Two-Way Automata Making Choices Only at the Endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2DFAs (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2DFAs. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2NFAs and 2DFAs can be obtained only for unrestricted 2NFAs using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2DFAs would imply LNL. In the same way, the alternating version of these machines is related to L =? NL =? P, the classical computational complexity problems.Comment: 23 page

Two-way automata making choices only at the endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2 nfas) by two-way deterministic automata (2 dfas) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2 dfas (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2 nfas. Here we use an opposite approach, increasing the power of 2 dfas to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2 dfas. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2 nfas and 2 dfas can be obtained only for unrestricted 2 nfas using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2 dfas would imply L 60 NL. In the same way, the alternating version of these machines is related to L over(=, ?) NL over(=, ?) P, the classical computational complexity problems. \ua9 2014 Elsevier Inc. All rights reserved

REGULAR LANGUAGES: TO FINITE AUTOMATA AND BEYOND - SUCCINCT DESCRIPTIONS AND OPTIMAL SIMULATIONS

\uc8 noto che i linguaggi regolari \u2014 o di tipo 3 \u2014 sono equivalenti agli automi a stati finiti. Tuttavia, in letteratura sono presenti altre caratterizzazioni di questa classe di linguaggi, in termini di modelli riconoscitori e grammatiche. Per esempio, limitando le risorse computazionali di modelli pi\uf9 generali, quali grammatiche context-free, automi a pila e macchine di Turing, che caratterizzano classi di linguaggi pi\uf9 ampie, \ue8 possibile ottenere modelli che generano o riconoscono solamente i linguaggi regolari. I dispositivi risultanti forniscono delle rappresentazioni alternative dei linguaggi di tipo 3, che, in alcuni casi, risultano significativamente pi\uf9 compatte rispetto a quelle dei modelli che caratterizzano la stessa classe di linguaggi. Il presente lavoro ha l\u2019obiettivo di studiare questi modelli formali dal punto di vista della complessit\ue0 descrizionale, o, in altre parole, di analizzare le relazioni tra le loro dimensioni, ossia il numero di simboli utilizzati per specificare la loro descrizione. Sono presentati, inoltre, alcuni risultati connessi allo studio della famosa domanda tuttora aperta posta da Sakoda e Sipser nel 1978, inerente al costo, in termini di numero di stati, per l\u2019eliminazione del nondeterminismo dagli automi stati finiti sfruttando la capacit\ue0 degli automi two-way deterministici di muovere la testina avanti e indietro sul nastro di input.It is well known that regular \u2014 or type 3 \u2014 languages are equivalent to finite automata. Nevertheless, many other characterizations of this class of languages in terms of computational devices and generative models are present in the literature. For example, by suitably restricting more general models such as context-free grammars, pushdown automata, and Turing machines, that characterize wider classes of languages, it is possible to obtain formal models that generate or recognize regular languages only. The resulting formalisms provide alternative representations of type 3 languages that may be significantly more concise than other models that share the same expressing power. The goal of this work is to investigate these formal systems from a descriptional complexity perspective, or, in other words, to study the relationships between their sizes, namely the number of symbols used to write down their descriptions. We also present some results related to the investigation of the famous question posed by Sakoda and Sipser in 1978, concerning the size blowups from nondeterministic finite automata to two-way deterministic finite automata

Two-wayness: Automata and Transducers

This PhD is about two natural extensions of Finite Automata (FA): the 2-way fa (2FA) and the 2-way transducers (2T). It is well known that 2FA s are computably equivalent to FAs, even in their nondeterministic (2nfa) variant. However, in the field of descriptional complexity, some questions remain. Raised by Sakoda and Sipser in 1978, the question of the cost of the simulation of 2NFA by 2DFA (the deterministic variant of 2FA) is still open. In this manuscript, we give an answer in a restricted case in which the nondeterministic choices of the simulated 2NFA may occur at the boundaries of the input tape only (2ONFA). We show that every 2ONFA can be simulated by a 2DFA of subexponential (but superpolynomial) size. Under the assumptions L=NL, this cost is reduced to the polynomial level. Moreover, we prove that the complementation and the simulation by a halting 2ONFA is polynomial. We also consider the anologous simulations for alternating devices. Providing a one-way write-only output tape to FAs leads to the notion of transducer. Contrary to the case of finite automata which are acceptor, 2-way transducers strictly extends the computational power of 1-way one, even in the case where both the input and output alphabets are unary. Though 1-way transducers enjoy nice properties and characterizations (algebraic, logical, etc. . . ), 2-way variants are less known, especially the nondeterministic case. In this area, this manuscript gives a new contribution: an algebraic characterization of the relations accepted by two-way transducers when both the input and output alphabets are unary. Actually, it can be reformulated as follows: each unary two-way transducer is equivalent to a sweeping (and even rotating) transducer. We also show that the assumptions made on the size of the alphabets are required, that is, sweeping transducers weakens the 2-way transducers whenever at least one of the alphabet is non-unary. On the path, we discuss on the computational power of some algebraic operations on word relations, introduced in the aim of describing the behavior of 2-way transducers or, more generally, of 2-way weighted automata. In particular, the mirror operation, consisting in reversing the input word in order to describe a right to left scan, draws our attention. Finally, we study another kind of operations, more adapted for binary word relations: the composition. We consider the transitive closure of relations. When the relation belongs to some very restricted sub-family of rational relations, we are able to compute its transitive closure and we set its complexity. This quickly becomes uncomputable when higher classes are considered