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Operational State Complexity of Block Languages

Guilherme Duarte, Nelma Moreira, Luca Prigioniero, Rogério Reis

TL;DR

This work addresses the deterministic and nondeterministic operational state complexities of standard operations on block languages, finite languages with fixed word length $\\ell$. It leverages bitmap representations to convert language operations into bitwise bitmap manipulations, enabling precise upper bounds and tightness results for reversal, word addition/removal, intersection, union, concatenation, block complement, and Kleene star/plus. The authors introduce witness families (notably MAX$_{\\ell}$ and L$_{k,d,x}$) to prove tightness and demonstrate that, despite being a subclass of finite languages, block languages often admit tighter bounds due to the fixed-length constraint. The results have implications for applications in areas like code theory and image processing, where homogeneous word-length constraints arise, and they provide a bitmap-driven framework for constructing minimal automata for block languages. Overall, the paper highlights both the parallels and the improvements over general finite-language state complexity when restricting to block languages.

Abstract

In this paper we consider block languages, namely sets of words having the same length, and study the deterministic and nondeterministic state complexity of several operations on these languages. Being a subclass of finite languages, the upper bounds of operational state complexity known for finite languages apply for block languages as well. However, in several cases, smaller values were found. Block languages can be represented as bitmaps, which are a good tool to study their minimal finite automata and their operations, as we illustrate here.

Operational State Complexity of Block Languages

TL;DR

This work addresses the deterministic and nondeterministic operational state complexities of standard operations on block languages, finite languages with fixed word length . It leverages bitmap representations to convert language operations into bitwise bitmap manipulations, enabling precise upper bounds and tightness results for reversal, word addition/removal, intersection, union, concatenation, block complement, and Kleene star/plus. The authors introduce witness families (notably MAX and L) to prove tightness and demonstrate that, despite being a subclass of finite languages, block languages often admit tighter bounds due to the fixed-length constraint. The results have implications for applications in areas like code theory and image processing, where homogeneous word-length constraints arise, and they provide a bitmap-driven framework for constructing minimal automata for block languages. Overall, the paper highlights both the parallels and the improvements over general finite-language state complexity when restricting to block languages.

Abstract

In this paper we consider block languages, namely sets of words having the same length, and study the deterministic and nondeterministic state complexity of several operations on these languages. Being a subclass of finite languages, the upper bounds of operational state complexity known for finite languages apply for block languages as well. However, in several cases, smaller values were found. Block languages can be represented as bitmaps, which are a good tool to study their minimal finite automata and their operations, as we illustrate here.
Paper Structure (12 sections, 28 theorems, 24 equations, 2 figures, 2 tables)

This paper contains 12 sections, 28 theorems, 24 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Block Languages and Bitmap Representation
  4. Operational Complexity
  5. Reversal
  6. Word Addition and Word Removal
  7. Intersection
  8. Union
  9. Concatenation
  10. Block Complement
  11. Kleene Star and Plus
  12. Conclusions

Key Result

Lemma 1

Let $L\subseteq\Sigma^\ell$ be block language over an alphabet of size $k$ and $\ell>0$. Then, we have

Figures (2)

  • Figure 1: Constraints in the widths of the ranks of a minimal DFA for a block language. Each rank (except the last and the first ones) is represented by a rectangle. The rightmost state is the dead-state ($\Omega$).
  • Figure 2: The minimal DFA accepting the language $\mathop{\mathrm{\mathsf{MAX}}}\nolimits_{\ell}$ for $\ell=5$. The sink-state is omitted, as well as all transitions from and to it.

Theorems & Definitions (52)

  • Lemma 1
  • proof : Proof (sketch).
  • Example 1
  • Lemma 2
  • proof
  • Example 2
  • Corollary 1
  • Proposition 1: dmpr24a
  • Example 3
  • Lemma 3
  • ...and 42 more