Two-way Automata Research Articles

Deterministic One-Way Simulation of Two-Way Deterministic Finite Automata Over Small Alphabets

It is shown that a two-way deterministic finite automaton (2DFA) with [Formula: see text] states over an alphabet [Formula: see text] can be transformed to an equivalent one-way automaton (1DFA) with [Formula: see text] states, where [Formula: see text]. This reflects the fact that, by keeping the last processed symbol [Formula: see text] in memory, the simulating 1DFA can remember one of [Formula: see text] states in which the automaton moves by [Formula: see text] to the right, and a function that maps [Formula: see text] states moving to the left to [Formula: see text] states moving to the right; cf. ca. [Formula: see text] functions in the classical construction. A close lower bound of [Formula: see text] states is established using a 2-symbol alphabet, with witness languages defined by direction-determinate 2DFA. The same lower bound is also achieved with witness languages defined by sweeping 2DFA, at the expense of using a 5-symbol alphabet. In addition, the complexity of transforming a sweeping or a direction-determinate 2DFA to a 1DFA is shown to be exactly [Formula: see text].

Lifting query complexity to time-space complexity for two-way finite automata

Time-space tradeoff has been studied in a variety of models, such as Turing machines, branching programs, and finite automata, etc. While communication complexity as a technique has been applied to study finite automata, it seems it has not been used to study time-space tradeoffs of finite automata. We design a new technique showing that separations of query complexity can be lifted, via communication complexity, to separations of time-space complexity of two-way finite automata. As an application, one of our main results exhibits the first example of a language L such that the time-space complexity of two-way probabilistic finite automata with a bounded error (2PFA) is Ω˜(n2), while of exact two-way quantum finite automata with classical states (2QCFA) is O˜(n5/3), that is, we demonstrate for the first time that exact quantum computing has an advantage in time-space complexity comparing to classical computing.

On the transformation of two-way finite automata to unambiguous finite automata

This paper estimates the number of states in an unambiguous finite automaton (UFA) that is sufficient and in the worst case necessary to simulate an n-state two-way deterministic finite automaton (2DFA) and an n-state two-way unambiguous finite automaton (2UFA). It is proved that a 2UFA with n states can be transformed to a UFA with fewer than 2n⋅n! states. On the other hand, for every n, there is a language recognized by an n-state 2DFA that requires a UFA with at least Ω((2+1)2n⋅n−1)=Ω(5.828n) states. The latter result is obtained as a lower bound on the rank of a certain matrix.

Converting nondeterministic two-way automata into small deterministic linear-time machines

In 1978 Sakoda and Sipser raised the question of the cost, in terms of size of representations, of the transformation of two-way and one-way nondeterministic automata into equivalent two-way deterministic automata. Despite all the attempts, the question has been answered only for particular cases, while it remains open in general, the best upper bound currently known being exponential. We present a new approach in which unrestricted nondeterministic automata are simulated by deterministic models extending two-way deterministic automata, paying only a polynomial increase of size. Indeed, we study the costs of the conversions of nondeterministic automata into some variants of one-tape deterministic Turing machines working in linear time; namely Hennie machines, weight-reducing Turing machines, and weight-reducing Hennie machines. All these variants are known to share the same computational power: they characterize the class of regular languages.

Two-way and one-way quantum and classical automata with advice for online minimization problems

We consider online algorithms. Typically, the model is investigated with respect to the competitive ratio. In this paper, we explore two-way automata and one-way automata as models for online algorithms. We focus on quantum and classical online algorithms. We show that there are problems that can be solved more efficiently by two-way automata with quantum and classical states than by classical two-way automata in the case of sublogarithmic memory (sublinear size), even if classical automata get advice bits. Additionally, we show that there are problems that can be solved more efficiently by one-way quantum automata than by classical one-way automata in the case of sublogarithmic memory (resp., sublinear size) and in the case of logarithmic memory (resp., linear size) even if classical automata get advice bits.

