scholarly journals New Results on the Minimum Amount of Useful Space

2016 ◽  
Vol 27 (02) ◽  
pp. 259-281 ◽  
Author(s):  
Zuzana Bednárová ◽  
Viliam Geffert ◽  
Klaus Reinhardt ◽  
Abuzer Yakaryilmaz
Keyword(s):  
Lower Bound ◽  
Minimum Amount ◽  
Turing Machines ◽  
Bounded Error ◽  
Space Requirements ◽  
Pushdown Automata ◽  
Classical States

We present several new results on minimal space requirements to recognize a nonregular language: (i) realtime nondeterministic Turing machines can recognize a nonregular unary language within weak log log n space, (ii) log log n is a tight space lower bound for accepting general nonregular languages on weak realtime pushdown automata, (iii) there exist unary nonregular languages accepted by realtime alternating one-counter automata within weak log n space, (iv) there exist nonregular languages accepted by two-way deterministic pushdown automata within strong log log n space, and, (v) there exist unary nonregular languages accepted by two-way one-counter automata using quantum and classical states with middle log n space and bounded error.

A SPACE LOWER BOUND FOR NAME-INDEPENDENT COMPACT ROUTING IN TREES

2007 ◽  
Vol 08 (03) ◽  
pp. 229-251
Author(s):  
KOFI A. LAING ◽  
RAJMOHAN RAJARAMAN
Keyword(s):  
Lower Bound ◽  
Routing Algorithm ◽  
Minimum Amount ◽  
Single Source ◽  
Compact Routing ◽  
Positive Edge ◽  
Routing Table ◽  
Space Requirements

Given a rooted n-node tree with arbitrary positive edge weights, and arbitrarily assigned node names, what is the minimum amount of space that a single-source compact routing algorithm could use in its largest routing table while achieving stretch 3? We show that the space requirements is [Formula: see text] bits in a port model that is more general than the fixed-port model, and note that this result also applies for all-pairs routing in trees.

Emptiness of Ordered Multi-Pushdown Automata is 2ETIME-Complete

2017 ◽  
Vol 28 (08) ◽  
pp. 945-975 ◽  
Author(s):  
Mohamed Faouzi Atig ◽  
Benedikt Bollig ◽  
Peter Habermehl
Keyword(s):  
Lower Bound ◽  
Turing Machine ◽  
Linear Ordering ◽  
Emptiness Problem ◽  
Pushdown Automata ◽  
Exponential Space

We consider ordered multi-pushdown automata, a multi-stack extension of pushdown automata that comes with a constraint on stack operations: a pop can only be performed on the first non-empty stack (which implies that we assume a linear ordering on the collection of stacks). We show that the emptiness problem for multi-pushdown automata is 2ETIME-complete. Containment in 2ETIME is shown by translating an automaton into a grammar for which we can check if the generated language is empty. The lower bound is established by simulating the behavior of an alternating Turing machine working in exponential space. We also compare ordered multi-pushdown automata with the model of bounded-phase (visibly) multi-stack pushdown automata, which do not impose an ordering on stacks, but restrict the number of alternations of pop operations on different stacks.

CLOSURE PROPERTY OF SPACE-BOUNDED TWO-DIMENSIONAL ALTERNATING TURING MACHINES, PUSHDOWN AUTOMATA, AND COUNTER AUTOMATA

2001 ◽  
Vol 15 (07) ◽  
pp. 1143-1165 ◽  
Author(s):  
TOKIO OKAZAKI ◽  
KATSUSHI INOUE ◽  
AKIRA ITO ◽  
YUE WANG
Keyword(s):  
Turing Machine ◽  
Nondecreasing Function ◽  
Closure Property ◽  
Two Dimensional ◽  
Turing Machines ◽  
Pushdown Automaton ◽  
Alternating Turing Machines ◽  
Positive Integers ◽  
Pushdown Automata

This paper investigates closure property of the classes of sets accepted by space-bounded two-dimensional alternating Turing machines (2-atm's) and space-bounded two-dimensional alternating pushdown automata (2-apda's), and space-bounded two-dimensional alternating counter automata (2-aca's). Let L(m, n): N2 → N (N denotes the set of all positive integers) be a function with two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). We show that (i) for any function f(m) = o( log m) (resp. f(m) = o( log m/ log log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional Turing machine (2-Tm) (resp. two-dimensional pushdown automaton (2-pda)), the class of sets accepted by L(m,n) space-bounded 2-atm's (2-apda's) is not closed under row catenation, row + or projection, and (ii) for any function f(m) = o(m/ log ) (resp. for any function f(m) such that log f(m) = o( log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional counter automaton (2-ca), the class of sets accepted by L(m, n) space-bounded 2-aca's is not closed under row catenation, row + or projection, where L(m, n) = f(m) + g(n) (resp. L(m, n) = f(m) × g(n)).

