456 research outputs found
Algebraic properties of structured context-free languages: old approaches and novel developments
The historical research line on the algebraic properties of structured CF
languages initiated by McNaughton's Parenthesis Languages has recently
attracted much renewed interest with the Balanced Languages, the Visibly
Pushdown Automata languages (VPDA), the Synchronized Languages, and the
Height-deterministic ones. Such families preserve to a varying degree the basic
algebraic properties of Regular languages: boolean closure, closure under
reversal, under concatenation, and Kleene star. We prove that the VPDA family
is strictly contained within the Floyd Grammars (FG) family historically known
as operator precedence. Languages over the same precedence matrix are known to
be closed under boolean operations, and are recognized by a machine whose pop
or push operations on the stack are purely determined by terminal letters. We
characterize VPDA's as the subclass of FG having a peculiarly structured set of
precedence relations, and balanced grammars as a further restricted case. The
non-counting invariance property of FG has a direct implication for VPDA too.Comment: Extended version of paper presented at WORDS2009, Salerno,Italy,
September 200
Tree-Structured Problems and Parallel Computation
Turing-Maschinen sind das klassische Beschreibungsmittel fĂŒr Wortsprachen und werden daher auch benĂŒtzt, um KomplexitĂ€tsklassen zu definieren. Dies geschieht zum Beispiel durch das EinschrĂ€nken des Platz- oder Zeitaufwandes der Berechnung zur Lösung eines Problems. FĂŒr sehr niedrige KomplexitĂ€t wie etwa sublineare Laufzeit, werden Schaltkreise verwendet. Schaltkreise können auf natĂŒrliche Art KomplexitĂ€ten wie etwa logarithmische Laufzeit modellieren. Ebenso können sie als eine Art paralleles Rechenmodell gesehen werden. Eine wichtige parallele KomplexitĂ€tsklasse ist NC1. Sie wird beschrieben durch Boolesche Schaltkreise logarithmischer Tiefe und beschrĂ€nktem Eingangsgrad der Gatter.
Eine initiale Beobachtung, die die vorliegende Arbeit motiviert, ist, dass viele schwere Probleme in NC1 eine Ă€hnliche Struktur haben und auf Ă€hnliche Art und Weise gelöst werden. Das Auswertungsproblem fĂŒr Boolesche Formeln ist eines der reprĂ€sentativsten Probleme aus dieser Klasse:
Gegeben ist hier eine aussagenlogische Formel samt Belegung fĂŒr die Variablen; gefragt ist, ob sie zu wahr oder zu falsch auswertet. Dieses Problem wird in NC1 gelöst durch den Algorithmus von Buss. Auf Ă€hnliche Art können arithmetische Formeln in #NC1 ausgewertet oder das Wortproblem fĂŒr Visibly-Pushdown-Sprachen gelöst werden. Zu besagter Klasse an Problemen gehört auch Courcelles Theorem, welches Berechnungen in Baumautomaten involviert. Zu bemerken ist, dass alle angesprochenen Probleme gemeinsam haben, dass sie aus Instanzen bestehen, die baumartig sind. Formeln sind BĂ€ume, Visibly-Pushdown-Sprachen enthalten als Wörter kodierte BĂ€ume und Courcelles Theorem betrachtet Graphen mit beschrĂ€nkter Baumweite, d.h. Graphen, die sich als Baum darstellen lassen. Insbesondere Letzteres ist ein Schema, das hĂ€ufiger auftritt. Zum Beispiel gibt es NP-vollstĂ€ndige Graphprobleme wie das Finden von Hamilton-Kreisen, welches unter beschrĂ€nkter Baumweite in P fĂ€llt. Neuere Analysen konnten diese Schranke weiter zu SAC1 verbessern, was eine parallele KomplexitĂ€tsklasse ist.