On the decidability of the valuedness problem for two-way finite transducers

A transducer is infinite-valued if the maximal number of different outputs for an input string is not bounded by any constant. For one-way finite transducers, sufficient and necessary conditions exist in terms of the structure of a transducer to characterize whether the transducer is infinite-valued or not, yielding the decidability result of the finite-valuedness problem. As crossing sequences in two-way automata often play similar roles as states in their one-way counterparts, we consider analogous criteria in the setting of crossing sequences to characterize the infinite-valuedness of two-way finite transducers. The characterization leads to a decidability proof for the valuedness problem of two-way finite transducers.

Minimal Size of Counters for (Real-Time) Multicounter Automata

We show that, for automata using a finite number of counters, the minimal space that is required for accepting a nonregular language is (log n)ɛ. This is required for weak space bounds on the size of their counters, for real-time and one-way, and for nondeterministic and alternating versions of these automata. The same holds for two-way automata, independent of whether they work with strong or weak space bounds, and of whether they are deterministic, nondeterministic, or alternating. (Here ɛ denotes an arbitrarily small—but fixed—constant; the “space” refers to the values stored in the counters, rather than to the lengths of their binary representation.) On the other hand, we show that the minimal space required for accepting a nonregular language is nɛ for multicounter automata with strong space bounds, both for real-time and one-way versions, independent of whether they are deterministic, nondeterministic, or alternating, and also for real-time and one-way deterministic multicounter automata with weak space bounds. All these bounds are optimal both for unary and general nonregular languages. However, for automata equipped with only one counter, it was known that one-way nondeterministic automata cannot recognize any unary nonregular languages at all, even if the size of the counter is not restricted, while, with weak space bound log n, we present a real-time nondeterministic automaton recognizing a binary nonregular language here.

On the Length of Shortest Strings Accepted by Two-way Finite Automata

Given a two-way finite automaton recognizing a non-empty language, consider the length of the shortest string it accepts, and, for each n ≥ 1, let f(n) be the maximum of these lengths over all n-state automata. It is proved that for n-state two-way finite automata, whether deterministic or nondeterministic, this number is at least Ω(10n/5) and less than (2nn+1), with the lower bound reached over an alphabet of size Θ(n). Furthermore, for deterministic automata and for a fixed alphabet of size m ≥ 1, the length of the shortest string is at least e(1+o(1))mn(log n− log m).

Two-dimensional pattern matching against local and regular-like picture languages

Given a two-dimensional array of symbols and a picture language over a finite alphabet, we investigate how to find rectangular subarrays that belong to the picture language. Two-dimensional pattern matching problems can be formulated by interpreting subarrays as matches and picture languages as patterns. We formulate four particular problems – finding maximum, minimum, any or all match(es) – and describe algorithms solving the problems for basic classes of picture languages, which include local picture languages and picture languages accepted by various types of deterministic two-dimensional finite automata such as four-way, three-way and two-way automata, on-line tessellation automata, and returning finite automata. We also prove that the pattern matching problems cannot be solved for the class of local picture languages in time linear in the input area unless the well known problem of triangle finding in a graph is solvable in quadratic time. This shows a fundamental difference between the complexity of one-dimensional and two-dimensional pattern matching.

Alternation in two-way finite automata

The study of alternation in two-way finite automata ( 2 fas) has been quite non-systematic. Since the 1970's, various authors with a variety of motivations have studied 2 fas with various types of alternation and a variety of names, creating a fairly long list of sporadic contributions with little internal consistency. This article attempts to organize the subject into a single unifying framework.We start with a detailed account of all contributions to date, that reveals the large variety of approaches and the lack of consistency between them. We then identify and name four types of automata that these contributions have really studied over the years: general two-way Boolean finite automata ( 2 b fas); monotone 2 b fas; basic 2 b fas; and (monotone basic, or) alternating 2 b fas ( 2 a fas). Next, we identify four different ways by which authors have described how such automata compute, each offering a distinct view onto their operation: the circuit view, where computation is modeled by a circuit of Boolean gates; the formula view, where computation is modeled by a Boolean formula over configuration-variables; the run view, where decisions are determined by the existence of appropriate trees of configuration-goal pairs, called “runs”; and the (most classic) tree view, where computation is modeled by a tree of configurations.After carefully defining each 2 b fa type and each view, we prove the following. First, that within each type, every two of the four views are equivalent to each other, in the strong sense that each of them closely mimics the other at every step of the computation. Second, that not all types of 2 b fas are equivalent: although general 2 b fas are as powerful as monotone 2 b fas, and basic 2 b fas are as powerful as 2 a fas (up to polynomial differences in the number of states), general 2 b fas may need exponentially fewer states than 2 a fas.

