New Results on the Minimum Amount of Useful Space

Bednárová, Zuzana; Geffert, Viliam; Reinhardt, Klaus; Yakaryılmaz, Abuzer

doi:10.1142/s0129054116400098

Cited by 8 publications

(8 citation statements)

References 37 publications

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“…It is known that in this case the height of the pushdown store should grow at least as the function log log n, with respect to the input length n [1]. Furthermore, this lower bound is optimal [3]. We show that in the unary case the optimal bound increases to a logarithmic function.…”

Section: An Optimal Lower Bound For Non-constant Heightmentioning

confidence: 91%

“…The answer to this question is already known and it derives from results on Turing machines: the height of the store should grow at least as a double logarithmic function [1]. This lower bound cannot be increased, because a matching upper bound recently obtained in [3]. As a consequence of the constructions presented in the second part of the paper, we are able to prove that in the unary case this lower bound is logarithmic.…”

Section: Introductionmentioning

confidence: 90%

“…For any recursive function f : N → N and for infinitely many integers n there exists a pda of size n accepting in constant height H(n), where H(n) cannot be bounded by f (n). 3 Proof. (outline) The argument is derived from [12,Prop.…”

Section: Undecidability and Non-recursive Boundsmentioning

confidence: 99%

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Pushdown Automata and Constant Height: Decidability and Bounds

Pighizzini

Prigioniero

2019

Descriptional Complexity of Formal Systems

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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 over a one-letter input alphabet, i.e., unary pushdown automata, the above property becomes decidable. Moreover, if the height is bounded by a constant in the input length, then it is at most exponential with respect to the size of the description of the pushdown automaton. This bound cannot be reduced. Finally, if a unary pushdown automaton uses nonconstant height to accept, then the height should grow at least as the logarithm of the input length. This bound is optimal.

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Section: An Optimal Lower Bound For Non-constant Heightmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 90%

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Pushdown Automata and Constant Height: Decidability and Bounds

Pighizzini

Prigioniero

2019

Descriptional Complexity of Formal Systems

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“…Until now, we focus on strong space bounds. On the other hand, we know that 2QCCAs can recognize the following nonregular unary language with bounded error in middle logarithmic space [5]: UPOWER2 = {a 2 n | n ≥ 0}.…”

Section: Quantum Modelsmentioning

confidence: 99%

Uncountable classical and quantum complexity classes

Dimitrijevs

Yakaryılmaz

2018

RAIRO-Theor. Inf. Appl.

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Polynomial-time constant-space quantum Turing machines (QTMs) and logarithmic-space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error Yakaryılmaz 2014, arXiv:1411.7647). In this paper, we investigate more restricted cases for both models to recognize uncountably many languages with bounded error. We show that double logarithmic space is enough for PTMs on unary languages in sweeping reading mode or logarithmic space for one-way head. On unary languages, for quantum models, we obtain middle logarithmic space for counter machines. For binary languages, arbitrary small non-constant space is enough for PTMs even using only counter as memory. For counter machines, when restricted to polynomial time, we can obtain the same result for linear space. For constant-space QTMs, we follow the result for a restricted sweeping head, known as restarting realtime.1 As another "probabilistic" but unconventional model, ultrametric automata can also define uncountably many languages with 2 states [6].

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“…Although alternating finite automata (AFAs) can only recognize regular languages, even when they can use a two-way input head, they can be more powerful when given other resources. For example, AFAs augmented with a counter (A1CAs) can recognize some unary nonregular languages [27,4], whereas their nondeterministic counterparts cannot recognize any unary nonregular language, even when allowed to pause the input head indefinitely on a symbol, and given a stack instead of a counter, which upgrades them to one-way nondeterministic pushdown automata (PDAs) [13].…”

Section: Introductionmentioning

confidence: 99%

Alternating, private alternating, and quantum alternating realtime automata

Demirci,

Hirvensalo,

Reinhardt

et al. 2014

Preprint

Self Cite

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We present new results on realtime alternating, private alternating, and quantum alternating automaton models. Firstly, we show that the emptiness problem for alternating one-counter automata on unary alphabets is undecidable. Then, we present two equivalent definitions of realtime private alternating finite automata (PAFAs). We show that the emptiness problem is undecidable for PAFAs. Furthermore, PAFAs can recognize some nonregular unary languages, including the unary squares language, which seems to be difficult even for some classical counter automata with two-way input. Regarding quantum finite automata (QFAs), we show that the emptiness problem is undecidable both for universal QFAs on general alphabets, and for alternating QFAs with two alternations on unary alphabets. On the other hand, the same problem is decidable for nondeterministic QFAs on general alphabets. We also show that the unary squares language is recognized by alternating QFAs with two alternations.

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New Results on the Minimum Amount of Useful Space

Cited by 8 publications

References 37 publications

Pushdown Automata and Constant Height: Decidability and Bounds

Pushdown Automata and Constant Height: Decidability and Bounds

Uncountable classical and quantum complexity classes

Alternating, private alternating, and quantum alternating realtime automata

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