Giovanni Pighizzini

• PhD
• Professor (Full) at University of Milan

Single-tape nondeterministic Turing machines that are allowed to replace the symbol in each tape cell only when it is scanned for the first time are also known as 1-limited automata. These devices characterize, exactly as finite automata, the class of regular languages. However, they can be extremely more succinct. Indeed, in the worst case, the si...

The descriptional complexity of basic operations on regular languages using 1-limited automata, a restricted version of one-tape Turing machines, is investigated. When simulating operations on deterministic finite automata with deterministic 1-limited automata, the sizes of the resulting devices are polynomial in the sizes of the simulated machines...

Single-tape nondeterministic Turing machines that are allowed to replace the symbol in each tape cell only when it is scanned for the first time are also known as 1-limited automata. These devices characterize, exactly as finite automata, the class of regular languages. However, they can be extremely more succinct. Indeed, in the worst case the siz...

We consider de Bruijn words and their recognition by finite automata. While on one-way nondeterministic automata the recognition of de Bruijn words of order k requires exponentially many states in k, we show a family of de Bruijn words such that the word \(w_k\) of order k, for \(k>0\), can be recognized by a deterministic two-way finite automaton...

We introduce and investigate forgetting 1-limited automata, which are single-tape Turing machines that, when visit a cell for the first time, replace the input symbol in it by a fixed symbol, so forgetting the original contents. These devices have the same computational power as finite automata, namely they characterize the class of regular languag...

It cannot be decided whether a one-counter automaton accepts each string in its language using a counter whose value is bounded by a constant. Furthermore, when the counter is bounded by a constant, its value cannot be limited by any recursive function in the size of the machine. We consider three measures: the costs of all computations (strong mea...

The descriptional complexity of basic operations on regular languages using 1-limited automata, a restricted version of one-tape Turing machines, is investigated. When simulating operations on deterministic finite automata with deterministic 1-limited automata, the sizes of the resulting devices are polynomial in the sizes of the simulated machines...

It cannot be decided whether a pushdown automaton accepts using a pushdown height, which does not depend on the input length, i.e., when it accepts using constant height. Furthermore, when a pushdown automaton accepts in constant height, the height can be arbitrarily large with respect to the size of the description of the machine, namely it does n...

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 bo...

Finite automata whose computations can be reversed, at any point, by knowing the last k symbols read from the input, for a fixed k, are considered. These devices and their accepted languages are called k-reversible automata and k-reversible languages, respectively. The existence of k-reversible languages which are not (k − 1)-reversible is known, f...

We continue the research on usefulness of information examining the effect of supplementary information on the complexity of solving a problem (see Rovan and Sádovský [1] for an overview). We use deterministic finite automata for a formal setting. Given a problem (a regular language) Lprob we measure the complexity of its solution — a DFA Aprob suc...

The descriptional complexity of basic operations on regular languages using 1-limited automata, a restricted version of one-tape Turing machines, is investigated. When simulating operations on deterministic finite automata with deterministic 1-limited automata, the sizes of the resulting devices are polynomial in the sizes of the simulated machines...

Descriptional complexity has historically been a multidisciplinary area of study, with contributions from automata theory, computational complexity, cryptography, information theory, probability, statistics, pattern recognition, machine learning, computational learning theory, computer vision, neural networks, formal languages and other fields. Som...

Non-self-embedding grammars are a subclass of context-free grammars which only generate regular languages. The size costs of the conversion of non-self-embedding grammars into equivalent finite automata are studied, by proving optimal bounds for the number of states of nondeterministic and deterministic automata equivalent to given non-self-embeddi...

It is well-known that one-tape Turing machines working in linear time are no more powerful than finite automata, namely they recognize exactly the class of regular languages. We prove that it is not decidable if a one-tape machine works in linear time, even if it is deterministic and restricted to use only the portion of the tape which initially co...

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 automa...

part : TC 1: Foundations of Computer Science

Non-self-embedding grammars are a restriction of context-free grammars which does not allow to describe recursive structures and, hence, which characterizes only the class of regular languages. A double exponential gap in size from non-self-embedding grammars to deterministic finite automata is known. The same size gap is also known from constant-h...

Limited automata are single-tape Turing machines with severe rewriting restrictions. They have been introduced in 1967 by Thomas Hibbard, who proved that they have the same computational power as pushdown automata. Hence, they provide an alternative characterization of the class of context-free languages in terms of recognizing devices. After that...

