Language Classes Defined by Generalized Quantum Turing Machine

2008 ◽  
Vol 15 (04) ◽  
pp. 383-396 ◽  
Author(s):  
Satoshi Iriyama ◽  
Masanori Ohya
Keyword(s):  
Turing Machine ◽  
Quantum Algorithm ◽  
Turing Machines ◽  
Completely Positive ◽  
Sat Problem ◽  
Language Classes ◽  
Generalize Quantum ◽  
Usual Quantum ◽  
Quantum Turing Machine

Ohya and Volovich proposed a quantum algorithm with chaotic amplification to solve the SAT problem, which went beyond the notion of the usual quantum algorithm. In this paper, we generalize quantum Turing machines by rewriting the usual quantum Turing automaton in terms of a channel transformation. Moreover, we define some computational classes of generalized quantum Turing machines and show that we can describe the Ohya-Volovich (OV) SAT algorithm with completely positive channels.

On Generalized Quantum Turing Machine and Its Applications

2009 ◽  
Vol 16 (02n03) ◽  
pp. 195-204
Author(s):  
Satoshi Iriyama ◽  
Masanori Ohya
Keyword(s):  
Computational Complexity ◽  
Turing Machine ◽  
Polynomial Time ◽  
Recursive Function ◽  
Quantum Algorithm ◽  
Partial Recursive Function ◽  
Sat Problem ◽  
Input Size ◽  
Amplification Process ◽  
Quantum Turing Machine

Ohya and Volovich discussed a quantum algorithm for the SAT problem with a chaos amplification process (OMV SAT algorithm) and showed that the number of steps it performed was polynomial in input size. In this paper, we define a generalized quantum Turing machine (GQTM) and related computational complexity. Then we show that there exists a GQTM which recognizes the SAT problem in polynomial time. Moreover, we discuss the problem of finding the quantum algorithm for a partial recursive function.

scholarly journals Revisiting the simulation of quantum Turing machines by quantum circuits

2019 ◽  
Vol 475 (2226) ◽  
pp. 20180767 ◽  
Author(s):  
Abel Molina ◽  
John Watrous
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Computational Models ◽  
Circuit Complexity ◽  
Simulation Method ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Turing Machines ◽  
Generated Family ◽  
Quantum Turing Machine

Yao's 1995 publication ‘Quantum circuit complexity’ in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science , pp. 352–361, proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t , and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t , rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of method described by Arright, Nesme and Werner in 2011 in Journal of Computer and System Sciences 77 , 372–378. ( doi:10.1016/j.jcss.2010.05.004 ), that allows for the localization of causal unitary evolutions.

Quantum turing machine and brain model represented by Fock space

2016 ◽  
Vol 14 (04) ◽  
pp. 1640008
Author(s):  
Satoshi Iriyama ◽  
Masanori Ohya
Keyword(s):  
Turing Machine ◽  
Quantum Channel ◽  
Fock Space ◽  
Adaptive Dynamics ◽  
Quantum Algorithm ◽  
Psychological Phenomena ◽  
Definition Of ◽  
Complex Phenomena ◽  
Math Phys ◽  
Quantum Turing Machine

The adaptive dynamics is known as a new mathematics to treat with a complex phenomena, for example, chaos, quantum algorithm and psychological phenomena. In this paper, we briefly review the notion of the adaptive dynamics, and explain the definition of the generalized Turing machine (GTM) and recognition process represented by the Fock space. Moreover, we show that there exists the quantum channel which is described by the GKSL master equation to achieve the Chaos Amplifier used in [M. Ohya and I. V. Volovich, J. Opt. B 5(6) (2003) 639., M. Ohya and I. V. Volovich, Rep. Math. Phys. 52(1) (2003) 25.]

