Comparison-Free Polyregular Functions.

This paper introduces a new automata-theoretic class of string-to-string functions with polynomialgrowth. Several equivalent definitions are provided: a machine model which is a restricted variant ofpebble transducers, and a few inductive definitions that close the class of regular functions undercertain operations. Our motivation for studying this class comes from another characterization,which we merely mention here but prove elsewhere, based on a λ-calculus with a linear type system.As their name suggests, these comparison-free polyregular functions form a subclass of polyregularfunctions; we prove that the inclusion is strict. We also show that they are incomparable withHDT0L transductions, closed under usual function composition – but not under a certain “map”combinator – and satisfy a comparison-free version of the pebble minimization theorem.On the broader topic of polynomial growth transductions, we also consider the recently introducedlayered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that afunction can be obtained by composing such transducers together if and only if it is polyregular,and that k-layered SSTs (or k-marble transducers) are closed under “map” and equivalent to acorresponding notion of (k + 1)-layered HDT0L systems

The many facets of string transducers

Regular word transductions extend the robust notion of regular languages from a qualitative to a quantitative reasoning. They were already considered in early papers of formal language theory, but turned out to be much more challenging. The last decade brought considerable research around various transducer models, aiming to achieve similar robustness as for automata and languages. In this paper we survey some older and more recent results on string transducers. We present classical connections between automata, logic and algebra extended to transducers, some genuine definability questions, and review approaches to the equivalence problem

Solution sets for equations over free groups are EDT0L languages

© World Scientific Publishing Company. We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to NSPACE(n log n) here

Equivalence of finite-valued streaming string transducers is decidable

In this paper we provide a positive answer to a question left open by Alur and and Deshmukh in 2011 by showing that equivalence of finite-valued copyless streaming string transducers is decidable

Implicit automata in typed λ\lambda-calculi II: streaming transducers vs categorical semantics

We characterize regular string transductions as programs in a linear $\lambda$-calculus with additives. One direction of this equivalence is proved by encoding copyless streaming string transducers (SSTs), which compute regular functions, into our $\lambda$-calculus. For the converse, we consider a categorical framework for defining automata and transducers over words, which allows us to relate register updates in SSTs to the semantics of the linear $\lambda$-calculus in a suitable monoidal closed category. To illustrate the relevance of monoidal closure to automata theory, we also leverage this notion to give abstract generalizations of the arguments showing that copyless SSTs may be determinized and that the composition of two regular functions may be implemented by a copyless SST. Our main result is then generalized from strings to trees using a similar approach. In doing so, we exhibit a connection between a feature of streaming tree transducers and the multiplicative/additive distinction of linear logic. Keywords: MSO transductions, implicit complexity, Dialectica categories, Church encodingsComment: 105 pages, 24 figure

Эффективные алгоритмы проверки эквивалентности для некоторых классов автоматов

Finite transducers, two-tape automata, and biautomata are related computational models descended from the concept of Finite-State Automaton. In these models an automaton controls two heads that read or write symbols on the tapes in the one-way mode. The computations of these three types of automata show many common features, and it is surprising that the methods for analyzing the behavior of automata developed for one of these models do not find suitable utilization in other models. The goal of this paper is to develop a uniform technique for building polynomial-time equivalence checking algorithms for some classes of automata (finite transducers, two-tape automata, biautomata, single-state pushdown automata) which exhibit certain features of the deterministic or unambiguous behavior. This new technique reduces the equivalence checking of automata to solvability checking of certain systems of equations over the semirings of languages or transductions. It turns out that such a checking can be performed by the variable elimination technique which relies on some combinatorial and algebraic properties of prefix-free regular languages. The main results obtained in this paper are as follows:1.            Using the algebraic approach a new algorithm for checking the equivalence of states of deterministic finite automata is constructed; time complexity of this algorithm is O(n log n).2.            A new class of prefix-free finite transducers is distinguished and it is shown that the developed algebraic approach provides the equivalence checking of transducers from this class in quadratic time (for real-time prefix-free transducers) and cubic (for prefix-free transducers with ɛ-transitions) relative to the sizes of analysed machines.3.            It is shown that the equivalence problem for deterministic two-tape finite automata can be reduced to the same problem for prefix-free finite transducers and solved in cubic time relative to the size of the analysed machines.4.            In the same way it is proved that the equivalence problem for deterministic finite biautomata can be solved in cubic time relative to the sizes of analysed machines.5.            By means of the developed approach an efficient equivalence checking algorithm for the class of simple grammars corresponding to deterministic single-state pushdown automata is constructed.Конечные преобразователи, двухленточные автоматы и биавтоматы — взаимосвязанные вычислительные модели, ведущие свое происхождение от концепции конечного автомата. В вычислениях этих машин проявляется много общих черт, и удивительно, что методы анализа, разработанные для одной из указанных моделей, не находят подходящего применения в других моделях. Целью данной статьи является разработка единой методики построения быстрых алгоритмов проверки эквивалентности для некоторых классов автоматов (конечных преобразователей, двухленточных автоматов, биавтоматов, магазинных автоматов), которые демонстрируют определенные черты детерминированного или однозначное поведение. Этот новый метод сводит проверку эквивалентности автоматов к проверке разрешимости систем уравнений над полукольцами языков или бинарных отношений. Как оказалось, такую проверку достаточно просто провести методом исключения переменных, используя некоторые комбинаторные и алгебраические свойства регулярных префиксных языков. Основные результаты, полученные в этой статье, таковы.1.            При помощи алгебраического метода построен новый алгоритм проверки эквивалентности детерминированных конечных автоматов, имеющий сложность по времени O(n log n).2.            Выделен новый класс префиксных конечных трансдьюсеров и показано, что проверка эквивалентности трансдьюсеров из этого класса может быть осуществлена новым методом за время, квадратичное (для префиксных трансдьюсеров реального времени) и кубическое (для префиксных трансдьюсеров с ɛ-переходами) относительно размеров анализируемых автоматов.3.            Показано, что проблема эквивалентности для детерминированных двухленточных конечных автоматов сводится к задаче проверки эквивалентности префиксных конечных трансдьюсеров и может быть решена за время, кубическое относительно их размеров.4.            Аналогичным образом установлена разрешимость проблемы эквивалентности для детерминированных конечных биавтоматов за время, кубическое относительно их размеров.5.            При помощи нового метода построен алгоритм проверки эквивалентности для простых грамматик, соответствующих детерминированным магазинным автоматам с одним состоянием

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science