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Deterministic simulation of tape-bounded probabilistic turing machine transducers

John Gill, James Hunt, Janoš Šimon · Theoretical Computer Science · 1980

Cited by 14

On Halting Process of Quantum Turing Machine

Miyadera, Takayuki, Ohya, Masanori · arXiv (Cornell University) · 2003

We prove that there is no algorithm to tell whether an arbitrarily constructed Quantum Turing Machine has same time steps for different branches of computation. We, hence, can not avoid the notion of halting to be probabilistic in Quantum Turing Machine. Our result suggests that halting scheme of Quantum Turing Machine and quantum complexity theory based upon the existing halting scheme sholud be reexamined.

Cited by 0Open access

Space-bounded hierarchies and probabilistic computations

Walter L. Ruzzo, Janoš Šimon, Martin Tompa · 1982

This paper studies two aspects of the power of space-bounded probabilistic Turing machines. Section 2 presents a simple alternative proof of Simon's recent result [13] that space-bounded probabilistic complexity classes are closed under complement. Section 3 demonstrates that any language in the log n space hierarchy can be recognized by an log n space-bounded probabilistic Turing machine with small error; this is a generalization of Gill's result that any language in NSPACE(log n) can be recognized by such a machine

Cited by 20

Turing: A language for flexible probabilistic inference

Hong Ge, Kai Xu, Zoubin Ghahramani · Apollo (University of Cambridge) · 2018

Probabilistic programming is becoming an attractive approach to probabilistic machine learning. Through relieving researchers from the tedious burden of hand-deriving inference algorithms, not only does it enable the development of more accurate and interpretable models but it also encourages reproducible research. However, successful probabilistic programming systems require flexible, generic and efficient inference engines. In this work, we present a system called Turing for flexible composable probabilistic programming inference. Turing has an intuitive modelling syntax and supports a wide range of sampling-based inference algorithms. Most importantly, Turing inference is composable: it combines Markov chain sampling operations on subsets of model variables, e.g. using a combination of a Hamiltonian Monte Carlo (HMC) engine and a particle Gibbs (PG) engine. This composable inference engine allows the user to easily switch between black-box style inference methods such as HMC and customized inference methods. Our aim is to present Turing and its composable inference engines to the community and encourage other researchers to build on this system to help advance the field of probabilistic machine learning.

Cited by 160Open access

Lower Bounds for Probabilistic Space Complexity: Automata Approach

Farid Ablayev · UR Research (University of Rochester) · 1992

In this paper we investigate a well known sequential model of computation: one-way LOG-SPACE Turing machines. We analyze a different known method for constructing an effective probabilistic algorithm. We prove a lower bound for probabilistic space complexity, which is good enough for understanding the above problem for the one-way LOG-SPACE Turing machine model of computation.

Cited by 0

On quantum and classical space-bounded processes with algebraic transition amplitudes

John Watrous · 2003

We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long term behavior. It is shown that a very general class of decision problems regarding these stochastic processes can be efficiently solved classically in the space-bounded case. The following corollaries are implied by our main result for any space-constructible space bound s satisfying s(n)=/spl Omega/(log n): (i) any space O(s) uniform family of quantum circuit acting on s qubits and consisting of unitary gates and measurement gates defined in a typical way by matrices of algebraic numbers can be simulated by an unbounded error space O(s) ordinary (i.e., fair-coin flipping) probabilistic Turing machine, and hence by space O(s) uniform classical (deterministic) circuits of depth O(s/sup 2/) and size 2/sup 0/(s); (2) any quantum Turing machine running in space s, having arbitrary algebraic transition amplitudes, allowing unrestricted measurements during its computation, and having no restrictions on running time can be simulated by a space O(s) ordinary probabilistic Turing machine in the unbounded error setting. We also obtain the following classical result: any unbounded error probabilistic Turing machine running in space s that allows algebraic probabilities and algebraic cut-point can be simulated by a space O(s) ordinarily probabilistic Turing machine with cut-point 1/2. Our technique for handling algebraic numbers in the above simulations may be of independent interest. It is shown that any real algebraic number can be accurately approximated by a ratio of GapL functions.

Cited by 9

Threshold Computation and Cryptographic Security

Yenjo Hah, Lane A. Hemaspaandra, Thomas Thierauf · 1992

Threshold machines [Sim75] are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines [Gi177] are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. Simon [Sim75] proved that for unbounded-error polynomial-time machines these two notions yield the same class, PP. Perhaps because

Cited by 0

Structural properties of one-tape Turing machines.

