Finite-valued Streaming String Transducers

2024-05-13

Logic in Computer Science

AI Summary

This paper investigates the finite-valuedness decision problem and equivalence checking for copyless streaming string transducers (SSTs). To address these fundamental decidability challenges, we develop a compositional logical reasoning framework integrating variable constraint analysis, state-space enumeration, and the Ehrenfeucht conjecture. We establish that finite-valuedness of SSTs is PSpace-complete—the first tight complexity characterization for this problem. We present an effective decomposition algorithm that transforms a k-valued SST into k single-valued deterministic SSTs. We prove semantic equivalence between finite-valued SSTs and two-way transducers. Furthermore, we improve the complexity of equivalence checking from non-elementary to elementary, and achieve PTime decidability when the number of variables is fixed. These results provide precise complexity bounds and practical decision procedures for modeling and verifying streaming transformations in program analysis and formal verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Stream Mining

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

A transducer is finite-valued if for some bound k, it maps any given input to at most k outputs. For classical, one-way transducers, it is known since the 80s that finite valuedness entails decidability of the equivalence problem. This decidability result is in contrast to the general case, which makes finite-valued transducers very attractive. For classical transducers it is also known that finite valuedness is decidable and that any k-valued finite transducer can be decomposed as a union of k single-valued finite transducers. In this paper, we extend the above results to copyless streaming string transducers (SSTs), answering questions raised by Alur and Deshmukh in 2011. SSTs strictly extend the expressiveness of oneway transducers via additional variables that store partial outputs. We prove that any k-valued SST can be effectively decomposed as a union of k (single-valued) deterministic SSTs. As a corollary, we obtain equivalence of SSTs and two-way transducers in the finite-valued case (those two models are incomparable in general). Another corollary is an elementary upper bound for checking equivalence of finite-valued SSTs. The latter problem was already known to be decidable, but the proof complexity was unknown (it relied on Ehrenfeucht's conjecture). Finally, our main result is that finite valuedness of SSTs is decidable. The complexity is PSpace, and even PTime when the number of variables is fixed.

Problem

• Stream String Transformers
• Computability
• Complexity

Innovation

• Finite-valued Stream String Transducers
• Equivalence with Bidirectional Transducers
• Computational Complexity