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Message complexity for unary multiautomata systems

Christian Choffrut

TL;DR

The paper analyzes unary multiautomata systems where multiple two-way automata operate synchronously on a unary input and can broadcast messages. It proves that if the total number of messages is bounded by a fixed integer $M$, the recognized languages are regular by encoding input lengths via Presburger formulas and by a detailed analysis of both single-automaton dynamics and inter-automaton communication through a finite set of first-order predicates and a global transition predicate $\Phi$. The main contributions include a constructive method to bound the time between broadcasting configurations, a formal reduction to Presburger definability, and the conclusion that bounded-message two-way multiautomata over unary inputs cannot exceed regular languages (with $M=0$ reducing to the classical two-way automata case). This clarifies the limits of distributed computation under tight communication constraints and connects automata theory with Presburger arithmetic in the unary setting.

Abstract

Finitely many two-way automata work independently and synchronously on a unary input. Some of their states are broadcasting, i.e., dispatched to all other automata. At each step of the computation, each automaton changes state and moves right, left or stay in place according to the current state and the possible messages dispatched. The input is recognized if the following occurs: starting from the initial configuration (the heads of all automata are positioned to the left end of the tape) one automaton reaches a final state when its head is positioned to the right end of the tape. We show that if the number of messages sent during the computation is bounded by some integer which is independent of the length of the input, then the language recognized is regular,

Message complexity for unary multiautomata systems

TL;DR

The paper analyzes unary multiautomata systems where multiple two-way automata operate synchronously on a unary input and can broadcast messages. It proves that if the total number of messages is bounded by a fixed integer , the recognized languages are regular by encoding input lengths via Presburger formulas and by a detailed analysis of both single-automaton dynamics and inter-automaton communication through a finite set of first-order predicates and a global transition predicate . The main contributions include a constructive method to bound the time between broadcasting configurations, a formal reduction to Presburger definability, and the conclusion that bounded-message two-way multiautomata over unary inputs cannot exceed regular languages (with reducing to the classical two-way automata case). This clarifies the limits of distributed computation under tight communication constraints and connects automata theory with Presburger arithmetic in the unary setting.

Abstract

Finitely many two-way automata work independently and synchronously on a unary input. Some of their states are broadcasting, i.e., dispatched to all other automata. At each step of the computation, each automaton changes state and moves right, left or stay in place according to the current state and the possible messages dispatched. The input is recognized if the following occurs: starting from the initial configuration (the heads of all automata are positioned to the left end of the tape) one automaton reaches a final state when its head is positioned to the right end of the tape. We show that if the number of messages sent during the computation is bounded by some integer which is independent of the length of the input, then the language recognized is regular,
Paper Structure (4 sections, 6 theorems, 27 equations, 5 figures)

This paper contains 4 sections, 6 theorems, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Premiminaries
  3. Dynamic of a single automaton
  4. Dynamic of the multiautomaton

Key Result

Theorem 1

For every integer $M$, the unary languages recognized by deterministic two-way multiautomata exchanging $M$ messages, are regular.

Figures (5)

  • Figure 1: $\lambda_{s}(s_{i_{1}})= \lambda_{s}(s_{i_{2}})= \lambda_{s}(s_{i_{3}})= \lambda_{s}(s_{i_{4}})= \lambda_{s}(s_{i_{5}})$. In the loop, the head passes $5$ times at position $p'$ (not $6$ times because it falls off before)
  • Figure 2: Illustration of RunRR
  • Figure 3: Illustration of RunRL
  • Figure 4: Illustration of RunLR
  • Figure 5: Illustration of RunLL

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 1 more