Eventually infinite time Turing machine degrees: infinite time decidable reals

2000 ◽  
Vol 65 (3) ◽  
pp. 1193-1203 ◽  
Author(s):  
P.D. Welch
Keyword(s):  
Turing Machine ◽  
Initial Segment ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Turing Machines ◽  
Jump Operator ◽  
Infinite Time

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

TWO TOPICS CONCERNING TWO-DIMENSIONAL AUTOMATA OPERATING IN PARALLEL

1992 ◽  
Vol 06 (02n03) ◽  
pp. 211-225 ◽  
Author(s):  
KATSUSHI INOUE ◽  
ITSUO SAKURAMOTO ◽  
MAKOTO SAKAMOTO ◽  
ITSUO TAKANAMI
Keyword(s):  
Turing Machine ◽  
Finite Automata ◽  
Space Complexity ◽  
Two Dimensional ◽  
Turing Machines ◽  
Small Space ◽  
Bottom Up ◽  
Constructible Function ◽  
Deterministic Turing Machine ◽  
Alternating Turing Machines

This paper deals with two topics concerning two-dimensional automata operating in parallel. We first investigate a relationship between the accepting powers of two-dimensional alternating finite automata (2-AFAs) and nondeterministic bottom-up pyramid cellular acceptors (NUPCAs), and show that Ω ( diameter × log diameter ) time is necessary for NUPCAs to simulate 2-AFAs. We then investigate space complexity of two-dimensional alternating Turing machines (2-ATMs) operating in small space, and show that if L (n) is a two-dimensionally space-constructible function such that lim n → ∞ L (n)/ loglog n > 1 and L (n) ≤ log n, and L′ (n) is a function satisfying L′ (n) =o (L(n)), then there exists a set accepted by some strongly L (n) space-bounded two-dimensional deterministic Turing machine, but not accepted by any weakly L′ (n) space-bounded 2-ATM, and thus there exists a rich space hierarchy for weakly S (n) space-bounded 2-ATMs with loglog n ≤ S (n) ≤ log n.

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)).

Computational speed-up by effective operators

1972 ◽  
Vol 37 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Albert R. Meyer ◽  
Patrick C. Fischer
Keyword(s):  
Turing Machine ◽  
Computable Function ◽  
Turing Machines ◽  
Complexity Measures ◽  
Computational Speed ◽  
Partial Recursive Functions ◽  
Speed Up ◽  
Recursive Equations ◽  
Computable Functions ◽  
Effective Operators

The complexity of a computable function can be measured by considering the time or space required to compute its values. Particular notions of time and space arising from variants of Turing machines have been investigated by R. W. Ritchie [14], Hartmanis and Stearns [8], and Arbib and Blum [1], among others. General properties of such complexity measures have been characterized axiomatically by Rabin [12], Blum [2], Young [16], [17], and McCreight and Meyer [10].In this paper the speed-up and super-speed-up theorems of Blum [2] are generalized to speed-up by arbitrary total effective operators. The significance of such theorems is that one cannot equate the complexity of a computable function with the running time of its fastest program, for the simple reason that there are computable functions which in a very strong sense have no fastest programs.Let φi be the ith partial recursive function of one variable in a standard Gödel numbering of partial recursive functions. A family Φ0, Φ1, … of functions of one variable is called a Blum measure on computation providing(1) domain (φi) = domain (Φi), and(2) the predicate [Φi(x) = m] is recursive in i, x and m.Typical interpretations of Φi(x) are the number of steps required by the ith Turing machine (in a standard enumeration of Turing machines) to converge on input x, the space or number of tape squares required by the ith Turing machine to converge on input x (with the convention that Φi(x) is undefined even if the machine fails to halt in a finite loop), and the length of the shortest derivation of the value of φi(x) from the ith set of recursive equations.

scholarly journals SOLVING THE WORD PROBLEM IN REAL TIME

2001 ◽  
Vol 63 (3) ◽  
pp. 623-639 ◽  
Author(s):  
DEREK F. HOLT ◽  
SARAH REES
Keyword(s):  
Recent Result ◽  
Real Time ◽  
Word Problem ◽  
Turing Machine ◽  
Linear Time ◽  
Nilpotent Groups ◽  
Hyperbolic Groups ◽  
Turing Machines ◽  
Geometrically Finite ◽  
Word Hyperbolic Groups

The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems.

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