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The Algebra of Nondeterministic Finite Automata

Roberto Gorrieri

TL;DR

A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA), and a representability theorem is proved: for each NFA, there exists a process algebraic term such that its semantics is an NFA isomorphic to $N.

Abstract

A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA, for short). We prove a representability theorem: for each NFA $N$, there exists a process algebraic term $p$ such that its semantics is an NFA isomorphic to $N$. Moreover, we provide a concise axiomatization of language equivalence: two NFAs $N_1$ and $N_2$ recognize the same language if and only if the associated terms $p_1$ and $p_2$, respectively, can be equated by means of a set of axioms, comprising 7 axioms plus 3 conditional axioms, only.

The Algebra of Nondeterministic Finite Automata

TL;DR

A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA), and a representability theorem is proved: for each NFA, there exists a process algebraic term such that its semantics is an NFA isomorphic to $N.

Abstract

A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA, for short). We prove a representability theorem: for each NFA , there exists a process algebraic term such that its semantics is an NFA isomorphic to . Moreover, we provide a concise axiomatization of language equivalence: two NFAs and recognize the same language if and only if the associated terms and , respectively, can be equated by means of a set of axioms, comprising 7 axioms plus 3 conditional axioms, only.
Paper Structure (16 sections, 21 theorems, 4 equations, 6 figures, 7 tables)

This paper contains 16 sections, 21 theorems, 4 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Nondeterministic Finite Automata
  3. SFM1: Syntax and Semantics
  4. Syntax
  5. Denotational Semantics
  6. Congruence
  7. Algebraic Properties
  8. The Folding Law
  9. Equational Deduction with Process Constants
  10. Axiomatization
  11. The Axioms
  12. Normal Forms, Unique Solutions and $\epsilon$-free Normal Forms
  13. Deterministic Normal Form
  14. Completeness for Observationally Guarded Processes
  15. Completeness for Unguarded Processes
  16. ...and 1 more sections

Key Result

theorem 1

For each SFM1 process $p$, $\llbracket p \rrbracket_{\emptyset}$ is a reduced NFA. Proof. By induction on the definition of $\llbracket p \rrbracket_{I}$. Then, the thesis follows for $I = \emptyset$. The first base case is for $\hbox{\bf 0}$ and the thesis is obvious, as, for any $I \subseteq \math

Figures (6)

  • Figure 1: Graphical description of the representability theorem, up to language equivalence
  • Figure 2: Graphical description of the representability theorem, up to isomorphism
  • Figure 3: An NFA and a DFA both recognizing the language $a^*b^*$
  • Figure 4: The NFA for $C \doteq b.D$, where $D \doteq a.(b.D + \hbox{\bf 1})$, of Example \ref{['ex-den-sfm1']}
  • Figure 5: Two NFAs for Example \ref{['rep-ex']}
  • ...and 1 more figures

Theorems & Definitions (35)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • remark 1
  • definition 5
  • theorem 1
  • theorem 2
  • corollary 1
  • definition 6
  • ...and 25 more