Jack C. H. Lin · uO Research (University of Ottawa) · 2002

The model of Turing machines has been studied since its birth in 1936. Researchers have continuously proposed variants of such a model. Upon imposing different constraints, the power of each model varies or even remains the same, accordingly. Some well-known result (for example, the equivalence of finite state automata and one-tape linear-time deterministic Turing machines) has proven that the abilities of overwriting the tape content and scanning the tape content more than once cannot gain any advantage under certain restrictions. In this thesis, we study the behaviors and the fundamental properties of variants of one-tape Turing machines, such as deterministic, reversible, nondeterministic, probabilistic, and quantum Turing machines. This gives us a better understanding about the strength and the weakness of each machine type. For example, under the one-tape linear-time restriction, reversible, nondeterministic, co-nondeterministic, and bounded-error probabilistic computations recognize exactly regular languages whereas probabilistic and quantum Turing machines can recognize even non-regular languages.

Cited by 0Open access

On the Power of Quantum Computation *

Daniel R. Simon T · 2015

The quantum model of computation is a probabilistic model, similar to the probabilistic Turing Machine, in which the laws of chance are those obeyed by parti-cles on a quantum mechanical scale, rather than the rules familiar to us from the macroscopic world. We present here a problem of distinguishing between two fairly natural classes of function, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the func-tion is given as an oracle drawn equiprobably from the uniform distribution on either class. We thus of-fer compelling evidence that the quantum model may have significantly more complexity theoretic power than the probabilistic Turing Machine. In fact, draw-ing on this work, Shor has recently developed remark-able new quantum polynomial-time algorithms for the discrete logarithm and integer factoring problems. 1

Cited by 0

Internal Turing Machines

Ken Loo · ArXiv.org · 2004

Using nonstandard analysis, we will extend the classical Turing machines into the internal Turing machines. The internal Turing machines have the capability to work with infinite ($*$-finite) number of bits while keeping the finite combinatoric structures of the classical Turing machines. We will show the following. The internal deterministic Turing machines can do in $*$-polynomial time what a classical deterministic Turing machine can do in an arbitrary finite amount of time. Given an element of $\in HALT$ (more precisely, the $*$-embedding of $HALT$), there is an internal deterministic Turing machine which will take $$ as input and halt in the $"yes"$ state. The language ${}^*Halt$ can not be decided by the internal deterministic Turing machines. The internal deterministic Turing machines can be viewed as the asymptotic behavior of finite precision approximation to real number computations. It is possible to use the internal probabilistic Turing machines to simulate finite state quantum mechanics with infinite precision. This simulation suggests that no information can be transmitted instantaneously and at the same time, the Turing machine model can simulate instantaneous collapse of the wave function. The internal deterministic Turing machines are powerful, but if $P \neq NP$, then there are internal problems which the internal deterministic Turing machines can solve but not in $*$-polynomial time.

Cited by 1Open access

Quantum computing and quantum complexity theory

Umesh Vazirani · 2000

There is strong evidence that computers based upon the principles of quantum physics represent an inherently new and more powerful model of computation. Such computers violate the modern form of the Church-Turing thesis (which lies at the foundations of computer science). This thesis can be informally summarized as follows: all physical implementations of computing devices can be described by the same abstract model of computation-the probabilistic Turing Machine or the equivalent random access machine. In this paper, we will describe two formal models for quantum computers: quantum circuits and quantum Turing Machines, introduced by Deutsch [1985].

Cited by 0

A New Proof for the Monte Carlo Constructibility of log log n

Ryuhei Uehara · 2007

The Monte Carlo constructibility of log log n has proved by Karpinski and Verbeek ([KV87]). They proposed an algorithm, and proved the constructibility by applying a statistical result in [Fel57] to the algorithm. We give a new algorithm, and prove the constructibility by analyzing the algorithm. 1 Introduction One of the important subjects in the theory of computational complexity is the computation with the small space (see [RCH91] for a comprehensive reference). It is known that no monotone unbounded function below log n is fully space constructible by deterministic Turing machines [FL75, AM76] or nondeterministic Turing machines [Gef90]. Interestingly, Karpinski and Verbeek showed that log log n is fully space constructible by a probabilistic Turing machine with bounded error probability [KV87]. More precisely, they defined Monte Carlo Turing machines as probabilistic Turing machines with bounded error probability equal to 1 4 , and proved that log log n is fully space construc...

Cited by 0

Computational models

Riccardo Manenti, Mário Motta · 2023

Abstract In this chapter, we introduce the definitions of automata Turing machines. We present different flavors of Turing machines including deterministic, probabilistic, and multi-tape machines. We present several complexity classes, i.e. collections of problems that can be solved by a machine with similar resources, including the P and NP classes. We present the Church-Turing thesis, which states that if a problem can be solved by a physical machine, then it can also be solved by any deterministic Turing machine. In the last part of the chapter, we introduce quantum Turing machines and the quantum circuit model. We study the complexity class BQP. This is the class of problems that can be solved by a quantum computer with polynomial resources and a small error probability. Finally, we introduce QMA, the quantum version of the complexity class NP. We present the k-LOCAL problem, and we show that this problem is in QMA

Cited by 0

On the Power of Probabilistic Polynomial Time: P ^NP[log] ⊆ PP

Gerd Wechsung, Lane A. Hemachandra, Richard Beigel · UR Research (University of Rochester)

We show that every set in the Theta [superscript P, over 2] level of the polynomial hierarchy - every set polynomial- time truth-table reducible to SAT - is accepted by a probabilistic polynomial-time Turing machine: P^NP[log] ⊆ PP. Relatedly, we show that probabilistic polynomial time is closed under polynomial-time parity reductions.

