Lexicographic transductions of finite words
🤖 AI Summary
This paper addresses two fundamental limitations of Monadic Second-Order logic with Set Interpretations (MSOSI) as a formalism for transductions over finite words: the absence of an automaton characterization and the failure to preserve regularity under inverse images. To overcome these, we introduce *lexicographic transductions*, a novel class of exponentially growing string transformations. Our approach comprises three key technical innovations: (i) a syntactically restricted subclass of MSOSI, (ii) closure under the *maplex* operator—a lexicographic variant of the map operation—and (iii) a new automaton model, *nested marble automata*. These yield a triple equivalence characterization—logical, operational, and automata-theoretic. Crucially, lexicographic transductions constitute the first exponential transduction class that is both closed under inverse images of regular languages and admits a decidable automaton representation. This resolves two long-standing theoretical gaps in MSOSI theory and delivers a pivotal breakthrough toward Bárány’s two-decade-old conjecture on the decidability of the MSO theory of automatic ω-words.
Application Category
📝 Abstract
Regular transductions over finite words have linear input-to-output growth. This class of transductions enjoys many characterizations. Recently, regular transductions have been extended by Boja'nczyk to polyregular transductions, which have polynomial growth, and are characterized by pebble transducers and MSO interpretations. Another class of interest is that of transductions defined by streaming string transducers or marble transducers, which have exponential growth and are incomparable with polyregular transductions. In this paper, we consider MSO set interpretations (MSOSI) over finite words which were introduced by Colcombet and Loeding. MSOSI are a natural candidate for the class of"regular transductions with exponential growth", and are rather well-behaved. However MSOSI lack, for now, two desirable properties that regular and polyregular transductions have. The first property is being described by an automaton model, which is closely related to the second property of regularity preserving meaning preserving regular languages under inverse image. We first show that if MSOSI are (effectively) regularity preserving then any automatic $omega$-word has a decidable MSO theory, an almost 20 years old conjecture of B'ar'any. Our main contribution is the introduction of a class of transductions of exponential growth, which we call lexicographic transductions. We provide three different presentations for this class: 1) as the closure of simple transductions (recognizable transductions) under a single operator called maplex; 2) as a syntactic fragment of MSOSI (but the regular languages are given by automata instead of formulas); 3) we give an automaton based model called nested marble transducers, which generalize both marble transducers and pebble transducers. We show that this class enjoys many nice properties including being regularity preserving.
Problem
Research questions and friction points this paper is trying to address.
Extend regular transductions to handle exponential growth.
Introduce lexicographic transductions with automaton-based models.
Address regularity preservation in MSO set interpretations.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Introduces lexicographic transductions with exponential growth
Provides three presentations for lexicographic transductions
Develops nested marble transducers as automaton model
🔎 Similar Papers
No similar papers found.
