Chapter 0
Mathematical Notions and Terminology
Alphabets and Strings
Topics
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Alphabets
Strings
Languages
Alphabets and Symbols
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Alphabet: any (nonempty) finite set
o Usually assume elements are some kind of characters
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Members of an alphabet are called symbols
o Symbols are usually letters and digits
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Example alphabets: {a}, {a, x}, {0}, {0, 1}
o Usually don't use quotes for characters (ie 'a')
o Usually use different font or context to recognize symbols
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Represent an alphabet with the Greek letters (eg Σ and Γ which are capital
Sigma and capital Gamma)
o Example: Σ = {0, 1}
o Example: let x be an element of Σ
Strings Over an Alphabet
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String: finite sequence of symbols over (ie from) an alphabet
o Symbols are usually written adjacent without commas
 Examples over Σ = {a,b,c,d}: aaa, bac
 Examples over Σ = {0,1}: 000, 01011
 acb is a shorthand for (a, c, b)
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Typically represent strings with letters like u, v, w, x, y, z
o Examples: u = acb, x = 10110
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Frequently the alphabet that strings are formed over is unspecifed and
assumed
o Example: 0 or more a's followed by a single b
Strings Operations
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We can define operation on strings
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The most common operations are
o Concatenation: Join two or more strings and/or symbols
 Example: x = cab, y = aba , xy = ??
 We freely concatenate any combination of strings and symbols
 Example: xay =
o
Superscript: xk
o
Reverse - Notation: wR
o
Length of string w:
 number of symbols in w
 written |w|
 |cab| = 3
The Empty String
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The string with no symbols
o If x is the empty string, then |x| = 0
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Conundrum: How do we write the empty string?
o Answer: A special symbol: ε
o Symbol λ also used
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Question: Can ε be a symbol in the alphabet we are using to form strings?
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How is ε represented in Ada and Java?
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Plays role of 0 or 1 in a number system
o wε = ?? = ??
o This is called ...
Lexicographic Ordering
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Need a standard order in which to list strings
Lexicographic ordering: shorter precede longer (eg {ε, 0, 1, 00, 01, 10, 11, 000,
...})
Assume symbols have an ordering
Languages
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Language: A set of strings (over an alphabet)
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Examples:
o Over alphabet {a, b, c}:
 L1 = {aaa, aba}
 L2 = {a, aa, aaa}
o
Over alphabet {0, 1}:
 L1 = {0}
 L2 = {0, 1, 010}
o
Frequently the alphabet that strings are formed over is unspecifed and
assumed
Empty Strings and Languages
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Are these two languages different? {00, 11} and {ε, 00, 11}
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Is this a language: {ε}
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Is this a language: {}
o Eventually we will see language specifications that turn out to define
empty languages, although that might not be obvious at first
o Notice the connection between the string (ie sequence of symbols) with
no symbols and the language (ie set of strings) with no elements
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Is this a valid alphabet: Σ = {}
o If it is, what are the strings over Σ?
o If it is, what are the languages over Σ?
o Is it valid? See page 13.
Infinite Languages
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Languages can be finite or infinite
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Infinite languages contain an infinite number of strings
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Example infinite languages:
o All binary strings that start with 0
 What's the shortest string in the language?
o
L = {wwR} | for any string w}
 What's wrong with this definition?
 What's the shortest string in the language?
 What language is this: Even length palindromes
 How do we specify odd length palindromes
Infinite Alphabets, Strings, and Languages
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Can an alphabet be infinite?
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Can a string be of infinite length?
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Look at infinite languages above: infinite number of finite strings
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In an infinite language, there is no limit to the length of a string, but the length
of each string is finite.
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Remember these definitions!!!
Tools for working with languages on Languages
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We construct 3 tools for working with languages:
o Automata:
o
Regular Expressions: Generate (describe) languages
o
Grammars: Generate languages
Some Questions when Working with Languages
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Each tool can be used to ask 2 questions:
o What strings are in the language?
o Is a given string in the language?
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