Reversibility of computations in graph-walking automata

Graph-walking automata (GWA) are finite-state devices that traverse graphs given as an input by following their edges; they have been studied both as a theoretical notion and as a model of pathfinding in robotics. If a graph is regarded as the set of memory configurations of a certain abstract machine, then various families of devices can be described as GWA: such are two-way finite automata, their multi-head and multi-tape variants, tree-walking automata and their extension with pebbles, picture-walking automata, space-bounded Turing machines, etc. This paper defines a transformation of an arbitrary deterministic GWA to a reversible GWA. This is done with a linear blow-up in the number of states, where the constant factor depends on the degree of the graphs being traversed. The construction directly applies to all basic models representable as GWA, and, in particular, subsumes numerous existing results for making individual models halt on every input.

Evaluating Datalog via Tree Automata and Cycluits

We investigate parameterizations of both database instances and queries that make query evaluation fixed-parameter tractable in combined complexity. We show that clique-frontier-guarded Datalog with stratified negation (CFG-Datalog) enjoys bilinear-time evaluation on structures of bounded treewidth for programs of bounded rule size. Such programs capture in particular conjunctive queries with simplicial decompositions of bounded width, guarded negation fragment queries of bounded CQ-rank, or two-way regular path queries. Our result is shown by translating to alternating two-way automata, whose semantics is defined via cyclic provenance circuits (cycluits) that can be tractably evaluated.

On the power of two-way multihead quantum finite automata

This paper introduces a variant of two-way quantum finite automata named two-way multihead quantum finite automata. A two-way quantum finite automaton is more powerful than classical two-way finite automata. However, the generalizations of two-way quantum finite automata have not been defined so far as compared to one-way quantum finite automata model. We have investigated the newly introduced automata from two aspects: the language recognition capability and its comparison with classical and quantum counterparts. It has been proved that a language which cannot be recognized by any one-way and multi-letter quantum finite automata can be recognized by two-way quantum finite automata. Further, it has been shown that a language which cannot be recognized by two-way quantum finite automata can be recognized by two-way multihead quantum finite automata with two heads. Furthermore, it has been investigated that quantum variant of two-way deterministic multihead finite automata takes less number of heads to recognize a language containing of all words whose length is a prime number.

Modeling of RNA secondary structures using two-way quantum finite automata

Quantum finite automata (QFA) play a crucial role in quantum information processing theory. The representation of ribonucleic acid (RNA) and deoxyribonucleic acid (DNA) structures using theory of automata had a great influence in the computer science. Investigation of the relationship between QFA and RNA structures is a natural goal. Two-way quantum finite automata (2QFA) is more dominant than its classical model in language recognition. Motivated by the concept of quantum finite automata, we have modeled RNA secondary structure loops such as, internal loop and double helix loop using two-way quantum finite automata. The major benefit of this approach is that these sequences can be parsed in linear time. In contrast, two-way deterministic finite automata (2DFA) cannot represent RNA secondary structure loops and two-way probabilistic finite automata (2PFA) can parse these sequences in exponential time. To the best of authors knowledge this is the first attempt to represent RNA secondary structure loops using two-way quantum finite automata.

New Size Hierarchies for Two Way Automata

We introduce a new type of nonuniform two-way automaton that can use a different transition function for each tape square. We also enhance this model by allowing to shuffle the given input at the beginning of the computation. Then we present some hierarchy and incomparability results on the number of states for the types of deterministic, nondeterministic, and bounded-error probabilistic models. For this purpose, we provide some lower bounds for all three models based on the numbers of subfunctions and we define two witness functions.

