18 research outputs found
Revisiting the Equivalence Problem for Finite Multitape Automata
The decidability of determining equivalence of deterministic multitape
automata (or transducers) was a longstanding open problem until it was resolved
by Harju and Karhum\"{a}ki in the early 1990s. Their proof of decidability
yields a co_NP upper bound, but apparently not much more is known about the
complexity of the problem. In this paper we give an alternative proof of
decidability, which follows the basic strategy of Harju and Karhumaki but
replaces their use of group theory with results on matrix algebras. From our
proof we obtain a simple randomised algorithm for deciding language equivalence
of deterministic multitape automata and, more generally, multiplicity
equivalence of nondeterministic multitape automata. The algorithm involves only
matrix exponentiation and runs in polynomial time for each fixed number of
tapes. If the two input automata are inequivalent then the algorithm outputs a
word on which they differ
Revisiting the Equivalence Problem for Finite Multitape Automata
Abstract. The decidability of determining equivalence of deterministic multitape automata was a longstanding open problem until it was resolved by Harju and Karhumäki in the early 1990s. Their proof of decidability yields a co-NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability which follows the basic strategy of Harju and Karhumäki, but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding equivalence of deterministic multitape automata, as well as automata with transition weights in the field of rational numbers. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ
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
On Rational Recursive Sequences
We study the class of rational recursive sequences (ratrec) over the rational
numbers. A ratrec sequence is defined via a system of sequences using mutually
recursive equations of depth 1, where the next values are computed as rational
functions of the previous values. An alternative class is that of simple ratrec
sequences, where one uses a single recursive equation, however of depth k: the
next value is defined as a rational function of k previous values.
We conjecture that the classes ratrec and simple ratrec coincide. The main
contribution of this paper is a proof of a variant of this conjecture where the
initial conditions are treated symbolically, using a formal variable per
sequence, while the sequences themselves consist of rational functions over
those variables. While the initial conjecture does not follow from this
variant, we hope that the introduced algebraic techniques may eventually be
helpful in resolving the problem.
The class ratrec strictly generalises a well-known class of polynomial
recursive sequences (polyrec). These are defined like ratrec, but using
polynomial functions instead of rational ones. One can observe that if our
conjecture is true and effective, then we can improve the complexities of the
zeroness and the equivalence problems for polyrec sequences. Currently, the
only known upper bound is Ackermanian, which follows from results on polynomial
automata. We complement this observation by proving a PSPACE lower bound for
both problems for polyrec. Our lower bound construction also implies that the
Skolem problem is PSPACE-hard for the polyrec class
Equivalence Testing of Weighted Automata over Partially Commutative Monoids
Motivated by equivalence testing of k-tape automata, we study the equivalence testing of weighted automata in the more general setting, over partially commutative monoids (in short, pc monoids), and show efficient algorithms in some special cases, exploiting the structure of the underlying non-commutation graph of the monoid.
Specifically, if the edge clique cover number of the non-commutation graph of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm for equivalence testing. As a corollary, we obtain the first deterministic quasi-polynomial time algorithms for equivalence testing of k-tape weighted automata and for equivalence testing of deterministic k-tape automata for constant k. Prior to this, the best complexity upper bound for these k-tape automata problems were randomized polynomial-time, shown by Worrell [James Worrell, 2013]. Finding a polynomial-time deterministic algorithm for equivalence testing of deterministic k-tape automata for constant k has been open for several years [Emily P. Friedman and Sheila A. Greibach, 1982] and our results make progress.
We also consider pc monoids for which the non-commutation graphs have an edge cover consisting of at most k cliques and star graphs for any constant k. We obtain a randomized polynomial-time algorithm for equivalence testing of weighted automata over such monoids.
Our results are obtained by designing efficient zero-testing algorithms for weighted automata over such pc monoids
Another approach to the equivalence of measure-many one-way quantum finite automata and its application
In this paper, we present a much simpler, direct and elegant approach to the
equivalence problem of {\it measure many one-way quantum finite automata}
(MM-1QFAs). The approach is essentially generalized from the work of Carlyle
[J. Math. Anal. Appl. 7 (1963) 167-175]. Namely, we reduce the equivalence
problem of MM-1QFAs to that of two (initial) vectors.
As an application of the approach, we utilize it to address the equivalence
problem of {\it Enhanced one-way quantum finite automata} (E-1QFAs) introduced
by Nayak [Proceedings of the 40th Annual IEEE Symposium on Foundations of
Computer Science, 1999, pp.~369-376]. We prove that two E-1QFAs
and over are equivalence if and only if they are
-equivalent where and are the numbers of states in
and , respectively.Comment: V 10: Corollary 3 is deleted, since it is folk. (V 9: Revised in
terms of the referees's comments) All comments, especially the linguistic
comments, are welcom
Scalable verification of probabilistic networks
This paper presents McNetKAT, a scalable tool for verifying
probabilistic network programs. McNetKAT is based on a
new semantics for the guarded and history-free fragment
of Probabilistic NetKAT in terms of finite-state, absorbing
Markov chains. This view allows the semantics of all programs to be computed exactly, enabling construction of an
automatic verification tool. Domain-specific optimizations
and a parallelizing backend enable McNetKAT to analyze
networks with thousands of nodes, automatically reasoning
about general properties such as probabilistic program equivalence and refinement, as well as networking properties such
as resilience to failures. We evaluate McNetKAT’s scalability using real-world topologies, compare its performance
against state-of-the-art tools, and develop an extended case
study on a recently proposed data center network design
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
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