Die angesprochenen Probleme kommen aus unterschiedlichen Bereichen und haben individuelle Lösungen. Hauptthese dieser Arbeit ist, dass sich diese Vielfalt vereinheitlichen lĂ€sst. Es wird ein generisches Lösungskonzept vorgestellt, welches darauf beruht, dass sich die Probleme auf ein Termevaluierungsproblem reduzieren lassen. KernstĂŒck ist daher ein Termevaluierungsalgorithmus, der unabhĂ€ngig von der Algebra, ĂŒber welche der Term evaluiert werden soll, ist.
Resultat ist, dass eine Vielzahl, darunter die oben angesprochenen Probleme, sich auf analoge Art lösen lassen, und dass sich ebenso leicht neue Resultate zeigen lassen. Diese Menge an Resultaten hÀtte sich ohne den vereinheitlichten Lösungsansatz nicht innerhalb des Rahmens einer Arbeit wie der vorliegenden zeigen lassen.
Der entwickelte Lösungsansatz fĂŒhrt stets zu Schaltkreisfamilien polylogarithmischer Tiefe. Es wird jedoch auch die Frage behandelt, wie mĂ€chtig Schaltkreisfamilien konstanter Tiefe noch bezĂŒglich Termevaluierung sind. Die Klasse AC0 ist hierfĂŒr ein natĂŒrlicher Kandidat; sie entspricht der Menge der Sprachen, die durch Logik erster Ordung beschreibbar sind. Um dieses Problem anzugehen, wird zunĂ€chst das Termevaluierungsproblem ĂŒber endlichen Algebren betrachtet. Dieses wiederum lĂ€sst sich in das Wortproblem von Visibly-Pushdown-Sprachen einbetten. Daher handelt dieser Teil der Arbeit vornehmlich von der Beschreibbarkeit von Visibly-Pushdown-Sprachen in Logik erster Ordnung. Hierbei treten ungelöste Probleme zu Tage, welche ein Indiz dafĂŒr sind, wie schlecht die KomplexitĂ€t konstanter Tiefe bisher noch verstanden ist, und das, trotz des Resultats von Furst, Saxe und Sipser, bzw. HĂ„stads.
Die bis jetzt beschrieben Inhalte sind Teil einer kontinuierlichen Entwicklung. Es gibt jedoch ein Thema in dieser Arbeit, das orthogonal dazu ist: Automaten und im speziellen Cost-Register-Automaten. Zum einen sind, wie oben angedeutet, Automaten Beispiele fĂŒr Anwendungen des hier entwickelten generischen Lösungsansatzes. Zum anderen können sie selbst zur Beschreibung von Termevaluierungsproblemen dienen; so können Visibly-Pushdown-Automaten Termevaluierung ĂŒber endlichen Algebren ausfĂŒhren. Um ĂŒber endliche Algebren hinauszugehen, benötigen die Automaten mehr Speicher. Visibly-Pushdown-Automaten haben einen Keller, der genau dafĂŒr geeignet ist, die Baumstruktur einer Eingabeformel zu verifizieren. FĂŒr nichtendliche Algebren eignet sich ein Modell, welches hier vorgestellt werden soll. Es kombiniert Visibly-Pushdown-Automaten mit Cost-Register-Automaten. Ein Cost-Register-Automat ist ein endlicher Automat, welcher mit zusĂ€tzlichen Registern ausgestattet ist. Die Register können Werte einer Algebra speichern und werden in jedem Schritt in AbhĂ€ngigkeit des Eingabezeichens und des Zustandes aktualisiert. Dieser Einwegdatenfluss von ZustĂ€nden zu Registern sorgt dafĂŒr, dass dieses Modell nicht nur entscheidbar bleibt, sondern, in AbhĂ€ngigkeit der Algebra, auch niedrige KomplexitĂ€t hat. Das neue Modell der Cost-Register-Visibly-Pushdown-Automaten kann nun Terme evaluieren. Es werden grundlegende Eigenschaften gezeigt, einschlieĂlich KomplexitĂ€tsaussagen
Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata
Lately, there have been intensive studies on strengths and limitations of
nonuniform families of promise decision problems solvable by various types of
polynomial-size finite automata families, where "polynomial-size" refers to the
polynomially-bounded state complexity of a finite automata family. In this line
of study, we further expand the scope of these studies to families of partial
counting and gap functions, defined in terms of nonuniform families of
polynomial-size nondeterministic finite automata, and their relevant families
of promise decision problems. Counting functions have an ability of counting
the number of accepting computation paths produced by nondeterministic finite
automata. With no unproven hardness assumption, we show numerous separations
and collapses of complexity classes of those partial counting and gap function
families and their induced promise decision problem families. We also
investigate their relationships to pushdown automata families of polynomial
stack-state complexity.Comment: (A4, 10pt, 21 pages) This paper corrects and extends a preliminary
report published in the Proceedings of the 24th International Symposium on
Fundamentals of Computation Theory (FCT 2023), Trier, Germany, September
18-24, 2023, Lecture Notes in Computer Science, vol. 14292, pp. 421-435,
Springer Cham, 202
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
Synchronizing Deterministic Push-Down Automata Can Be Really Hard
The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata.
