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Nondeterminism makes unary 1-limited automata concise

Bruno Guillon, Luca Prigioniero, Javad Taheri

TL;DR

This work probes the descriptional complexity of unary 1-limited automata and their common-guess variants, establishing strong separations between nondeterministic and deterministic models. By building unary witness languages from a full binary sequence and related constructions, it proves exponential lower bounds for simulating 2dfas+cg by 1-las and 2nfas, and a doubly-exponential lower bound for simulating 2dfas+cg by 1dfas, ultimately closing a unary case open question. The paper also demonstrates an exponential lower bound for complementing unary 2dfas+cg (and unary 1-las) and develops a sequence of tight results using a structured family of languages (IterSuffFullBinSeq_n and M_n). Together, these results clarify the role of nondeterminism in the size of descriptions for regular languages and show substantial gaps between nondeterministic and deterministic/one-way models in unary settings, with implications for the state-efficiency of simulations and determinization efforts.

Abstract

We investigate the descriptional complexity of different variants of 1-limited automata (1-las), an extension of two-way finite automata (2nfas) characterizing regular languages. In particular, we consider 2nfas with common-guess (2nfas+cg), which are 2nfas equipped with a new kind of nondeterminism that allows the device to initially annotate each input symbol, before performing a read-only computation over the resulting annotated word. Their deterministic counterparts, namely two-way deterministic finite automata with common-guess (2dfas+cg), still have a nondeterministic annotation phase and can be considered as a restriction of 1-las. We prove exponential lower bounds for the simulations of 2dfas+cg (and thus of 1-las) by deterministic 1-las and by 2nfas. These results are derived from a doubly exponential lower bound for the simulation of 2dfas+cg by one-way deterministic finite automata (1dfas). Our lower bounds are witnessed by unary languages, namely languages defined over a singleton alphabet. As a consequence, we close a question left open in [Pighizzini and Prigioniero. Limited automata and unary languages. Inf. Comput., 266:60-74], about the existence of a double exponential gap between 1-las and 1dfas in the unary case. Lastly, we prove an exponential lower bound for complementing unary 2dfas+cg (and thus unary 1-las).

Nondeterminism makes unary 1-limited automata concise

TL;DR

This work probes the descriptional complexity of unary 1-limited automata and their common-guess variants, establishing strong separations between nondeterministic and deterministic models. By building unary witness languages from a full binary sequence and related constructions, it proves exponential lower bounds for simulating 2dfas+cg by 1-las and 2nfas, and a doubly-exponential lower bound for simulating 2dfas+cg by 1dfas, ultimately closing a unary case open question. The paper also demonstrates an exponential lower bound for complementing unary 2dfas+cg (and unary 1-las) and develops a sequence of tight results using a structured family of languages (IterSuffFullBinSeq_n and M_n). Together, these results clarify the role of nondeterminism in the size of descriptions for regular languages and show substantial gaps between nondeterministic and deterministic/one-way models in unary settings, with implications for the state-efficiency of simulations and determinization efforts.

Abstract

We investigate the descriptional complexity of different variants of 1-limited automata (1-las), an extension of two-way finite automata (2nfas) characterizing regular languages. In particular, we consider 2nfas with common-guess (2nfas+cg), which are 2nfas equipped with a new kind of nondeterminism that allows the device to initially annotate each input symbol, before performing a read-only computation over the resulting annotated word. Their deterministic counterparts, namely two-way deterministic finite automata with common-guess (2dfas+cg), still have a nondeterministic annotation phase and can be considered as a restriction of 1-las. We prove exponential lower bounds for the simulations of 2dfas+cg (and thus of 1-las) by deterministic 1-las and by 2nfas. These results are derived from a doubly exponential lower bound for the simulation of 2dfas+cg by one-way deterministic finite automata (1dfas). Our lower bounds are witnessed by unary languages, namely languages defined over a singleton alphabet. As a consequence, we close a question left open in [Pighizzini and Prigioniero. Limited automata and unary languages. Inf. Comput., 266:60-74], about the existence of a double exponential gap between 1-las and 1dfas in the unary case. Lastly, we prove an exponential lower bound for complementing unary 2dfas+cg (and thus unary 1-las).

Paper Structure

This paper contains 17 sections, 22 theorems, 11 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our results.
  3. Outline.
  4. Preliminaries
  5. Common-guess.
  6. Size of models.
  7. ${\textsc{2}\xspace}{\textsc{n}{\textsc{fa}\xspace}\xspace}\xspace{\text{s}\xspace}\mathsf{+cg}$ underlying 2nfas.
  8. Relating ${\textsc{2}\xspace}{\textsc{n}{\textsc{fa}\xspace}\xspace}\xspace{\text{s}\xspace}\mathsf{+cg}$ and ${\textsc{2}\xspace}{\textsc{d}{\textsc{fa}\xspace}\xspace}\xspace{\text{s}\xspace}\mathsf{+cg}$ to 1/̄las.
  9. Full binary sequence
  10. Exponential gap from d1/̄las and ${\textsc{2}\xspace}{\textsc{d}{\textsc{fa}\xspace}\xspace}\xspace{\text{s}\xspace}\mathsf{+cg}$ to 2nfas
  11. Exponential gap from ${\textsc{2}\xspace}{\textsc{d}{\textsc{fa}\xspace}\xspace}\xspace{\text{s}\xspace}\mathsf{+cg}$ to d1/̄las and 2nfas
  12. A small ${\textsc{2}\xspace}{\textsc{d}{\textsc{fa}\xspace}\xspace}\xspace{}\mathsf{+cg}$ for $\mathsf{IterSuffFullBinSeq}_{n}\xspace$
  13. Lower bounds
  14. Complement of $\mathsf{M}\xspace_n$
  15. First proof using the extended fooling set technique.
  16. ...and 2 more sections

Key Result

Proposition 4

If a language $\mathsf{L}\xspace\subseteq\Gamma^*$ is recognized by a 2nfa$\mathcal{A}\xspace$, then the unary language ${\pi_0\left(\mathsf{L}\xspace\right)}$ is recognized by a ${\textsc{2}\xspace}{\textsc{n}{\textsc{fa}\xspace}\xspace}\xspace{}\mathsf{+cg}$$\mathcal{B}\xspace$ with the same state

Figures (1)

  • Figure 1: The $11$/̄state automaton $F$ recognizing $X_\sigma$, for each $\sigma\in\set{0,1,\sharp\xspace,\$,\bot}$ using the corresponding state $q_\sigma$ (represented as a purple-filled rectangle); the trap state $q_\bot$ (entered from every state $q_\sigma$ or $p_\sharp\xspace$ by reading $\sharp\xspace$) is not depicted.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Definition 5
  • Proposition 6
  • Lemma 7
  • Proposition 8
  • Lemma 9
  • Lemma 10
  • ...and 16 more