scholarly journals Computational limitations of affine automata and generalized affine automata

2021 ◽  
Author(s):  
Mika Hirvensalo ◽  
Etienne Moutot ◽  
Abuzer Yakaryılmaz
Keyword(s):  
Rational Numbers ◽  
Error Model ◽  
Computational Power ◽  
Transition Matrices ◽  
Logarithmic Space ◽  
Unary Languages ◽  
Bounded Error

AbstractWe present new results on the computational limitations of affine automata (AfAs). First, we show that using the endmarker does not increase the computational power of AfAs. Second, we show that the computation of bounded-error rational-valued AfAs can be simulated in logarithmic space. Third, we identify some logspace unary languages that are not recognized by algebraic-valued AfAs. Fourth, we show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of AfAs in the unbounded-error model. When focusing only the rational values, we obtain the the same result also for bounded error. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only.

scholarly journals Uncountable Realtime Probabilistic Classes

2019 ◽  
Vol 30 (08) ◽  
pp. 1317-1333
Author(s):  
Maksims Dimitrijevs ◽  
Abuzer Yakaryılmaz
Keyword(s):  
Linear Space ◽  
Logarithmic Space ◽  
Unary Languages ◽  
Bounded Error

We investigate the minimal cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On non-unary case, we obtain the same result for double logarithmic space, which is also tight. When replacing the work tape with a few counters, we can still achieve similar results for unary linear-space two-counter automata, unary sublinear-space three-counter automata, and non-unary sublinear-space two-counter automata. We also show how to slightly improve the sublinear-space constructions by using more counters.

scholarly journals Uncountable classical and quantum complexity classes

2018 ◽  
Vol 52 (2-3-4) ◽  
pp. 111-126
Author(s):  
Maksims Dimitrijevs ◽  
Abuzer Yakaryılmaz
Keyword(s):  
Linear Space ◽  
Time Constant ◽  
Complexity Classes ◽  
Turing Machines ◽  
Small Space ◽  
Logarithmic Space ◽  
Unary Languages ◽  
Bounded Error ◽  
Counter Machines ◽  
Quantum Complexity

It is known that poly-time constant-space quantum Turing machines (QTMs) and logarithmic-space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error (A.C. Cem Say and A. Yakaryılmaz, Magic coins are useful for small-space quantum machines. Quant. Inf. Comput. 17 (2017) 1027–1043). 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 obtain the result for a restricted sweeping head, known as restarting realtime.

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 Matchgate and space-bounded quantum computations are equivalent

2009 ◽  
Vol 466 (2115) ◽  
pp. 809-830 ◽  
Author(s):  
Richard Jozsa ◽  
Barbara Kraus ◽  
Akimasa Miyake ◽  
John Watrous
Keyword(s):  
Quantum Computation ◽  
Polynomial Time ◽  
Perfect Matchings ◽  
Spin Chains ◽  
Quantum Gates ◽  
Nearest Neighbour ◽  
Computational Power ◽  
One Dimensional ◽  
Logarithmic Space ◽  
Algebraic Constraints

Matchgates are an especially multiflorous class of two-qubit nearest-neighbour quantum gates, defined by a set of algebraic constraints. They occur for example in the theory of perfect matchings of graphs, non-interacting fermions and one-dimensional spin chains. We show that the computational power of circuits of matchgates is equivalent to that of space-bounded quantum computation with unitary gates, with space restricted to being logarithmic in the width of the matchgate circuit. In particular, for the conventional setting of polynomial-sized (logarithmic-space generated) families of matchgate circuits, known to be classically simulatable, we characterize their power as coinciding with polynomial-time and logarithmic-space-bounded universal unitary quantum computation.

P AUTOMATA WITH RESTRICTED POWER

2014 ◽  
Vol 25 (04) ◽  
pp. 391-408 ◽  
Author(s):  
ERZSÉBET CSUHAJ-VARJÚ ◽  
GYÖRGY VASZIL
Keyword(s):  
Turing Machines ◽  
Computational Power ◽  
Membrane Systems ◽  
Infinite Hierarchy ◽  
Logarithmic Space ◽  
Language Class

P automata are variants of symport/antiport membrane systems which describe string languages by applying a mapping to the sequence of multisets entering the system during computations. In this paper, we study their computational power in the case when the input mapping associates each input multiset with the set of strings consisting of all permutations of its elements. We show that P automata of this type are strictly less powerful than so-called restricted logarithmic space Turing machines, and we also exhibit a strict infinite hierarchy within the accepted language class based on the number of membranes present in the system.

The Joint Universality for L-Functions of Elliptic Curves

2004 ◽  
Vol 9 (4) ◽  
pp. 331-348
Author(s):  
V. Garbaliauskienė
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

Productly linearly independent sequences

2015 ◽  
Vol 52 (3) ◽  
pp. 350-370
Author(s):  
Jaroslav Hančl ◽  
Katarína Korčeková ◽  
Lukáš Novotný
Keyword(s):  
Rational Numbers ◽  
Infinite Products ◽  
Linearly Independent ◽  
New Concepts ◽  
Infinite Sequences

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

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