There are several interpretations of what synchronizability should mean for deterministic push-down automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference
Alternating and empty alternating auxiliary stack automata
AbstractWe consider variants of alternating auxiliary stack automata and characterize their computational power when the number of alternations is bounded by a constant or unlimited. In this way we get new characterizations of NP, the polynomial hierarchy, PSpace, and bounded query classes like co-DP=NLăNP[1]ă and Î2P=PNP[O(logn)], in a uniform framework
History-deterministic Parikh Automata
Parikh automata extend finite automata by counters that can be tested for
membership in a semilinear set, but only at the end of a run. Thereby, they
preserve many of the desirable properties of finite automata. Deterministic
Parikh automata are strictly weaker than nondeterministic ones, but enjoy
better closure and algorithmic properties. This state of affairs motivates the
study of intermediate forms of nondeterminism. Here, we investigate
history-deterministic Parikh automata, i.e., automata whose nondeterminism can
be resolved on the fly. This restricted form of nondeterminism is well-suited
for applications which classically call for determinism, e.g., solving games
and composition. We show that history-deterministic Parikh automata are
strictly more expressive than deterministic ones, incomparable to unambiguous
ones, and enjoy almost all of the closure and some of the algorithmic
properties of deterministic automata.Comment: arXiv admin note: text overlap with arXiv:2207.0769
Branching-time model checking of one-counter processes
One-counter processes (OCPs) are pushdown processes which operate only on a
unary stack alphabet. We study the computational complexity of model checking
computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from
the modal mu-calculus for this problem. First, we analyze the periodic
behaviour of CTL over OCPs and derive a model checking algorithm whose running
time is exponential only in the number of control locations and a syntactic
notion of the formula that we call leftward until depth. Thus, model checking
fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This
generalizes a result of the first author, Mayr, and To for the expression
complexity of CTL's fragment EF. Second, we prove that already over some fixed
OCP, CTL model checking is PSPACE-hard. Third, we show that there already
exists a fixed CTL formula for which model checking of OCPs is PSPACE-hard. To
obtain the latter result, we employ two results from complexity theory: (i)
Converting a natural number in Chinese remainder presentation into binary
presentation is in logspace-uniform NC^1 and (ii) PSPACE is AC^0-serializable.
We demonstrate that our approach can be used to obtain further results. We show
that model-checking CTL's fragment EF over OCPs is hard for P^NP, thus
establishing a matching lower bound and answering an open question of the first
author, Mayr, and To. We moreover show that the following problem is hard for
PSPACE: Given a one-counter Markov decision process, a set of target states
with counter value zero each, and an initial state, to decide whether the
probability that the initial state will eventually reach one of the target
states is arbitrarily close to 1. This improves a previously known lower bound
for every level of the Boolean hierarchy by Brazdil et al
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