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 cla...

We resolve an old problem, namely to design a ‘natural’ machine model for accepting the complements of recursively enumerable languages. The model we present is based on Turing machines with ‘onesided’ advice, which are Turing machines with advice with a restricted scenario for the use of their advice during computations. We show that Turing machines with one-sided advice accept precisely the c...

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 ...

The Church-Turing thesis 2 states that to be computable is to be computable by a Turing machine and the modern discipline in computational complexity theory states that to be efficiently computable is to be computable by a Turing machine within polynomial steps in the length of the input data. However, Feynman pointed out that a Turing machine cannot simulate a quantum mechanical process effici...

The theory of Kolmogorov complexity is based on the discovery, by Alan Turing in 1936, of the universal Turing machine. After proposing the Turing machine as an explanation of the notion of a computing machine, Turing found that there exists one Turing machine which can simulate any other Turing machine. Complexity, according to Kolmogorov, can be measured by the length of the shortest program ...

In this brief note, we give a simple information-theoretic proof that 2-state 3-color universal Turing machines don’t exist. Let us suppose that there exists a 2-state 3-color universal Turing machine U (without a halting state). Then it must be possible to program U to emulate any Turing machine T (without a halting state) by initializing the tape of U with a string of colors that specify both...

Turing machines have been well studided in the context of Computability theory, looking at computations starting from a very specific state and a given input. We intend to study here Turing machines from the point of view of dynamical systems, that is consider Turing machines starting from any input, and any possible state. This model is very interesting from this point of view, and intimately ...

A Turing machine has an important role in education in the field of computer science, as it is a milestone in courses related to automata theory, theory of computation and computer architecture. Its value is also recognized in the Computing Curricula proposed by the Association for Computing Machinery (ACM) and IEEE Computer Society. In this paper we present a physical implementation of the Tur...

Computing Machine Models To gage the computational power of a family of mechanical computers, we will use a widely known abstract computational model known as the Turing machine, defined in this section. The Turing machine. The Turing machine model formulated by Alan Turing (1937) was the first complete mathematical model of an abstract computing machine that possessed universal computing power...