Terminology

Formal languages are our models for the data manipulated by computers.

• A symbol (or letter) is an undefined term.

• An alphabet is a nonempty, finite set of symbols, e.g., if Σ = {a, b} then Σ is the alphabet while a and b are symbols.

• A string (word, sentence) is a finite list of symbols chosen from an alphabet, e.g., ⟨a, b, a⟩, usually written aba.

Terminology

• The length of string w, denoted ∣w∣, is the length of the list.

• The empty string ɛ has length 0.

• Formal language theory allows infinite-length strings; we don't.

• If ∣w∣ = n, we write w as w₁w₂…w_n. E.g., letting w =  aba, we have w₁ = w₃ =  a, and w₂ =  b.

• For any string w and symbol a, we write ∣w∣_a to denote the number of times the symbol a occurs in string w. E.g.,

  • |aba|_a = 2

  • |aba|_b = 1

  • |aba|_c = 0

Terminology

• Let x = x₁x₂…x_m and y = y₁y₂…y_n be strings. The concatenation of x and y, written xy, is the string x₁x₂…x_my₁y₂…y_n of length m + n that results from appending y to the end of x, e.g., concatenating back and bone gives backbone.

• String concatenation operation is associative, and ɛ is the identity element, i.e., ɛw = wɛ = w for any string w.
  ∴  the set of all strings over an alphabet is a monoid under concatenation.

Terminology

• If w is a string and n is a positive integer, we write wⁿ to mean the concatenation of n copies of w. The notation w⁰ is defined to be ɛ.

• A string y is a substring (or subword) of string w if there exist strings x, z such that w = xyz.

• A string x is a prefix of string w if there exists a string y such that w = xy.

• A string y is a suffix of string w if there exists a string x such that w = xy.

• By definition,

  • an empty string is a substring, prefix, and suffix of any string

  • any string is a substring, prefix, and suffix of itself

Terminology

• String x is a subsequence of string y if x is obtained by striking out 0 or more symbols from y. E.g., bat is a subsequence of habitat.

• Let w = w₁w₂…w_n be a string of length n. By the reverse of w, notated w^R, we mean the string w_nw_n − 1…w₁. For example, star^R = rats.

• A string w is a palindrome if w^R = w. Examples of palindromes are eve, madam, racecar, deified, rotator.

• Given alphabet Σ, define Σ^* to be the set of all strings over Σ. E.g., if Σ = {a, b} then Σ^* = {ɛ, a, b, aa, ab, ba, bb, aaa, …}.

• The listing of strings above is in shortlex order (string order, radix order), i.e., ordered like in a dictionary, except that a shorter string always precedes a longer one.

Exercises

• Define precisely the less than relation < for dictionary order (lexicographic order).

• Define precisely the less than relation < for shortlex order (string order, radix order).

• What is the position of the string ab, when the strings of {a, b}^* are arranged in dictionary order? in shortlex order?

Languages

• A language over the alphabet Σ is any subset of Σ^*.

• Some example languages:

  1. The set of all strings with an odd number of a.

  2. The set of all palindromes.

  3. The set of all strings of "balanced" left and right parentheses.

  4. The set of all strings with equal numbers of a, b, and c.

  5. The set of all binary strings that represent prime numbers.

  6. The set of all graphs with a Hamiltonian cycle, where the graph is encoded as a string.

  7. ∅ and {ɛ} are different languages.

Remarks

• The subject matter of this course is languages and machines that recognize/compute them!

• Finite languages are trivial.

• A lone letter like a is ambiguous. It either represents a symbol or a string of length 1. Context decides which meaning is intended.

• The concepts of "string", "concatenation", "string length", "string reversal", etc., can be defined inductively.

Language Operations

• Set Operations:  ∪ ,  ∩ ,  \ , △, complement Ā of language A

• Concatenation: The concatenation of two languages A and B is AB, i.e., the set of all strings xy where x ∈ A and y ∈ B. When precision is desired, concatenation is denoted by  ∘ , e.g., x ∘ y, A ∘ B.

• Let O = {all strings of odd length}, E = {all strings of even length}, and N = {a}. Find ON, OE, and EE.

Answer:
    ON = { all strings of even length ending in a }  
    OE = O  
    EE = E

Language Operations

• Power: For any language A, language A⁰ denotes {ɛ}; languages Aⁱ denotes AA^i − 1 whenever i > 0.

• Kleene Closure: A^* = ⋃_i = 0^∞Aⁱ. E.g., ∅^* = {ɛ}. Note how this definition of ^* agrees nicely with our previous definition of ^* in Σ^* if we identify a string of length one with the symbol contained in it.

• Positive Closure: A⁺ = ⋃_i = 1^∞Aⁱ.

Exercises

• Is it true that A⁺ = A^* \ {ɛ} for every language A?.

• Which ones of the seven example languages satisfy A = A^*?

• Characterize languages A that satisfy A^* = A⁺?

• Describe these languages: A∅, A{ɛ}, A ∪ ∅, A ∪ {ɛ}.

• The  ∪  and the  ∘  operators for languages are comparable to the  +  and the  ×  operators for numbers, respectively.

  • What is the identity element for  ∪ ? for  ∘ ?

  • What rules governing  +  and  ×  are also obeyed by  ∪  and  ∘ ?