automata_worksheet3

Introduction to Finite-State Automata and Languages

Languages can be defined using either finite-state automata (FSA) or regular expressions, leading to the emergence of automaton regular languages and specification regular languages.
These two classes of languages are equivalent, as shown in upcoming discussions (Worksheet 4).
There exist languages that are not regular, which will be explored after an introduction to generating functions.

For any language ( L ), define ( c_L(n) = |{ w \in L : |w| = n }| ): the number of words in ( L ) of length ( n ).
The collection ( c_L(n) ) for ( n = 0, 1, 2, \ldots ) is referred to as the profile of the language.
The generating function for language ( L ) can be formed as:
- Generating Function: ( g_L(z) = \sum_{n=0}^{\infty} z^n c_L(n) )
- This function can be viewed as the structure generating function for language ( L ).

For ( L = { \varepsilon, a, aa, ab, bb } ) over alphabet ( { a, b } ):
- The generating function will be computed as part of Exercises 3.1 and 3.2.
When ( L ) consists of all possible words over ( A = { a, b } ), we know:
- ( c_L(n) = |A^n| = |A|^n = 2^n ). Thus, the generating function for this language:
- Generating Function: ( g_L(z) = \sum_{n=0}^{\infty} z^n 2^n = \frac{1}{1 - 2z} )

Let ( A ) be a finite-state automaton with states ( S ), accepting states ( S' ), input alphabet ( I ), and transition function ( T ).
The languages accepted by the automaton from an initial state ( i ) are given by ( L_i = L(A, i) ).
The generating function for the language accepted from state ( i ) can be defined as:
- ( g_i(z) = g_{L_i}(z) = \sum_{n=0}^{\infty} z^n c_{L_i}(n) ).
This relationship must satisfy a system of linear equations:
- Transition Matrix Representation:
  - Define ( M_{ij} = |{ \alpha \in I : T(i, \alpha) = j }| ): the number of inputs transitioning from state ( i ) to state ( j ).
  - Transition Matrix: M = (M_{ij}) for each state pair.
- The acceptance vector ( v_i ) is defined:
  - ( v_i = 1 ) if ( i \in S' ) or ( v_i = 0 ) otherwise.

Establishes a system of linear equations as follows:
- ( g_i(z) = v_i + z \sum_{j \in S} M_{ij} g_j(z) )
- Or in vector form:
  - ( g(z) = v + z M g(z) )

Applying Theorem 3.4, an example automaton yields its transition matrix ( M ) and acceptance vector ( v ).
The relationships found lead to specific equations for ( g_1(z), g_2(z), g_3(z) ).

Exercise 3.5: Prove ( \sum_{j \in S} M_{ij} = |I| ) to relate state transitions and inputs.
Exercise 3.6: Show how constants ( c_{\alpha L}(n) ) relate to ( c_L(n) ) for language transformations.
Exercise 3.7: For disjoint languages ( L_1 ) and ( L_2 ), establish:
- Generating Function: ( g_{L_1 \cup L_2}(z) = g_{L_1}(z) + g_{L2}(z) ).

Exercises 3.8: Derive the relationships from the acceptance of words starting with symbols and adjustments of the state transition processes.

Exercise 3.9 involves analyzing an automaton leading to the Fibonacci sequence.
Relates the generating function to a ratio of polynomials:
- Demonstrates that automaton-generating functions typically yield rational functions.

Theorem 3.10 states: The generating function for any finite-state automaton's accepted language is a ratio of polynomials.
If ( I - Mz ) is invertible for values of ( z ), this theorem holds.

Explore matrices and eigenvalues to analyze invertibility conditions in larger contexts.
Countability discussions show there are many languages that are not regular despite being constrained by rational polynomials.
Exercises pushing the understanding of language profiles, showing the existence of many irregular languages.

This worksheet effectively connects finite-state automata, generating functions, and language profiles, leading to broader implications in discrete mathematics regarding the nature of regular languages.