It cannot be decided whether a pushdown automaton accepts using constant pushdown height, with respect to the input length, or not. Furthermore, in the case of acceptance in constant height, the height cannot be bounded by any recursive function in the size of the description of the machine. In contrast, in the restricted case of pushdown automata...

Limited automata are one-tape Turing machines that can rewrite the contents of tape cells only in the first d visits, for a fixed d. When d=1 these models characterize regular languages. We show an exponential gap between the size of limited automata accepting unary languages and the size of equivalent finite automata. Despite this gap, there are u...

Non-self-embedding grammars are a restriction of context-free grammars which does not allow to describe recursive structures and, hence, which characterizes only the class of regular languages. A double exponential gap in size from non-self-embedding grammars to deterministic finite automata is known. The same size gap is also known from constant-h...

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...

Finite automata whose computations can be reversed, at any point, by knowing the last k symbols read from the input, for a fixed k, are considered. These devices and their accepted languages are called k-reversible automata and k-reversible languages, respectively. The existence of k-reversible languages which are not (k-1)-reversible is known, for...

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. When \(d=1\) these models characterize regular languages. An exponential gap between the size of limited automata accepting unary languages and the size of equivalent finite automata is proved. S...

A condition characterizing the class of regular languages which have several nonisomorphic minimal reversible automata is presented. The condition concerns the structure of the minimum automaton accepting the language under consideration. It is also observed that there exist reduced reversible automata which are not minimal, in the sense that all t...

A condition characterizing the class of regular languages which have several nonisomorphic minimal reversible automata is presented. The condition concerns the structure of the minimum automaton accepting the language under consideration. It is also observed that there exist reduced reversible automata which are not minimal, in the sense that all t...

Nondeterministic finite automata with don't care states, namely states which neither accept nor reject, are considered. A characterization of deterministic automata compatible with such a device is obtained. Furthermore, an optimal state bound for the smallest compatible deterministic automata is provided. It is proved that the problem of minimizin...

In 1965 Hennie proved that one-tape deterministic Turing machines working in linear time are equivalent to finite automata, namely they characterize regular languages. This result has been improved in different directions, by obtaining optimal lower bounds for the time that one-tape deterministic and nondeterministic Turing machines need to recogni...

The investigation of automata and languages defined over a one letter alphabet shows interesting differences with respect to the case of alphabets with at least two letters. Probably, the oldest example emphasizing one of these differences is the collapse of the classes of regular and context-free languages in the unary case (Ginsburg and Rice, 196...

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 ≥ 2. Furthermore, for d = 2 deterministic limited automata are equiva...

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...

Two language operations that can be expressed by suitably combining complement with concatenation and star, respectively, are introduced. The state complexity of those operations on regular languages is investigated. In the deterministic case, optimal exponential state gaps are proved for both operations. In the nondeterministic case, for one opera...

Let L/poly and NL be the standard complexity classes, of languages recognizable in logarithmic space by Turing machines which are deterministic with polynomially-long advice and nondeterministic without advice, respectively. We recast the question whether L/poly ⊇ NL in terms of deterministic and nondeterministic two-way finite automata (2DFAs and...

Nondeterministic finite automata with don't care states, namely states which neither accept nor reject, are considered. A characterization of deterministic automata compatible with such a device is obtained. Furthermore, an optimal state bound for the smallest compatible deterministic automata is provided. Finally, it is proved that the problem of...

Limited automata are one-tape Turing machines which are allowed to rewrite each tape cell only in the first d visits, for a given constant d. For each d >= 2, these devices characterize the class of context-free languages. We investigate the equivalence between 2-limited automata and pushdown automata, comparing the relative sizes of their descript...

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. In the case d=1, namely, when a rewriting is possible only during the first visit to a cell, these models have the same power of finite state automata. We prove state upper and lower bounds for t...

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. In the case d = 1, namely, when a rewriting is possible only during the first visit to a cell, these models have the same power of finite state automata. We prove state upper and lower bounds for...

We investigate the descriptional complexity and decidability of obliviousness for two-way finite automata. In particular, we consider the simulation of two-way deterministic finite automata (2DFAs) by oblivious 2DFAs, the simulation of oblivious 2DFAs by sweeping 2DFAs and one-way nondeterministic finite automata (1NFAs) as well as the simulation o...

We investigate, under Parikh equivalence, the state complexity of some language operations which preserve regularity. For union, concatenation, Kleene star, complement, intersection, shuffle, and reversal, we obtain a polynomial state complexity over any fixed alphabet, in contrast to the intrinsic exponential state complexity of some of these oper...

