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Nondeterministic tree-walking automata are not closed under complementation

Olga Martynova, Alexander Okhotin

TL;DR

This work settles Bojańczyk–Colcombet’s question by proving that the class of tree languages recognized by nondeterministic tree-walking automata is not closed under complementation. The authors construct a separating language $L$ recognizable by an NTWA yet whose complement cannot be recognized by any NTWA, using a pattern-based framework with Δ-n patterns and a faulty Δ-2M' pattern to force a contradiction. They also demonstrate that UTWAs are strictly weaker than NTWAs, while a deterministic one-pebble tree-walking automaton can recognize $L$, making its complement recognizable by a DTWA and highlighting non-deterministic limitations. Together, these results refine the landscape of tree-walking automata, imply that NTWA cannot be determinized, and raise further questions about the relative power and closure properties of unambiguous and graph-walking variants.

Abstract

It is proved that the family of tree languages recognized by nondeterministic tree-walking automata is not closed under complementation, solving a problem raised by Bojańczyk and Colcombet ("Tree-walking automata do not recognize all regular languages", SIAM J. Comp. 38 (2008) 658--701). In addition, it is shown that nondeterministic tree-walking automata are stronger than unambiguous tree-walking automata.

Nondeterministic tree-walking automata are not closed under complementation

TL;DR

This work settles Bojańczyk–Colcombet’s question by proving that the class of tree languages recognized by nondeterministic tree-walking automata is not closed under complementation. The authors construct a separating language recognizable by an NTWA yet whose complement cannot be recognized by any NTWA, using a pattern-based framework with Δ-n patterns and a faulty Δ-2M' pattern to force a contradiction. They also demonstrate that UTWAs are strictly weaker than NTWAs, while a deterministic one-pebble tree-walking automaton can recognize , making its complement recognizable by a DTWA and highlighting non-deterministic limitations. Together, these results refine the landscape of tree-walking automata, imply that NTWA cannot be determinized, and raise further questions about the relative power and closure properties of unambiguous and graph-walking variants.

Abstract

It is proved that the family of tree languages recognized by nondeterministic tree-walking automata is not closed under complementation, solving a problem raised by Bojańczyk and Colcombet ("Tree-walking automata do not recognize all regular languages", SIAM J. Comp. 38 (2008) 658--701). In addition, it is shown that nondeterministic tree-walking automata are stronger than unambiguous tree-walking automata.

Paper Structure

This paper contains 11 sections, 13 theorems, 5 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Trees and tree-walking automata
  3. Separating language
  4. Tools from the paper by Bojańczyk and Colcombet and their further properties
  5. Patterns $\Delta_n$ and $\Delta_{2M}'$
  6. The correct pattern $\Delta_{2M}$ and a run of type $(q_{\text{start}},1,q_{\text{finish}},0)$ through it
  7. Shrinking and stretching proper steps to bypass the fault
  8. How to move through the faulty pattern $\Delta_{2M}'$ without noticing the fault
  9. UTWA are weaker than NTWA
  10. A deterministic one-pebble tree-walking automaton recognizing $L$
  11. Conclusion

Key Result

Theorem 1

There is a nondeterministic tree-walking automaton that recognizes the language $L$. No nondeterministic tree-walking automaton can recognize the complement of $L$.

Figures (15)

  • Figure 1: (left) A tree in $L$; (right) a tree not in $L$.
  • Figure 2: NTWA accepting a tree from $L$, executing the algorithm from Section \ref{['section_separating_language']}.
  • Figure 3: The general form of accepting computation of NTWA $A_L$.
  • Figure 4: Inner loop $p \to_{\varepsilon} q$.
  • Figure 5: Simplifying a transfer: (left) two consecutive transfers through the same element; (middle) inserting $\Delta_1$ at port $y$; (right) the resulting transfer of type $(r, x, t, z)$.
  • ...and 10 more figures

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Definition 5: Bojańczyk and Colcombet BojanczykColcombet_det, BojanczykColcombet_reg
  • Lemma 1: Bojańczyk and Colcombet BojanczykColcombet_det, BojanczykColcombet_reg
  • Lemma 2: Bojańczyk and Colcombet BojanczykColcombet_det, BojanczykColcombet_reg
  • Lemma 3: Bojańczyk and Colcombet BojanczykColcombet_reg
  • Definition 6: Bojańczyk and Colcombet BojanczykColcombet_reg
  • ...and 36 more