1 integration of tools

1.1 context free languages (CFL) - properties

• CFL is closed under union, concatenation, and Kleene closure
• CFL is not closed under intersection and complement

1.2 CFL \subset CSL

CSL’s normal form where \delta_1 = \delta_2 = \epsilon is a context free grammar.

1.3 Equivalence of classes

Equivalent are classes of finite automata, regular expressions, regular grammars

1.4 Myhill-Nerode Theorem

The following conditions are equivalent:

1. L \in \Sigma^* is accepted by some DFA
2. L is union of some equivalence classes of some equivalence relation \rho of finite index which is right-invariant¹
3. R_L has finite index

1.4.1 Proof

1.4.1.1 1. \implies 2.

Assume that A=(Q, \Sigma, \delta, q_0, F) is DFA and accepts L. Let x, y \in \Sigma^*, x \rho y \equiv \text{ computation for both ends in the same state }. It is an equivalence relation, has finite index, and is right-invariant.

1.4.1.2 2. \implies 3.

x \rho y \implies x R_L y. Each equivalence class of \rho is included in some equivalence class of R_L. If two languages are related then they are in the same equivalence class and thus are both inside or both outside of L.

1.4.1.3 3. \implies 1.

Assume that A=(Q, \Sigma, \delta, q_0, F) is DFA and accepts L.

• q_0 \equiv q_{[\epsilon]}
• F states labeled by classes included in the language
• \delta(q_{[\epsilon]}, a) = q_{[wa]} where w \in \Sigma^* and a \in \Sigma

1.5 inner operations in regular languages

• union, concatenation, Kleene closure
  • +, \circ, ^* - operations on regular expressions are the corresponding operations on languages
• complement, intersection

Let L_1 and L_2 be regular languages. Let A_k = (Q_k, \Sigma, \delta_k, q_0^k, F_k) be their automaton.

• \bar A_k = (Q_k, \Sigma, \delta_k, q_0^k, Q_k \backslash F_k) accepts \bar L_k.
• A_1 \times A_2 = (Q_1 \times Q_2, \Sigma, (\delta_1, \delta_2), (q_0^1, q_0^2), F_1 \times F_2) accepts L_1 \cap L_2

1.5.1 preposition

• The class of finite automata is equivalent to the class of regular expressions
• The class of finite automata is equivalent to the class of regular grammars

1.6 RgL \subsetneq CFL

• Finite automata are special cases of push down automata.
• Regular grammars are special cases of context free grammars.
• L = \text{same amount of 'a' as 'b'} - it is context free but not regular

1.7 CSL \subsetneq RkL

Let G = (V, T, P, S). Let V = \{v_1, \cdots, v_m\}, T = \{t_1, \cdots, t_n\}.