One-Tape Turing Machine Variants and Language Recognition

We present two restricted versions of one-tape Turing machines. Both characterize the class of context-free languages. In the first version, proposed by Hibbard in 1967 and called limited automata, each tape cell can be rewritten only in the first $d$ visits, for a fixed constant $d\geq 2$. Furthermore, for $d=2$ deterministic limited automata are equivalent to deterministic pushdown automata, namely they characterize deterministic context-free languages. Further restricting the possible operations, we consider strongly limited automata. These models still characterize context-free languages. However, the deterministic version is less powerful than the deterministic version of limited automata. In fact, there exist deterministic context-free languages that are not accepted by any deterministic strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of the September 2015 issue of SIGACT New

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

Elements of computability, decidability, and complexity (Third edition)

These lecture notes are intended to introduce the reader to the basic notions of computability theory, decidability, and complexity. More information on these subjects can be found in classical books such as [Cut80,Dav58,Her69,HoU79,Rog67]. The results reported in these notes are taken from those books and in various parts we closely follow their style of presentation. The reader is encouraged to look at those books for improving his/her knowledge on these topics. Some parts of the chapter on complexity are taken from the lecture notes of a beautiful course given by Prof. Leslie Valiant at Edinburgh University, Scotland, in 1979. It was, indeed, a very stimulating and enjoyable course. For the notions of Predicate Calculus we have used in this book the reader may refer to [Men87]. I would like to thank Dr. Maurizio Proietti at IASI-CNR (Roma, Italy), my colleagues, and my students at the University of Roma Tor Vergata and, in particular, Michele Martone. They have been for me a source of continuous inspiration and enthusiasm. Finally, I would like to thank Dr. Gioacchino Onorati and Lorenzo Costantini of the Aracne Publishing Company for their helpful cooperation

Elements of computability, decidability, and complexity (Third edition)

These lecture notes are intended to introduce the reader to the basic notions of computability theory, decidability, and complexity. More information on these subjects can be found in classical books such as [Cut80,Dav58,Her69,HoU79,Rog67]. The results reported in these notes are taken from those books and in various parts we closely follow their style of presentation. The reader is encouraged to look at those books for improving his/her knowledge on these topics. Some parts of the chapter on complexity are taken from the lecture notes of a beautiful course given by Prof. Leslie Valiant at Edinburgh University, Scotland, in 1979. It was, indeed, a very stimulating and enjoyable course. For the notions of Predicate Calculus we have used in this book the reader may refer to [Men87]. I would like to thank Dr. Maurizio Proietti at IASI-CNR (Roma, Italy), my colleagues, and my students at the University of Roma Tor Vergata and, in particular, Michele Martone. They have been for me a source of continuous inspiration and enthusiasm. Finally, I would like to thank Dr. Gioacchino Onorati and Lorenzo Costantini of the Aracne Publishing Company for their helpful cooperation

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 (e.g., restrictions of the class of simulated automata or of the class of simulating automata). However the problem remains open in the general case, the best-known upper bound being exponential. We present a new approach in which unrestricted nondeterministic finite automata are simulated by deterministic models extending two-way deterministic finite automata, paying a polynomial increase of size only. Indeed, we study the costs of the conversions of nondeterministic finite 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

Automata theory and formal languages

These lecture notes present some basic notions and results on Automata Theory, Formal Languages Theory, Computability Theory, and Parsing Theory. I prepared these notes for a course on Automata, Languages, and Translators which I am teaching at the University of Roma Tor Vergata. More material on these topics and on parsing techniques for context-free languages can be found in standard textbooks such as [1, 8, 9]. The reader is encouraged to look at those books. A theorem denoted by the triple k.m.n is in Chapter k and Section m, and within that section it is identified by the number n. Analogous numbering system is used for algorithms, corollaries, definitions, examples, exercises, figures, and remarks. We use ‘iff’ to mean ‘if and only if’. Many thanks to my colleagues of the Department of Informatics, Systems, and Production of the University of Roma Tor Vergata. I am also grateful to my stu- dents and co-workers and, in particular, to Lorenzo Clemente, Corrado Di Pietro, Fulvio Forni, Fabio Lecca, Maurizio Proietti, and Valerio Senni for their help and encouragement. Finally, I am grateful to Francesca Di Benedetto, Alessandro Colombo, Donato Corvaglia, Gioacchino Onorati, and Leonardo Rinaldi of the Aracne Publishing Com- pany for their kind cooperation