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

Streamability of nested word transductions

We consider the problem of evaluating in streaming (i.e., in a single left-to-right pass) a nested word transduction with a limited amount of memory. A transduction T is said to be height bounded memory (HBM) if it can be evaluated with a memory that depends only on the size of T and on the height of the input word. We show that it is decidable in coNPTime for a nested word transduction defined by a visibly pushdown transducer (VPT), if it is HBM. In this case, the required amount of memory may depend exponentially on the height of the word. We exhibit a sufficient, decidable condition for a VPT to be evaluated with a memory that depends quadratically on the height of the word. This condition defines a class of transductions that strictly contains all determinizable VPTs

Bisimilarity of Pushdown Systems is Nonelementary

Given two pushdown systems, the bisimilarity problem asks whether they are bisimilar. While this problem is known to be decidable our main result states that it is nonelementary, improving EXPTIME-hardness, which was the previously best known lower bound for this problem. Our lower bound result holds for normed pushdown systems as well

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

Revisiting Membership Problems in Subclasses of Rational Relations

We revisit the membership problem for subclasses of rational relations over finite and infinite words: Given a relation R in a class C_2, does R belong to a smaller class C_1? The subclasses of rational relations that we consider are formed by the deterministic rational relations, synchronous (also called automatic or regular) relations, and recognizable relations. For almost all versions of the membership problem, determining the precise complexity or even decidability has remained an open problem for almost two decades. In this paper, we provide improved complexity and new decidability results. (i) Testing whether a synchronous relation over infinite words is recognizable is NL-complete (PSPACE-complete) if the relation is given by a deterministic (nondeterministic) omega-automaton. This fully settles the complexity of this recognizability problem, matching the complexity of the same problem over finite words. (ii) Testing whether a deterministic rational binary relation is recognizable is decidable in polynomial time, which improves a previously known double exponential time upper bound. For relations of higher arity, we present a randomized exponential time algorithm. (iii) We provide the first algorithm to decide whether a deterministic rational relation is synchronous. For binary relations the algorithm even runs in polynomial time

Decision Problems for Subclasses of Rational Relations over Finite and Infinite Words

We consider decision problems for relations over finite and infinite words defined by finite automata. We prove that the equivalence problem for binary deterministic rational relations over infinite words is undecidable in contrast to the case of finite words, where the problem is decidable. Furthermore, we show that it is decidable in doubly exponential time for an automatic relation over infinite words whether it is a recognizable relation. We also revisit this problem in the context of finite words and improve the complexity of the decision procedure to single exponential time. The procedure is based on a polynomial time regularity test for deterministic visibly pushdown automata, which is a result of independent interest.Comment: v1: 31 pages, submitted to DMTCS, extended version of the paper with the same title published in the conference proceedings of FCT 2017; v2: 32 pages, minor revision of v1 (DMTCS review process), results unchanged; v3: 32 pages, enabled hyperref for Figure 1; v4: 32 pages, add reference for known complexity results for the slenderness problem; v5: 32 pages, added DMTCS metadat

Inkdots as advice for finite automata

We examine inkdots placed on the input string as a way of providing advice to finite automata, and establish the relations between this model and the previously studied models of advised finite automata. The existence of an infinite hierarchy of classes of languages that can be recognized with the help of increasing numbers of inkdots as advice is shown. The effects of different forms of advice on the succinctness of the advised machines are examined. We also study randomly placed inkdots as advice to probabilistic finite automata, and demonstrate the superiority of this model over its deterministic version. Even very slowly growing amounts of space can become a resource of meaningful use if the underlying advised model is extended with access to secondary memory, while it is famously known that such small amounts of space are not useful for unadvised one-way Turing machines.Comment: 14 page

Monadic Decomposability of Regular Relations

Monadic decomposibility - the ability to determine whether a formula in a given logical theory can be decomposed into a boolean combination of monadic formulas - is a powerful tool for devising a decision procedure for a given logical theory. In this paper, we revisit a classical decision problem in automata theory: given a regular (a.k.a. synchronized rational) relation, determine whether it is recognizable, i.e., it has a monadic decomposition (that is, a representation as a boolean combination of cartesian products of regular languages). Regular relations are expressive formalisms which, using an appropriate string encoding, can capture relations definable in Presburger Arithmetic. In fact, their expressive power coincide with relations definable in a universal automatic structure; equivalently, those definable by finite set interpretations in WS1S (Weak Second Order Theory of One Successor). Determining whether a regular relation admits a recognizable relation was known to be decidable (and in exponential time for binary relations), but its precise complexity still hitherto remains open. Our main contribution is to fully settle the complexity of this decision problem by developing new techniques employing infinite Ramsey theory. The complexity for DFA (resp. NFA) representations of regular relations is shown to be NLOGSPACE-complete (resp. PSPACE-complete)