falsifiable
= Not valid
Contradictory
Invalid
Figure 1: The Universe of Formulas
Structural Induction Example
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 17
Define binary trees as follows: 1. is a binary tree. 2. If Land
R are binary trees, then (L, R) is a binary tree. 3.Nothing else
except one that satisfies 1 or 2 above is a
binary tree.
Structural Induction Example
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 17
Define binary trees as follows: 1. is a binary tree. 2. If Land
R are binary trees, then (L, R) is a binary tree. 3.Nothing else
except one that satisfies 1 or 2 above is a
binary tree. Let the set of binary trees be B. Define the height
function
H : B N as follows: H() = 0;H((L, R)) = max(H(L), H(R)) + 1.
Structural Induction Example
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 17
Define binary trees as follows: 1. is a binary tree. 2. If Land
R are binary trees, then (L, R) is a binary tree. 3.Nothing else
except one that satisfies 1 or 2 above is a
binary tree. Let the set of binary trees be B. Define the height
function
H : B N as follows: H() = 0;H((L, R)) = max(H(L), H(R)) + 1.
Similarly define the size function S : B N as follows:S() = 0;
S((L, R)) = S(L) + S(R) + 1.
Structural Induction Example
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 17
Define binary trees as follows: 1. is a binary tree. 2. If Land
R are binary trees, then (L, R) is a binary tree. 3.Nothing else
except one that satisfies 1 or 2 above is a
binary tree. Let the set of binary trees be B. Define the height
function
H : B N as follows: H() = 0;H((L, R)) = max(H(L), H(R)) + 1.
Similarly define the size function S : B N as follows:S() = 0;
S((L, R)) = S(L) + S(R) + 1.
To Prove:
H(T) S(T) 2H(T) 1 (1)
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 18
H() = 0; H((L, R)) = max(H(L), H(R)) + 1.
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 18
H() = 0; H((L, R)) = max(H(L), H(R)) + 1.
S() = 0; S((L, R)) = S(L) + S(R) + 1.
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 18
H() = 0; H((L, R)) = max(H(L), H(R)) + 1.
S() = 0; S((L, R)) = S(L) + S(R) + 1.
To Prove: T = H(T) + 1 S(T) 2H(T) 1
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 18
H() = 0; H((L, R)) = max(H(L), H(R)) + 1.
S() = 0; S((L, R)) = S(L) + S(R) + 1.
To Prove: T = H(T) + 1 S(T) 2H(T) 1
Proof: By Structural Induction. Base Case: For T0 = it
istrivially true; or for T1 = (, ),H(T1) = max(0, 0) + 1 = 0 + 1 =
1 andS(T1) = 0 + 0 + 1 = 1, hence the result is true.
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 18
H() = 0; H((L, R)) = max(H(L), H(R)) + 1.
S() = 0; S((L, R)) = S(L) + S(R) + 1.
To Prove: T = H(T) + 1 S(T) 2H(T) 1
Proof: By Structural Induction. Base Case: For T0 = it
istrivially true; or for T1 = (, ),H(T1) = max(0, 0) + 1 = 0 + 1 =
1 andS(T1) = 0 + 0 + 1 = 1, hence the result is true.
Induction Hypothesis: Assume that for T = Tk, 1 k h,such that
H(Tk) = k, the result is true.
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 19
To Prove: H(T) S(T) 2H(T) 1
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 19
To Prove: H(T) S(T) 2H(T) 1
Proof: By Structural Induction. Base Case: For T0 = orT1 = (, ),
the result is true.
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 19
To Prove: H(T) S(T) 2H(T) 1
Proof: By Structural Induction. Base Case: For T0 = orT1 = (, ),
the result is true.
Induction Hypothesis: Assume that for T = Tk, 1 k h,such that
H(Tk) = k, the result is true.
S.I. contd...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 19
To Prove: H(T) S(T) 2H(T) 1
Proof: By Structural Induction. Base Case: For T0 = orT1 = (, ),
the result is true.
Induction Hypothesis: Assume that for T = Tk, 1 k h,such that
H(Tk) = k, the result is true.
Induction Step: Let T = Th+1, with H(Th+1) = h + 1. Sinceh >
0, = Th+1 = (L, R). h + 1 = max(H(L), H(R)) + 1, H(L) h and H(R) h;
and, H(L) = h H(R) = h. LetH(L) = h and 0 H(R) h. By the I.H., h
S(L) 2h 1
and 0 S(R) 2h 1. Now S(Th+1) = S(L) + S(R) + 1.Combining with
the inequalities just proved,
h + 0 + 1 S(Th+1) 2h
1 + 2h
1 + 1, hence(h + 1) S(Th+1) 2
(h+1) 1. Thus assuming the result forTk, 1 k h leads to the
result for Th+1.
More Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 20
Let us take a subset C B: T C T = (T = (L, R) ({L, R} C) (|H(L)
H(R)| 1)). Such trees arehistorically called AVL trees, but if that
makes you skitter
about frantically to google up acronyms, confusing you moreand
leading to a stampede and a scare, we might as well callthem LVA
trees or Yranib trees. The name does not matter, arose with the
name esor would also smell like a rose andlook like a rose.
More Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 20
Let us take a subset C B: T C T = (T = (L, R) ({L, R} C) (|H(L)
H(R)| 1)). Such trees arehistorically called AVL trees, but if that
makes you skitter
about frantically to google up acronyms, confusing you moreand
leading to a stampede and a scare, we might as well callthem LVA
trees or Yranib trees. The name does not matter, arose with the
name esor would also smell like a rose andlook like a rose.
To prove: T C H(T) c log(S(T) + 1) for some c > 0,not
dependent on H(T) (or for that matter not dependent anyother
quantity related to T; dependent only on the fact that Tis in
C).
More Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 20
Let us take a subset C B: T C T = (T = (L, R) ({L, R} C) (|H(L)
H(R)| 1)). Such trees arehistorically called AVL trees, but if that
makes you skitterabout frantically to google up acronyms, confusing
you moreand leading to a stampede and a scare, we might as well
callthem LVA trees or Yranib trees. The name does not matter, arose
with the name esor would also smell like a rose andlook like a
rose.
To prove: T C H(T) c log(S(T) + 1) for some c > 0,not
dependent on H(T) (or for that matter not dependent anyother
quantity related to T; dependent only on the fact that Tis in
C).
Proof: A little harder, because c is not known. So whats
thetrick now?
More Structural Induction
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 20
Let us take a subset C B: T C T = (T = (L, R) ({L, R} C) (|H(L)
H(R)| 1)). Such trees arehistorically called AVL trees, but if that
makes you skitterabout frantically to google up acronyms, confusing
you moreand leading to a stampede and a scare, we might as well
callthem LVA trees or Yranib trees. The name does not matter, arose
with the name esor would also smell like a rose andlook like a
rose.
To prove: T C H(T) c log(S(T) + 1) for some c > 0,not
dependent on H(T) (or for that matter not dependent anyother
quantity related to T; dependent only on the fact that Tis in
C).
Proof: A little harder, because c is not known. So whats
thetrick now?
For the base case, take or (, ). Any one will do. Whatnext?
More S. I. ...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 21
I. H. : For Tk C, 0 k n, such that S(Tk) = k,H(Tk) c log(k +
1).
More S. I. ...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
Symbols
Syntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 21
I. H. : For Tk C, 0 k n, such that S(Tk) = k,H(Tk) c log(k +
1).
I. S. : Tn+1 = (L, R) such that S(Tn+1) = n + 1. Let
H(Tn+1) = h. Now Tn+1 C B, max(H(L), H(R)) = H(Tn+1) 1 = h 1, as
alsomin(H(L), H(R)) H(Tn+1) 2 = h 2. Nown + 1 max(S(L), S(R)) + 1
hence the I.H. applies to bothL, R. Let hL, hR be the heights and
sL, sR be the sizes ofL, R respectively.
More S. I. ...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
SymbolsSyntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 21
I. H. : For Tk C, 0 k n, such that S(Tk) = k,H(Tk) c log(k +
1).
I. S. : Tn+1 = (L, R) such that S(Tn+1) = n + 1. Let
H(Tn+1) = h. Now Tn+1 C B, max(H(L), H(R)) = H(Tn+1) 1 = h 1, as
alsomin(H(L), H(R)) H(Tn+1) 2 = h 2. Nown + 1 max(S(L), S(R)) + 1
hence the I.H. applies to bothL, R. Let hL, hR be the heights and
sL, sR be the sizes ofL, R respectively.
Thus h 2 min(hL, hR) max(hL, hR) = h 1. By I.H.h 2 c log(min(sL,
sR) + 1) andh 1 c log(max(sL, sR) + 1).
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More S. I. ...
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Handout Part II
Introduction
Logic and Common Sense
Formal Languages
SymbolsSyntax
Syntax
Interpretation
Interpretation
Symmetry
A Proof
The first proof
The first proof
Another Proof
The second proof
Ramprasad Joshi, September 12, 2013 Header - p. 21
I. H. : For Tk C, 0 k n, such that S(Tk) = k,H(Tk) c log(k +
1).
I. S. : Tn+1 = (L, R) such that S(Tn+1) = n + 1. Let
H(Tn+1) = h. Now Tn+1 C B, max(H(L), H(R)) = H(Tn+1) 1 = h 1, as
alsomin(H(L), H(R)) H(Tn+1) 2 = h 2. Nown + 1 max(S(L), S(R)) + 1
hence the I.H. applies to bothL, R. Let hL, hR be the heights and
sL, sR be the sizes ofL, R respectively.
Thus h 2 min(hL, hR) max(hL, hR) = h 1. By I.H.h 2 c log(min(sL,
sR) + 1) andh 1 c log(max(sL, sR) + 1).
min(sL, sR) 2h2
c 1 and max(sL, sR) 2h1
c .
But 2 min(sL, sR) + 1 n + 1 2 max(sL, sR) + 1. Simplealgebra
gives h c log(n + 2) c + 2 h + 1. This proves
the result for any c 2, independent of n.
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