Exact Lower Bounds on the Descriptional Complexity of Determinizing Input-Driven Pushdown Automata

Core Concepts

The exact number of states and stack symbols needed to determinize nondeterministic input-driven pushdown automata (NIDPDA) is determined. It is proved that in the worst case, 2n^2 states and |Σ+1|(2n^2 - 1) stack symbols are necessary to determinize an n-state NIDPDA.

Abstract

The paper investigates the exact descriptional complexity of determinizing nondeterministic input-driven pushdown automata (NIDPDA). It makes the following key contributions:

For NIDPDA with a fixed 4-symbol input alphabet and a 2-symbol stack alphabet, it is proved that 2n^2 states are necessary in the worst case to determinize an n-state NIDPDA. This is the first precise lower bound on the state complexity of NIDPDA determinization with a bounded alphabet.

For NIDPDA with only one left bracket in the input alphabet, it is shown that 2n^2 - 1 stack symbols are necessary in the worst case to determinize an n-state NIDPDA. This improves upon the previous asymptotic lower bound.

The exact lower bound is then established for NIDPDA with any number of left brackets less than 2n^2: 2n^2 states and |Σ+1|(2n^2 - 1) stack symbols are necessary in the worst case to determinize an n-state NIDPDA. The witness automata use a stack alphabet growing linearly in n and logarithmically in the number of left brackets.

The paper provides tight lower bounds on both the state complexity and the stack alphabet size required for determinizing NIDPDA, improving upon the previous asymptotic results.

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Key Insights Distilled From

Exact descriptional complexity of determinization of input-driven pushdown automata

by Olga Martyno... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10516.pdf

Deeper Inquiries

How do the lower bounds on NIDPDA determinization compare to the upper bounds achieved by previous determinization constructions

The lower bounds on NIDPDA determinization presented in the paper are significant as they provide precise information on the minimum resources required for determinizing these automata. In comparison to previous determinization constructions that achieved upper bounds, the lower bounds established in the paper offer a clear understanding of the inherent complexity of the determinization process. The upper bounds achieved by previous constructions may have been based on specific techniques or assumptions that allowed for a more efficient determinization process. However, the lower bounds reveal the fundamental complexity of determinizing NIDPDA, showcasing the minimum number of states and stack symbols needed in the worst-case scenario.

Are there any practical implications of these tight lower bounds on the resources required for determinizing NIDPDA

The tight lower bounds on the resources required for determinizing NIDPDA have several practical implications. Firstly, they provide valuable insights for designing efficient algorithms and systems that work with pushdown automata. Understanding the exact resources needed for determinization can help in optimizing the performance of automata-based applications. Additionally, the lower bounds serve as a benchmark for evaluating the efficiency of existing determinization algorithms. If an algorithm surpasses the lower bounds, it indicates a more resource-efficient approach to determinization. Moreover, the lower bounds can guide researchers and practitioners in developing strategies to handle the complexity of NIDPDA determinization effectively.

Can the techniques used in this paper be extended to analyze the descriptional complexity of determinizing other variants of pushdown automata beyond the classical input-driven model

The techniques used in the paper to analyze the descriptional complexity of determinizing NIDPDA can be extended to other variants of pushdown automata beyond the classical input-driven model. By adapting the methodology to different types of pushdown automata, researchers can explore the determinization process in various contexts and gain insights into the resource requirements for different automata models. The approach of constructing witness automata, analyzing state transitions, and establishing lower bounds can be applied to study determinization in event-clock input-driven pushdown automata, probabilistic pushdown automata, or other variants. This extension of techniques can contribute to a comprehensive understanding of determinization complexities across diverse pushdown automata models.