Accepting runs in a two-way finite automaton

An accepting run in a two-way finite automaton M is a sequence of states that M enters during some accepting computation. The set of all such runs is denoted by Lrun,M. We study the complexity of Lrun,M when M is a 2NFA (2DFA). We also look at the complexity of “position sampling” (the sequence of states that M enters in specified positions of some accepted input) in a 2NFA. In particular, we give some language properties of sampled runs of 2NFAs augmented with restricted unbounded storage.

On the Size of Two-Way Reasonable Automata for the Liveness Problem

The existence of a substantial gap between deterministic and nondeterministic two-way automata is one of the most famous open problems in automata theory. This problem is also related to the fundamental DLOG vs. NLOG question. An exponential gap between the number of states of two-way nondeterministic automata (2NFAS) and their deterministic counterparts (2DFAS) has been proved only for some restrictions of 2DFAS up to now. It seems that the hardness of this problem lies in the fact that, when trying to prove lower bounds, we must consider every possible automaton, without imposing any particular structure or meaning to the states, while when designing a specific automaton, we always assign an unambiguous interpretation to the states. In an attempt to capture the concept of meaning of states, a new model of two-way automata, namely reasonable automaton (RA), was introduced in [6]. In a RA, each state is associated with a logical formula expressing some properties of the input word, and transitions are designed to maintain consistency within this setting. In this paper we extend the study, started in [6], of the descriptional complexity of RAS solving the liveness problem, showing several lower and upper bounds with respect to the logic used for describing the meaning of the individual states.

Descriptional complexity of limited automata

A k-limited automaton is a linear bounded automaton that may rewrite each tape cell only in the first k visits, where k≥0 is a fixed constant. It is known that these automata accept context-free languages only. We investigate the descriptional complexity of limited automata. Since the unary languages accepted are necessarily regular, we first study the cost in the number of states when finite automata simulate a unary k-limited automaton. For the conversion of a 4n-state deterministic 1-limited automaton into one-way or two-way deterministic or nondeterministic finite automata, we show a lower bound of n⋅F(n) states, where F denotes Landau's function. So, even the ability to deterministically rewrite any cell only once gives an enormous descriptional power. For the simulation cost for removing the ability to rewrite each cell k≥1 times, more precisely, the cost for the simulation of sweeping unary k-limited automata by deterministic finite automata, we obtain a lower bound of n⋅F(n)k. The upper bound of the cost for the simulation by two-way deterministic finite automata is a polynomial whose degree is quadratic in k. If the k-limited automaton is rotating, the upper bound reduces to O(nk+1) and the lower bound derived is Ω(nk+1) even for nondeterministic two-way finite automata. So, for rotating k-limited automata, the trade-off for the simulation is tight in the order of magnitude. Finally, we consider the simulation of k-limited automata over general alphabets by pushdown automata. It turns out that the cost is an exponential blow-up of the size. Furthermore, an exponential size is also necessary.

On the state complexity of operations on two-way finite automata

The paper investigates the effect of basic language-theoretic operations on the number of states in two-way deterministic finite automata (2DFAs). If m and n are the number of states in the 2DFAs recognizing the arguments of the following operations, then their result requires the following number of states: at least m+n−o(m+n) and at most 4m+n+const for union; at least m+n−o(m+n) and at most m+n+1 for intersection; at least Ω(mn)+2Ω(n)log⁡m and at most 2mm+1⋅2nn+1 for concatenation; at least 1n2n2−1 and at most 2O(nn+1) for Kleene star, square and projections; between n+1 and n+2 for reversal; exactly 2n for inverse homomorphisms. All results are obtained by first establishing high lower bounds on the number of states in any 1DFAs recognizing these languages, and then using these bounds to reason about the size of any equivalent 2DFAs.

Two-way representations and weighted automata

We study the series realized by weighted two-way automata, that are strictly more powerful than weighted one-way automata. To this end, we consider the Hadamard product and the Hadamard iteration of formal power series. We introduce two-way representations and show that the series they realize are the solutions of fixed-point equations. In rationally additive semirings, we prove that two-way automata are equivalent to two-way representations, and, for some specific classes of two-way automata, rotating and sweeping automata, we give a characterization of the series that can be realized.