Self-Verifying Pushdown and Queue Automata

2021 ◽  
Vol 180 (1-2) ◽  
pp. 1-28
Author(s):  
Henning Fernau ◽  
Martin Kutrib ◽  
Matthias Wendlandt
Keyword(s):  
Recursive Function ◽  
Input Word ◽  
Closure Properties ◽  
Descriptional Complexity ◽  
Trade Offs ◽  
Deterministic Automata ◽  
Context Free ◽  
Pushdown Automata ◽  
Computation Path

We study the computational and descriptional complexity of self-verifying pushdown automata (SVPDA) and self-verifying realtime queue automata (SVRQA). A self-verifying automaton is a nondeterministic device whose nondeterminism is symmetric in the following sense. Each computation path can give one of the answers yes, no, or do not know. For every input word, at least one computation path must give either the answer yes or no, and the answers given must not be contradictory. We show that SVPDA and SVRQA are automata characterizations of so-called complementation kernels, that is, context-free or realtime nondeterministic queue automaton languages whose complement is also context free or accepted by a realtime nondeterministic queue automaton. So, the families of languages accepted by SVPDA and SVRQA are strictly between the families of deterministic and nondeterministic languages. Closure properties and various decidability problems are considered. For example, it is shown that it is not semidecidable whether a given SVPDA or SVRQA can be made self-verifying. Moreover, we study descriptional complexity aspects of these machines. It turns out that the size trade-offs between nondeterministic and self-verifying as well as between self-verifying and deterministic automata are non-recursive. That is, one can choose an arbitrarily large recursive function f, but the gain in economy of description eventually exceeds f when changing from the former system to the latter.

ON THE DESCRIPTIONAL COMPLEXITY OF METALINEAR CD GRAMMAR SYSTEMS

2005 ◽  
Vol 16 (05) ◽  
pp. 1011-1025
Author(s):  
BETTINA SUNCKEL
Keyword(s):  
Recursive Function ◽  
Trade Off ◽  
Descriptional Complexity ◽  
Grammar System ◽  
Trade Offs ◽  
Grammar Systems ◽  
Context Free ◽  
Context Free Grammars

Metalinear CD grammar systems are context-free CD grammar systems where each component consists of metalinear productions. The maximal number of nonterminals in all starting productions is referred to as the width of a CD grammar system. It is shown that between the class of CD grammar systems of width m + 1 and of width m there are savings concerning the size of the grammar system not bounded by any recursive function. This is called a non-recursive trade-off. Furthermore, it is proven that there are non-recursive trade-offs between the class of metalinear CD grammar systems of width m and the class of (2m - 1)-linear context-free grammars. In addition, some decidability results are presented.

Descriptional Complexity of the Forever Operator

2019 ◽  
Vol 30 (01) ◽  
pp. 115-134 ◽  
Author(s):  
Michal Hospodár ◽  
Galina Jirásková ◽  
Peter Mlynárčik
Keyword(s):  
Regular Language ◽  
Finite Automata ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Descriptional Complexity ◽  
Trade Offs ◽  
Number Formula ◽  
Deterministic Automata ◽  
Initial States ◽  
Nondeterministic State

We examine the descriptional complexity of the forever operator, which assigns the language [Formula: see text] to a regular language [Formula: see text], and we investigate the trade-offs between various models of finite automata. We consider complete and partial deterministic finite automata, nondeterministic finite automata with single or multiple initial states, alternating, and Boolean finite automata. We assume that the argument and the result of this operation are accepted by automata belonging to one of these six models. We investigate all possible trade-offs and provide a tight upper bound for 32 of 36 of them. The most interesting result is the trade-off from nondeterministic to deterministic automata given by the Dedekind number [Formula: see text]. We also prove that the nondeterministic state complexity of [Formula: see text] is [Formula: see text] which solves an open problem stated by Birget [The state complexity of [Formula: see text] and its connection with temporal logic, Inform. Process. Lett. 58 (1996) 185–188].

