311 research outputs found
Converting two-way nondeterministic unary automata into simpler automata
AbstractWe show that, on inputs of length exceeding 5n2, any n-state unary two-way nondeterministic finite automaton (2nfa) can be simulated by a (2n+2)-state quasi-sweeping 2nfa. Such a result, besides providing a “normal form” for 2nfa's, enables us to get a subexponential simulation of unary 2nfa's by two-way deterministic finite automata (2dfa's). In fact, we prove that any n-state unary 2nfa can be simulated by a sweeping 2dfa with O(n⌈log2(n+1)+3⌉) states
Translation from Classical Two-Way Automata to Pebble Two-Way Automata
We study the relation between the standard two-way automata and more powerful
devices, namely, two-way finite automata with an additional "pebble" movable
along the input tape. Similarly as in the case of the classical two-way
machines, it is not known whether there exists a polynomial trade-off, in the
number of states, between the nondeterministic and deterministic pebble two-way
automata. However, we show that these two machine models are not independent:
if there exists a polynomial trade-off for the classical two-way automata, then
there must also exist a polynomial trade-off for the pebble two-way automata.
Thus, we have an upward collapse (or a downward separation) from the classical
two-way automata to more powerful pebble automata, still staying within the
class of regular languages. The same upward collapse holds for complementation
of nondeterministic two-way machines.
These results are obtained by showing that each pebble machine can be, by
using suitable inputs, simulated by a classical two-way automaton with a linear
number of states (and vice versa), despite the existing exponential blow-up
between the classical and pebble two-way machines
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
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
Parikh Image of Pushdown Automata
We compare pushdown automata (PDAs for short) against other representations.
First, we show that there is a family of PDAs over a unary alphabet with
states and stack symbols that accepts one single long word for
which every equivalent context-free grammar needs
variables. This family shows that the classical algorithm for converting a PDA
to an equivalent context-free grammar is optimal even when the alphabet is
unary. Moreover, we observe that language equivalence and Parikh equivalence,
which ignores the ordering between symbols, coincide for this family. We
conclude that, when assuming this weaker equivalence, the conversion algorithm
is also optimal. Second, Parikh's theorem motivates the comparison of PDAs
against finite state automata. In particular, the same family of unary PDAs
gives a lower bound on the number of states of every Parikh-equivalent finite
state automaton. Finally, we look into the case of unary deterministic PDAs. We
show a new construction converting a unary deterministic PDA into an equivalent
context-free grammar that achieves best known bounds.Comment: 17 pages, 2 figure
Syntactic Minimization Of Nondeterministic Finite Automata
Nondeterministic automata may be viewed as succinct programs implementing deterministic automata, i.e. complete specifications. Converting a given deterministic automaton into a small nondeterministic one is known to be computationally very hard; in fact, the ensuing decision problem is PSPACE-complete. This paper stands in stark contrast to the status quo. We restrict attention to subatomic nondeterministic automata, whose individual states accept unions of syntactic congruence classes. They are general enough to cover almost all structural results concerning nondeterministic state-minimality. We prove that converting a monoid recognizing a regular language into a small subatomic acceptor corresponds to an NP-complete problem. The NP certificates are solutions of simple equations involving relations over the syntactic monoid. We also consider the subclass of atomic nondeterministic automata introduced by Brzozowski and Tamm. Given a deterministic automaton and another one for the reversed language, computing small atomic acceptors is shown to be NP-complete with analogous certificates. Our complexity results emerge from an algebraic characterization of (sub)atomic acceptors in terms of deterministic automata with semilattice structure, combined with an equivalence of categories leading to succinct representations
Logics for Unranked Trees: An Overview
Labeled unranked trees are used as a model of XML documents, and logical
languages for them have been studied actively over the past several years. Such
logics have different purposes: some are better suited for extracting data,
some for expressing navigational properties, and some make it easy to relate
complex properties of trees to the existence of tree automata for those
properties. Furthermore, logics differ significantly in their model-checking
properties, their automata models, and their behavior on ordered and unordered
trees. In this paper we present a survey of logics for unranked trees
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
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
- …