A Regular and Complete Notion of Delay for Comparing Executions of Streaming String Transducers

Core Concepts

The authors introduce a novel notion of delay that can be used to regularly and completely compare executions of streaming string transducers, a powerful class of string-to-string transducers.

Abstract

The paper introduces a new notion of delay for comparing executions of streaming string transducers (SSTs). The key insights are:

The delay between two SST executions should not depend on the order in which periodic output blocks are produced, but only on the positions where the periodic structure changes.

The authors define a delay measure delayℓ that compares the number of output positions produced up to the end of each block of length at most ℓ.

They prove that this delay notion is regular - there exists a finite automaton that can check whether the delay between two SST executions is bounded by a given constant.

The authors also prove that their delay notion is complete - two SSTs are equivalent if and only if their executions have a bounded delay. This result extends to other transducer models like deterministic two-way transducers and MSO transducers.

The regularity and completeness of the delay notion enable decidability of several problems for SSTs, like equivalence up to a fixed delay bound, which are undecidable in the general case.

The paper provides a robust and machine-independent way to compare the executions of powerful string-to-string transducers, with important applications in transducer theory and verification.

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Key Insights Distilled From

A Regular and Complete Notion of Delay for Streaming String Transducers

by Emma... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2205.04287.pdf

Deeper Inquiries

How can the delay notion introduced in this paper be extended or adapted to compare executions of other types of transducers or automata beyond SSTs

The delay notion introduced in the paper for comparing executions of Streaming String Transducers (SSTs) can be extended or adapted to compare executions of other types of transducers or automata by considering the underlying principles of the delay measure. One way to extend this notion is to apply it to compare executions of different types of transducers, such as deterministic two-way transducers or monadic second-order transducers, which are known to be equivalent to SSTs. By adapting the delay measure to account for the specific characteristics and behaviors of these transducer models, it can be used to analyze and compare their executions in a similar manner to SSTs. Additionally, the delay notion can be extended to compare executions of more complex automata models by incorporating origin information and weight differences to measure the similarity between their outputs. This extension would involve defining appropriate origin functions for the automata and adapting the delay measure to capture the differences in their output productions.

Are there other applications of the regular and complete delay measure beyond the decision problems discussed in the paper, for example in program analysis or verification

The regular and complete delay measure introduced in the paper has various applications beyond the decision problems discussed. One potential application is in program analysis and verification, particularly in the context of string transformations and formal language processing. The delay notion can be utilized to analyze the behavior of programs that involve string manipulations, such as compilers, interpreters, or string processing algorithms. By applying the delay measure, one can compare the outputs of different program executions and identify discrepancies or similarities in their behavior. This can be valuable for debugging, testing, and verifying the correctness of string manipulation algorithms or programs. Furthermore, the regularity and completeness properties of the delay measure can be leveraged in automated verification tools to ensure the correctness and consistency of string transformations in software systems.

Can the techniques used to prove the regularity of the delay notion be applied to develop regular notions for comparing executions of other complex computational models

The techniques used to prove the regularity of the delay notion in the context of SSTs can be applied to develop regular notions for comparing executions of other complex computational models. By adapting the approach taken in the paper to different types of automata or transducers, one can establish regularity properties for comparing their executions based on origin information and weight differences. This methodology can be extended to various computational models, such as pushdown automata, Turing machines, or other abstract machines, by defining appropriate origin functions and weight measures specific to each model. By proving the regularity of delay notions for these models, one can enhance the understanding of their behaviors and facilitate the comparison of their executions in a systematic and structured manner.