The investigation of automata and languages defined over a one letter alphabet shows interesting differences with respect to the case of alphabets with at least two letters. Probably, the oldest example emphasizing one of these differences is the collapse of the classes of regular and context-free languages in the unary case (Ginsburg and Rice, 196...

We investigate the conversion of nondeterministic finite automata and context-free grammars into Parikh equivalent deterministic finite automata, from a descriptional complexity point of view. We prove that for each nondeterministic automaton with n states there exists a Parikh equivalent deterministic automaton with \(e^{O(\sqrt{n \cdot \ln n})}\)...

We investigate the possibility of extending Chrobak normal form to the probabilistic case. While in the nondeterministic case a unary automaton can be simulated by an automaton in Chrobak normal form without increasing the number of the states in the cycles, we show that in the probabilistic case the simulation is not possible by keeping the same n...

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...

A two-way deterministic finite automaton with r(n) reversals performs ≤r(n) input head reversals on every n-long input. Let 2D[r(n)] be all families of problems solvable by such automata of size polynomial in the index of the family. Then the reversal hierarchy 2D[0] ⊆ 2D[1] ⊆ 2D[2] ⊆ ⋯ is strict, but 2D[O(1)] = 2D[o(n)]. Moreover, the inner-revers...

A two-way deterministic finite automaton with r(n) reversals performs ≤r(n) input head reversals on every n-long input. Let 2D[r(n)] be all families of problems solvable by such automata of size polynomial in the index of the family. Then the reversal hierarchy 2D[0] ⊆ 2D[1] ⊆ 2D[2] ⊆ ⋯ is strict, but 2D[O(1)] = 2D[o(n)]. Moreover, the inner-revers...

The notion of two-way automata was introduced at the very beginning of automata theory. In 1959, Rabin and Scott and, independently, Shepherdson, proved that these models, both in the deterministic and in the nondeterministic versions, have the same power of one-way automata, namely, they characterize the class of regular languages. In 1978, Sakoda...

We investigate the descriptional complexity of some inverse language operations applied to languages accepted by finite automata. For instance, the inverse Kleene star operation for a language L asks for the smallest language S such that S * is equal to L, if it exists [J. Brzozowski. Roots of star events. J. ACM 14, 1967]. Other inverse operations...

Let L/poly and NL be the standard complexity classes, of languages recognizable in logarithmic space by Turing machines which are deterministic with polynomially-long advice and nondeterministic without advice, respectively. We recast the question whether L/poly⊇NL in terms of deterministic and nondeterministic two-way finite automata (2DFAs and 2N...

Let L/poly and NL be the standard complexity classes, of languages recognizable in logarithmic space by Turing machines which are deterministic with polynomially-long advice and nondeterministic without advice, respectively. We recast the question whether L/poly⊇NL in terms of deterministic and nondeterministic two-way finite automata (2DFAs and 2N...

It is well known that for each context-free language there exists a regular language with the same Parikh image. We investigate this result from a descriptional complexity point of view, by proving tight bounds for the size of deterministic automata accepting regular languages Parikh equivalent to some kinds of context-free languages. First, we pro...

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...

Self-verifying automata are a special variant of finite automata with a symmetric kind of nondeterminism. We study the conversion of self-verifying automata to deterministic automata from a descriptional complexity point of view. The main result is the exact cost, in terms of the number of states, of such a simulation.

We investigate and compare the descriptional power of unary probabilistic and nondeterministic automata (pfa's and nfa's, respectively). We show the existence of a family of languages hard for pfa's in the following sense: For any positive integer d, there exists a unary d-cyclic language such that any pfa accepting it requires d states, as the sma...

We strengthen previously known connections between the size complexity of two-way finite automata (2fas) and the space complexity of Turing machines. We prove that • every s-state 2nfa can be simulated on all poly(s)-long inputs by some poly(s)-state 2dfa if and only if NL⊆ L/poly and • every s-state 2nfa can be simulated on all 2 poly(s)-lo...

Descriptional Complexity of Formal Systems (DCFS) is the successor workshop and the merger of two related workshops, Descriptional Complexity of Automata, Grammars and Related Structures (DCAGRS) and Formal Descriptions and Software Reliability (FDSR). The DCAGRS workshop took place in Magdeburg, Germany (1999), London, Ontario, Canada (2000), and...

The 12th annual workshop, Descriptional Complexity of Formal Systems 2010, is taking place in Saskatoon, Canada, on August 8-10, 2010. It is jointly organized by the IFIP Working Group 1.2 on Descriptional Complexity and by the Department of Computer Science at the University of Saskatchewan. This volume contains the papers of the invited lectures...