QUANTUM COMPUTATION WITH RESTRICTED AMPLITUDES

2003 ◽  
Vol 14 (05) ◽  
pp. 853-870 ◽  
Author(s):  
HARUMICHI NISHIMURA
Keyword(s):  
Quantum Computation ◽  
Polynomial Time ◽  
Exact Algorithms ◽  
Quantum Computers ◽  
Turing Machines ◽  
Computational Power ◽  
Polynomial Time Algorithms ◽  
Exact Quantum ◽  
Bounded Error ◽  
Transition Amplitudes

In this paper, we explore the power of quantum computers with restricted transition amplitudes. In 1997 Adleman, DeMarrais, and Huang showed that quantum Turing machines (QTMs) with the amplitudes from [Formula: see text] are computationally equivalent to ones with the polynomial-time computable amplitudes as machines implementing bounded-error polynomial-time algorithms. We show that QTMs with the amplitudes from [Formula: see text] is polynomial-time equivalent to deterministic Turing machines as machines implementing exact algorithms, i.e., algorithms that output correct answers with certainty. By extending this result, it is shown that exact quantum computers with rational biased coins are equivalent to classical computers. Moreover, we discuss the computational power of exact quantum computers with multiple types of coins. We also show that, from the viewpoint of zero-error polynomial-time algorithms, [Formula: see text] is not more powerful than [Formula: see text] as the set of amplitudes taken by QTMs; however, it is sufficient to solve the factoring problem.

scholarly journals Magic coins are useful for small-space quantum machines

2017 ◽  
Vol 17 (11&12) ◽  
pp. 1027-1043
Author(s):  
A.C. Cem Say ◽  
Abuzer Yakaryilmaz
Keyword(s):  
Polynomial Time ◽  
Transition Probabilities ◽  
Alternative Proof ◽  
Turing Machines ◽  
Small Space ◽  
Interactive Proof ◽  
Quantum Bits ◽  
Proof Systems ◽  
Language L ◽  
Bounded Error

Although polynomial-time probabilistic Turing machines can utilize uncomputable transition probabilities to recognize uncountably many languages with bounded error when allowed to use double logarithmic space, it is known that such “magic coins” give no additional computational power to constant-space versions of those machines. We show that adding a few quantum bits to the model changes the picture dramatically. For every language L, there exists such a two-way quantum finite automaton (2qcfa) that recognizes a language of the same Turing degree as L with bounded error in polynomial time. When used as verifiers in public-coin interactive proof systems, such automata can verify membership in all languages with bounded error, outperforming their classical counterparts, which are known to fail for the palindromes language. Corollaries demonstrate even faster machines when one is allowed to use a counter as memory, and an alternative proof of the uncountability of stochastic languages.

EFFICIENT DETECTORS AND CONSTRUCTORS FOR SIMPLE LANGUAGES

1991 ◽  
Vol 02 (03) ◽  
pp. 183-205 ◽  
Author(s):  
Dung T. Huynh
Keyword(s):  
Finite Automata ◽  
Boolean Circuits ◽  
Turing Machines ◽  
Constant Depth ◽  
Pushdown Automaton ◽  
Polynomial Size ◽  
Regular Sets ◽  
Context Free ◽  
Pushdown Automata ◽  
Nondeterministic Logspace

In this paper, we investigate the complexity of computing the detector, constructor and lexicographic constructor functions for a given language. The following classes of languages will be considered: (1) context-free languages, (2) regular sets, (3) languages accepted by one-way nondeterministic auxiliary pushdown automata, (4) languages accepted by one-way nondeterministic logspace-bounded Turing machines, (5) two-way deterministic pushdown automaton languages, (6) languages accepted by uniform families of constant-depth polynomial-size Boolean circuits, and (7) languages accepted by multihead finite automata. We show that for the classes (1)–(4), efficient detectors, constructors and lexicographic constructors exist, whereas for (5)– (7) polynomial-time computable detectors, constructors and lexicographic constructors exist iff there are no sparse sets in NP−P (or equivalently, E=NE). Our results provide sharp boundaries between classes of languages which have efficient detectors, constructors and classes of languages for which efficient detectors and constructors do not appear to exist.

On the Power of One-Way Automata with Quantum and Classical States

2015 ◽  
Vol 26 (07) ◽  
pp. 895-912 ◽  
Author(s):  
Maria Paola Bianchi ◽  
Carlo Mereghetti ◽  
Beatrice Palano
Keyword(s):  
Lower Bound ◽  
Regular Languages ◽  
Direct Approach ◽  
Computational Power ◽  
Cut Point ◽  
Classical States

We consider the model of one-way automata with quantum and classical states (QCFAs) introduced in [28]. We show, by a direct approach, that QCFAs with isolated cut-point accept regular languages only, thus characterizing their computational power.Concerning descriptional power, we quickly overview a size lower bound for QCFAs accepting regular languages, and address its optimality. Then, we explicitly build QCFAs accepting the word quotients and inverse homomorphic images of languages accepted by given QCFAs with isolated cut-point, maintaining the same cut-point, isolation, and only polynomially increasing the size.

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