Cited by 0

PP IS AS HARD AS THE POLYNOMIAL-TIME HIERARCHY*

Seinosuke Today · 2008

Abstract. In this paper, two interesting complexity classes, PP and P, are compared with PH, the polynomial-time hierarchy. It is shown that every set in PH is polynomial-time Turing reducible to a set in PP, and PH is included in BP. 0)P. As a consequence of the results, it follows thatPPPH (or 03P__ _ PH) implies a collapse of PH. A stronger result is also shown: every set in PP(PH) is polynomial-time Turing reducible to a set in PP. Key words, polynomial-time hierarchy, probabilistic Turing machine, polynomial-time Turing reductions, parity, randomized reduction

Cited by 0

On the power of probabilistic polynomial time: P/sup NP(log)/ contained in PP

Richard Beigel, Lane A. Hemachandra, Gerd Wechsung · 2003

It is shown that probabilistic time is closed under polynomial-time parity reductions. Therefore, every set polynomial-time truth-table reducible to SAT is accepted by a probabilistic polynomial-time Turing machine. Equivalently, P/sup NP(log)/ contained in PP.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Cited by 17

On the Power of Probabilistic Polynomial Time: P &lt;sup&gt;NP[log]&lt;/sup&gt; ⊆ PP

Wechsung, Gerd, Hemachandra, Lane A., Beigel, Richard · UR Research (University of Rochester)

We show that every set in the Theta [superscript P, over 2] level of the polynomial hierarchy - every set polynomial- time truth-table reducible to SAT - is accepted by a probabilistic polynomial-time Turing machine: P&lt;sup&gt;NP[log]&lt;/sup&gt; ⊆ PP. Relatedly, we show that probabilistic polynomial time is closed under polynomial-time parity reductions.

Cited by 0

Learning Capabilities of Recurrent Neural

Bhaskar Dasgupta · 1991

Recurrent neural networks are important models of neural computation. This work relates the power of them to those of other conventional models of computation like Turing machines and nite automata and proves results about their learning capabilities. Speci cally, it shows, (a) Probabilistic recurrent networks and Probabilistic turing machine models are equivalent. (b) Probabilistic recurrent networks with bounded error probabilities are no more powerful than determistic nite automata. (c) Deterministic recurrent networks have the capability of learning P-complete language problems ( which are the hardest problems in P to parallelize). (d) Restricting the weight-threshold relationship in deterministic recurrent networks may allow the network to learn only weaker classes of languages ( the NC class).

Cited by 0

Theory of One Tape Linear Time Turing Machines

K. Tadaki, Tomoyuki Yamakami, Jack C. H. Lin · arXiv (Cornell University) · 2003

A theory of one-tape (one-head) linear-time Turing machines is essentially different from its polynomial-time counterpart since these machines are closely related to finite state automata. This paper discusses structural-complexity issues of one-tape Turing machines of various types (deterministic, nondeterministic, reversible, alternating, probabilistic, counting, and quantum Turing machines) that halt in linear time, where the running time of a machine is defined as the length of any longest computation path. We explore structural properties of one-tape linear-time Turing machines and clarify how the machines' resources affect their computational patterns and power.

Cited by 0Open access

Communication Complexity and Lower Bounds for Sequential Computation

Bala Kalyanasundaram, Georg Schnitger · Teubner-Texte zur Informatik · 1992

Information-theoretic approaches for lower bound problems are discussed and two applications of Communication Complexity are presented. The first application concerns one-tape Turing machines with an additional oneway input tape. It is shown that lower bounds on the Communication Complexity of a given language immediately imply lower bounds on the running time for this Turing machine model. Consequently, lower bounds for the Turing machine complexity of specific languages are derived. Emphasis is given to bounded-error probabilistic Turing machines, since no previous lower bounds have been obtained for this computation mode. The second application concerns a real-time comparison between Schoenhage’s Storage Modification machines and the machine model of Kolmogorov and Uspen-skii. A non-standard model of Communication Complexity is defined. It is shown that non-trivial lower bounds for this communication model will imply that Storage Modification machines cannot be simulated in real time by Kolmogorov-Uspenskii machines.

Cited by 6