The 12th annual workshop, Descriptional Complexity of Formal Systems 2010, is taking place in Saskatoon, Canada, on August 8-10, 2010. It is jointly organized by the IFIP Working Group 1.2 on Descriptional Complexity and by the Department of Computer Science at the University of Saskatchewan. This volume contains the papers of the invited lectures...

For each sufficiently large n, there exists a unary regular language L such that both L and its complement L (c) are accepted by unambiguous nondeterministic automata with at most n states, while the smallest deterministic automata for these two languages still require a superpolynomial number of states, at least e(Omega)((3)root n.ln(2) n. Actuall...

We show that, for any ε > 0, there exists a language accepted in strong ε log n space by a 2-way deterministic Turing machine working with a single binary worktape, that cannot be accepted in sublogarithmic weak space by any pebble machine (i.e., a 2-way nondeterministic Turing machine with one pebble that can be moved onto the input cell...

We show that, for any ε>0, there exists a language accepted in strong ε-log n space by a 2-way deterministic Turing machine working with a single binary worktape, that cannot be accepted in sublogarithmic weak space by any pebble machine (i.e., a 2-way nondeterministic Turing machine with one pebble that can be moved onto the input cells). Moreover...

We consider some questions about formal languages that arise when inverses of letters, words and languages are defined. The reduced representation of a language over the free monoid is its unique equivalent representation in the free group. We show that the class of regular languages is closed under taking the reduced representation, while the clas...

The 11th workshop, Descriptional Complexity of Formal Systems 2009, is taking place in Magdeburg, Germany, on July 6-9, 2009. It is jointly organized by the IFIP Working Group 1.2 on Descriptional Complexity and by the Faculty of Computer Science at the Otto von Guericke University Magdeburg. This volume contains the papers of the invited lectures...

The 11th workshop, Descriptional Complexity of Formal Systems 2009, is taking place in Magdeburg, Germany, on July 6-9, 2009. It is jointly organized by the IFIP Working Group 1.2 on Descriptional Complexity and by the Faculty of Computer Science at the Otto von Guericke University Magdeburg. This volume contains the papers of the invited lectures...

The simulation of deterministic pushdown automata defined over a one-letter alphabet by finite state automata is investigated from a descriptional complexity point of view. We show that each unary deterministic pushdown automaton of size s can be simulated by a deterministic finite automaton with a number of states that is exponential in s. We prov...

In this paper we consider the time and the crossing sequence complexities of one-tape off-line Turing machines. We show that the running time of each nondeterministic machine accepting a nonregular language must grow at least as n\log n, in the case all accepting computations are considered (accept measure). We also prove that the maximal length of...

Finite-turn pushdown automata (PDA) are investigated concerning their descriptional complexity. It is known that they accept exactly the class of ultralinear context-free languages. Furthermore, the increase in size when converting arbitrary PDAs accepting ultralinear languages to finite-turn PDAs cannot be bounded by any recursive function. The la...

A pumping condition for ultralinear languages is proved. Some examples are given using that condition to state lower bounds on the rank of some ultralinear languages and to prove that a certain language is not ultralinear. The restriction to the metalinear case is also investigated.

We study the relationship between the sizes of two-way finite automata accepting a language and its complement. In the deterministic case, for a given automaton (2dfa) with n states, we build an automaton accepting the complement with at most 4n states, independently of the size of the input alphabet. Actually, we show a stronger result, by present...

In this paper we characterize concurrent alphabets for which every recognizable trace language admits a minimum finite state asynchronous automaton. Furthermore, we consider the equivalence problem for unambiguous regular trace languages, and prove that in some cases it is decidable even if the concurrency relation is not transitive.

In this paper we analyze some intrusion detection strategies proposed in the literature and we show that they represent the various facets of a well known formal languages problem: computing the distance between a string x and a language L. In particular, the main differences among the various approaches adopted for building intrusion detection sys...

We study the relationship between the sizes of two-way finite automata accepting a language and its complement. In the deterministic case, by adapting Sipser’s method, for a given automaton (2dfa) with n states we build an automaton accepting the complement with at most 4n states, independently of the size of the input alphabet. Actually, we show a...

We compare the nondeterministic state complexity of unary regular languages and that of their complements: if a unary language L has a succinct nondeterministic finite automaton, then nondeterminism is useless in order to recognize its complement, namely, the smallest nondeterministic automaton accepting the complement of L has as many states as th...