Formal Languages

Phrase Structure Grammars
1. (V, A, <s>, P)
  1. Finite set V
  2. A subset A of V
  3. An element <s> of V\A
  4. Finite subset P ⊂ (V*\A*) x V*
    1. In comparison, for CFG, P ⊂ (V\A) x V*
2. A production rule in a PSG r → w has a LHS, r, that may contain more than one nonterminal. It is required to contain at least one nonterminal.
3. For example, if A = {0, 1} and <s> is the start symbol in a PSG, 0<s>0<s>0 → 00010 would be an acceptable production rule, but NOT in a CFG.
4. Noam Chomsky
Regular Languages
1. Introduced by Stephen Kleene in 1951. Sometimes known as a finite-state language, as a language is regular iff it can be recognised by an FSA.
2. The regular languages over A constitute the smallest collection, C, of the subsets of A*, satisfying that:
  1. 1. All finite subsets of A* belong to C.
    1. 2. C is closed under the Kleene star operation (if M⊆A* is inside C, i.e. if M ∈ C, then M* ∈ C)
      1. 3. C is closed under concatenation and union
3. Let A be a finite set, and let A* be the set of words over the alphabet A.
  1. A subset, L, of A*, is called a regular language over the alphabet A, if L = Lm for some finite sequence L1, L2, ..., Lm of subsets of A* with the property that ∀i, 1 ≤ i ≤ m, Li satisfies one of the following:
  2. Example
    1. Let A = {0, 1}. Let L = {0^m0^n | m, n ∈ N, m ≥ 0, n ≥ 0}.
      1. L is a regular language by way of these statements
      2. 0^m1^n stands for m 0's followed by n 1's i.e. 0^m ◦ 1^n
        So, let us examine:
        L' = {0^m | m ∈ N, m ≥ 0}
        L'' = {1^n | n ∈ N, n ≥ 0}
      3. L, however, is NOT a regular language if the power of 0 and 1 were the same. This language consists of strings of 0's, followed by an equal number of strings of 1's.
        For a machine to decide that the string is inside the language, it must store the number of 1's, as it examines the number of 0's or vice versa
        The number of strings of the type 0m1n is not finite, however, so a finite-state machine cannot recognise this language.
Finite State Acceptors
1. Components
  1. Set of states
    1. Initial state
    2. The subset F ⊆ S of finishing states
  2. Transition mapping [s' = t(s, a)]
  3. FSA = (S, F, i, t, A)
  4. Alphabet
2. Recognition and Acceptance
  1. A word, w, over alphabet A is recognised or accepted by the FSA if each letter, in order, is found in the FSA states
  2. A language, L, over the alphabet A, is said to be recognised or accepted by the FSA if L is the set consisting of all words recognised by the FSA
  3. Myhill-Nerode theorem
    1. Let L be a language over the alphabet A. If the set L/N of equivalence classes in L is infinite, then L is NOT a regular language.
3. Two Types:
  1. Deterministic
    1. every states has exactly one transition for each possible input (function)
  2. Non-deterministric
    1. an input can lead to one, more than one, or no transition for a given state (not a function)
Automata Theory
1. An automaton is a mathematical model of a computing device.
Regular Grammars
1. A context-free grammar is given by (V, A, <s>, P) is called a regular grammar if every production rule in P is of one of the three forms:
  1. 1. <A> → b<B>
  2. 2. <A> → b
  3. 3. <A> → ε
  4. Normal Form
    1. Lemma: Any language generated by a regular grammar may be generated by a regular grammar in normal form, if you replace rules of type 2 (<A> → b) with two rules, <A> → b<F> and <F> → ε.
  5. Left-Regular Grammar
  6. Right-Regular Grammar
    1. Rule: <A> → <B>b
2. Example
Regular Expressions
1. ∅, ε, and all elements of A are regular expressions
  1. ε = language of a single string (the empty string)
    1. ∅ = language that doesn't contain any strings
2. A language is regular ⇐⇒ some regular expression describes it
The Pumping Lemma
1. If L is a regular language, then there is a number p (the pumping length) where if w is any word in L of length at least p, then w = xuy for words x, y, and u satisfying:
  1. 1. u != ε
  2. 2.|xu| ≤ p
  3. 3. x u^n y ∈ L, ∀n ≥ 0
Basics
1. Let A be a finite set
  1. A = alphabet
    1. string in form a1, a2, ..., an = word
      1. A^n = set of all words of length n over the alphabet A
      2. A^+ = set of all words of positive length
      3. |w| = length of word
      4. w1◦w2 = concatenation
        Associative
        only concatenative if A has only one letter
      5. A^0 = empty word {ε}
      6. A* (Kleene star) = A^+ ∪ A^0
  2. Elements of A = letters
  3. L = language over A (subset of A*)
    1. L = formal language if ∃ a finite set of rules or algorithm that generates exactly L
Formal Grammars
1. A set of production rules for strings in a language. To generate such a language, we use:
  1. 1. the set A (alphabet)
  2. 2. a start symbol <s>
  3. 3. a set of production rules
2. Example
  1. A = {0, 1}
    1. <s>
      1. <s> → 0<s>1 <s> → 01
        all strings of form 0^m1^m
        terminals (strings of just elements of A i.e. 01)
        nonterminals (strings that contain elements which aren't part of A i.e. 00<s>11)
3. Context-Free Grammar (V, A, <s>, P)
  1. V: set of terminals and nonterminals, A: set of terminals
    1. P ⊂ (V\A) x V*
  2. Ambiguous: if same string can be generated in more than one way
  3. A language generated by a CFG is known as a context-free language.
4. Context-Sensitive Grammars

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