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On the Equivalence Checking Problem for Deterministic Top-Down Tree Automata

Zhibo Deng, Vladimir A. Zakharov

TL;DR

The paper tackles the problem of determining whether two states of a deterministic top-down finite tree automaton recognize the same tree language. It introduces an algebraic approach that reduces equivalence to the solvability of a system of language equations and employs a two-stage procedure with variable elimination. Stage 1 constructs a system $oldsymbol{\mathcal{E}}_0$ of equations using state variables $X_q$ and transitions, augmented by the constraint $X_{q'}=X_{q''}$; Stage 2 iteratively applies substitution, conflict detection, and restoration to decide solvability. The authors prove termination and correctness and establish a worst-case time bound of $O(n^2)$ under a RAM with pointers, highlighting practical implications for analyzing tree-structured computations and suggesting extensions to related models such as top-down tree transducers.

Abstract

We present an efficient algorithm for checking language equivalence of states in top-down deterministic finite tree automata (DFTAs). Unlike string automata, tree automata operate over hierarchical structures, posing unique challenges for algorithmic analysis. Our approach reduces the equivalence checking problem to that of checking the solvability of a system of language-theoretic equations, which specify the behavior of a DFTA. By constructing such a system of equations and systematically manipulating with it through substitution and conflict detection rules, we develop a decision procedure that determines whether two states accept the same tree language. We formally prove the correctness and termination of the algorithm and establish its worst-case time complexity as $O(n^2)$ under the RAM (Random Access Machine) model of computation augmented with pointers.

On the Equivalence Checking Problem for Deterministic Top-Down Tree Automata

TL;DR

The paper tackles the problem of determining whether two states of a deterministic top-down finite tree automaton recognize the same tree language. It introduces an algebraic approach that reduces equivalence to the solvability of a system of language equations and employs a two-stage procedure with variable elimination. Stage 1 constructs a system of equations using state variables and transitions, augmented by the constraint ; Stage 2 iteratively applies substitution, conflict detection, and restoration to decide solvability. The authors prove termination and correctness and establish a worst-case time bound of under a RAM with pointers, highlighting practical implications for analyzing tree-structured computations and suggesting extensions to related models such as top-down tree transducers.

Abstract

We present an efficient algorithm for checking language equivalence of states in top-down deterministic finite tree automata (DFTAs). Unlike string automata, tree automata operate over hierarchical structures, posing unique challenges for algorithmic analysis. Our approach reduces the equivalence checking problem to that of checking the solvability of a system of language-theoretic equations, which specify the behavior of a DFTA. By constructing such a system of equations and systematically manipulating with it through substitution and conflict detection rules, we develop a decision procedure that determines whether two states accept the same tree language. We formally prove the correctness and termination of the algorithm and establish its worst-case time complexity as under the RAM (Random Access Machine) model of computation augmented with pointers.

Paper Structure

This paper contains 7 sections, 4 theorems, 8 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalence checking algorithm
  4. Stage 1: Constructing the system of equations.
  5. Stage 2: Checking the solvability of the equation system ${\mathcal{E}}_0$.
  6. Proofs of algorithm \ref{['alg']} and complexity analysis
  7. Conclusion

Key Result

Theorem 1

For a top-down DFTA $\mathcal{A}$, algorithm alg_empty always correctly determines whether $L_{\mathcal{A}}(q) \neq \emptyset$. And it runs in time $O(n)$, where $n$ is the total size of $\mathcal{A}$.

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof