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.

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 ).

Example: Generating Function for a Specific Language

• 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} )

Finite-State Automata and Generating Functions

• 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.

Theorem 3.4: Generating Functions for Automata

• 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) )

Example Application of Theorem 3.4

• 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) ).

Exercises on Generating Functions

Key Concepts

• 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) ).

Proof Strategy for Theorem 3.4

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

Fibonacci and Automata

• 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.

Consequences of Theorem 3.10

• 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.

Exercises on Linear Algebra and Countability

• 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.

Summary

• 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.