Strings

1
Strings

• A string over a set A is a finite sequence of
  elements from A.

• The set of elements from which the strings are
  built is called
• an alphabet.

Definition. An alphabet is a nonempty, finite set
of indivisible symbols. We are going to denote
it by ?.

• Any program is a string of keywords, variable
  names, and
• permissible symbols.

• A programming language should satisfy general
  rules (grammar)
• to be understood by computer (compiler). These
  rules are studied
• by the formal theory of programming languages.

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Definition. A string w over an alphabet ? is a
sequence of symbols, w a1 a2 an, where each
ai? ?, 1? i ? n.

• The number of symbols is called the length of
  the string, wn.
• There is one special string that has zero
  length (contains no symbols).
• It is called the empty string and has special
  notation , ?.
• w0 ? w ?.
• ? is not an element of any alphabet, ?? ?.

Example. Let ? a, bb, c. Find all possible
strings with length less or equal 3 built from
?.
Length 0 ?
Length 1 a bb c
Length 2 aa abb ac bba bbbb bbc ca cbb cc
Length 3 aaa aabb aac abba abbbb abbc
aca acbb
3

• Two strings u and v can be concatenated to form
  a single
• string uv, that consists of the symbols of
  string u, followed by
• symbols of of string v. The length uv
  u v .

• ?u u? u

• Concatenation is associative (uv)w u(vw),
• but not commutative uv ? vu (the order is
  important!).

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Definition. Any set of strings over some alphabet
is called a language.
Examples Set of all executable computer programs
is a language.
Alphabet itself is a language as well (the
language of all one-symbol words).
Since languages are sets, we can apply all set
operations to languages union, intersection and
set difference. There is one operation
specific for languages concatenation of two
languages L1?L2 uv u ? L1 and v ? L2
A L1?L2 ? L2?L1
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Example. Take the alphabet ? a, b, c.
Consider two languages over alphabet ? L1 a,
ab and L2 b, bc, c. Find L1?L2 and L2?L1.
We need to take every string from L1 and
concatenate with every string from L2 . In this
way we get L1?L2 strings ab, abc, ac,
abb, abbc, abc. Note, that not all strings are
distinct, like abc. L1?L2 ab, abc, ac, abb,
abbc .
In the same way L2?L1 ba, bab, bca, bcab,
ca, cab.
The cardinality L1?L2 is the number of
distinct strings, resulting from concatenation .
In general, L1?L2 ? L1 ? L2 and L1?L2
? L2?L1 In the example L1?L2 5lt
L1 ? L2 6.
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In particular, we can consider the concatenation
of an alphabet ? with itself ??? is the
language of all two-symbol words. Notation ???
?2
Example ?a, b, ??? ?2 aa, ab, ba, bb
Similarly, ?3 ?2??, the language that consists
of all 3-symbol words ?3 aaa, aba, baa, bba,
aab, abb, bab, bbb.
So, we can define recursively for any ngt1 ?n
?n-1??
To make this recursive definition agree with the
basis case n 1, ? ?0?? , zero power ?0 is
defined as ?0 ?, (no matter what is ? ).
Then ??? ? x x ? ? x x ? ? ?
What is ? ? ?2?
What is ? ? ?2 ? ?3 ? ? ?n ?
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Kleene star notation ? ?0 ? ?1 ? ?2 ?
So, ? is the (infinite) set of all possible
words over alphabet ?, including empty string ?.
Example. ? 0, 1. ? is an infinite set of all
possible bit strings. (or all binary numbers
including numbers with leading 0s and empty
string).
Any language L over alphabet ? is a subset of ?
, L ? ? .
Note that ??? , because ? ? ? ?1,
?0.
A language L may contain ? , or may not.
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Example. Consider two languages over alphabet ?
a L1aa, L2?, aa, aaaa. What is L1?
By definition of Kleene star L1 L10 ? L11 ?
L12 ? ??aa ?aaaa ?aaaaaa ?
?, aa, aaaa, aaaaaa, infinite set of
strings of even length build from symbol a.
What is L2?
L2 L20 ? L21 ? L22 ? ???, aa,
aaaa ??, aa, aaaa, aaaaaa, aaaaaaaa?
?, aa, aaaa, aaaaaa,
L1
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Definition. A string u is called a substring of v
if there exist two strings x and y, such that v
xuy, and x, y ? ?
Definition. A string u is called a prefix of v if
there exists a string x ? ?, such that v ux.
Similarly, a string u is called a suffix of v if
there exists a string y ? ?, such that v yu.
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Theorem 1. Let A, B and C be sets of strings.
Then (A?B)?C A?C?B?C
Proof. a) We need to prove the equality of two
sets of strings. We can do it by
double-inclusion, i. e. to show that i)
(A?B)?C ? A?C?B?C and ii) A?C?B?C ? (A?B)?C
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i) To prove (A?B)?C ? A?C?B?C, its suffices to
show that for any string w, w?(A?B)?C ?
w?A?C ?B?C
Take any w? (A?B)?C
(dfn of concat)
??x, y, such that w xy and x?(A?B) and y?C
? (x?A or x?B) and y ?C (dfn of ? )
? (x?A and y?C) or (x?B and y?C) (distributive
property)
? w? A?C or w?B?C
(dfn of concat)
? w? A?C ?B?C
(dfn of ? )
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ii) To prove that A?C?B?C ?(A?B)?C, we need to
show that for any string w, w? A?C?B?C ? w
?(A?B)?C
Take any w ? A?C ?B?C
? w? A?C or w? B?C (dfn of ?)
??x, y, such that w xy and (x ?A and y ?C) or
(x ?B and y ?C) (dfn of concat)
So we can have two cases. In the first case, (x
?A and y ?C) implies that (x ?A?B and y ?C)
because A ?(A?B).
In the second case, (x ?B and y ?C) implies that
(x ?A?B and y ?C) because B ?(A?B). So, in
either case we have
? w ?(A?B)?C (dfn of concat)
So, we proved A?C?B?C ?(A?B)?C and
(A?B)?C?A?C?B?C, that means (A?B)?C A?C?B?C
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Theorem 2. Let A, B and C be sets of strings.
Then (A?B)?C ? A?C?B?C
Proof. To prove subset relation we need to show
that for any string w, w?(A?B)?C ? w?A?C?B?C.
Why not to prove A?C?B?C ? (A?B)?C as well?
Lets try. Take arbitrary w?A?C?B?C ? w?A?C and
w?B?C .
?(?x, y, wxy, x?A and y?C) and (?u,v, wuv, u?B
and v?C)
Can we imply xyuv ? x u ?
No, because the same string abc may come from
a?bc and ab?c
Example. A a, B ab, C c, bc.
Then A?B, (A?B)?C.
A?Cac, abc
B?Cabc, abbc
abc ?A?C?B?C, but we can not imply that abc
?(A?B)?C