scholarly journals Good-for-games $\omega$-Pushdown Automata

2022 ◽  
Vol Volume 18, Issue 1 ◽  
Author(s):  
Karoliina Lehtinen ◽  
Martin Zimmermann
Keyword(s):  
Infinite Games ◽  
Closure Properties ◽  
New Class ◽  
Deterministic Automata ◽  
Good For ◽  
Pushdown Automata

We introduce good-for-games $\omega$-pushdown automata ($\omega$-GFG-PDA). These are automata whose nondeterminism can be resolved based on the input processed so far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata. Our main results are that $\omega$-GFG-PDA are more expressive than deterministic $\omega$- pushdown automata and that solving infinite games with winning conditions specified by $\omega$-GFG-PDA is EXPTIME-complete. Thus, we have identified a new class of $\omega$-contextfree winning conditions for which solving games is decidable. It follows that the universality problem for $\omega$-GFG-PDA is in EXPTIME as well. Moreover, we study closure properties of the class of languages recognized by $\omega$-GFG- PDA and decidability of good-for-gameness of $\omega$-pushdown automata and languages. Finally, we compare $\omega$-GFG-PDA to $\omega$-visibly PDA, study the resources necessary to resolve the nondeterminism in $\omega$-GFG-PDA, and prove that the parity index hierarchy for $\omega$-GFG-PDA is infinite.

scholarly journals Iterative arrays with self-verifying communication cell

2020 ◽  
Author(s):  
Martin Kutrib
Keyword(s):  
Cellular Automata ◽  
Time Complexity ◽  
Input Word ◽  
Closure Properties ◽  
Multiplicative Factor ◽  
Computational Capacity ◽  
The Family ◽  
Computation Path

Abstract We study the computational capacity of self-verifying iterative arrays ($${\text {SVIA}}$$ SVIA ). A self-verifying device is a nondeterministic device whose nondeterminism is symmetric in the following sense. Each computation path can give one of the answers yes, no, or do not know. For every input word, at least one computation path must give either the answer yes or no, and the answers given must not be contradictory. It turns out that, for any time-computable time complexity, the family of languages accepted by $${\text {SVIA}}$$ SVIA s is a characterization of the so-called complementation kernel of nondeterministic iterative array languages, that is, languages accepted by such devices whose complementation is also accepted by such devices. $${\text {SVIA}}$$ SVIA s can be sped-up by any constant multiplicative factor as long as the result does not fall below realtime. We show that even realtime $${\text {SVIA}}$$ SVIA are as powerful as lineartime self-verifying cellular automata and vice versa. So they are strictly more powerful than the deterministic devices. Closure properties and various decidability problems are considered.

Set Automata

2016 ◽  
Vol 27 (02) ◽  
pp. 187-214 ◽  
Author(s):  
Martin Kutrib ◽  
Andreas Malcher ◽  
Matthias Wendlandt
Keyword(s):  
Finite Automata ◽  
Point Of View ◽  
Storage Medium ◽  
Deterministic Finite Automaton ◽  
Closure Properties ◽  
Language Class ◽  
Trade Offs ◽  
Double Exponential ◽  
Order Of Magnitude ◽  
Context Free

We consider the model of deterministic set automata which are basically deterministic finite automata equipped with a set as an additional storage medium. The basic operations on the set are the insertion of elements, the removing of elements, and the test whether an element is in the set. We investigate the computational power of deterministic set automata and compare the language class accepted with the context-free languages and classes of languages accepted by queue automata. As result the incomparability to all classes considered is obtained. Furthermore, we examine the closure properties under several operations. Then we show that deterministic set automata may be an interesting model from a practical point of view by proving that their regularity problem as well as the problems of emptiness, finiteness, infiniteness, and universality are decidable. Finally, the descriptional complexity of deterministic and nondeterministic set automata is investigated. A conversion procedure that turns a deterministic set automaton accepting a regular language into a deterministic finite automaton is developed which leads to a double exponential upper bound. This bound is proved to be tight in the order of magnitude by presenting also a double exponential lower bound. In contrast to these recursive bounds we obtain non-recursive trade-offs when nondeterministic set automata are considered.

ON THE DESCRIPTIONAL COMPLEXITY OF THE WINDOW SIZE FOR DELETING RESTARTING AUTOMATA

2013 ◽  
Vol 24 (06) ◽  
pp. 831-846 ◽  
Author(s):  
MARTIN KUTRIB ◽  
FRIEDRICH OTTO
Keyword(s):  
Upper Bound ◽  
Recursive Function ◽  
Window Size ◽  
Input Word ◽  
Trade Off ◽  
Descriptional Complexity ◽  
The Impact ◽  
Current Word ◽  
Restarting Automaton

The restarting automaton was inspired by the technique of ‘analysis by reduction’ from linguistics. A restarting automaton processes a given input word through a sequence of cycles. In each cycle the current word on the tape is scanned from left to right and a single local simplification (a rewrite) is executed. One of the essential parameters of a restarting automaton is the size of its read/write window. Here we study the impact of the window size on the descriptional complexity of several types of deterministic and nondeterministic restarting automata. For all k ≥ 4, we show that the savings in the economy of descriptions of restarting automata that can only delete symbols but not rewrite them (that is, the so-called R- and RR-automata) cannot be bounded by any recursive function, when changing from window size k to window size k + 1. This holds for deterministic as well as for nondeterministic automata, and for k ≥ 5, it even holds for the stateless variants of these automata. However, the trade-off between window sizes two and one is recursive for deterministic devices. In addition, a polynomial upper bound is given for the trade-off between RRWW-automata with window sizes k + 1 and k for all k ≥ 2.

scholarly journals Digging input-driven pushdown automata

2021 ◽  
Vol 55 ◽  
pp. 6
Author(s):  
Martin Kutrib ◽  
Andreas Malcher
Keyword(s):  
Input Symbol ◽  
Closure Properties ◽  
Descriptional Complexity ◽  
Finite State Transducers ◽  
Finite State ◽  
The Impact ◽  
Pushdown Automata

Input-driven pushdown automata (IDPDA) are pushdown automata where the next action on the pushdown store (push, pop, nothing) is solely governed by the input symbol. Nowadays such devices are usually defined such that popping from the empty pushdown does not block the computation but continues it with empty pushdown. Here, we consider IDPDAs that have a more balanced behavior concerning pushing and popping. Digging input-driven pushdown automata (DIDPDA) are basically IDPDAs that, when forced to pop from the empty pushdown, dig a hole of the shape of the popped symbol in the bottom of the pushdown. Popping further symbols from a pushdown having a hole at the bottom deepens the current hole furthermore. The hole can only be filled up by pushing symbols previously popped. We study the impact of the new behavior of DIDPDAs on their power and compare their capacities with the capacities of ordinary IDPDAs and tinput-driven pushdown automata which are basically IDPDAs whose input may be preprocessed by length-preserving finite state transducers. It turns out that the capabilities are incomparable. We address the determinization of DIDPDAs and their descriptional complexity, closure properties, and decidability questions.

WHEN CHURCH-ROSSER BECOMES CONTEXT FREE

2007 ◽  
Vol 18 (06) ◽  
pp. 1293-1302 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER
Keyword(s):  
Linear Time ◽  
Membership Problem ◽  
Closure Properties ◽  
Context Free Language ◽  
Arithmetic Hierarchy ◽  
Context Free ◽  
Free Language

We investigate the intersection of Church-Rosser languages and (strongly) context-free languages. The intersection is still a proper superset of the deterministic context-free languages as well as of their reversals, while its membership problem is solvable in linear time. For the problem whether a given Church-Rosser or context-free language belongs to the intersection we show completeness for the second level of the arithmetic hierarchy. The equivalence of Church-Rosser and context-free languages is Π1-complete. It is proved that all considered intersections are pairwise incomparable. Finally, closure properties under several operations are investigated.

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