Related papers: Small weakly universal Turing machines
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…
We present a simple algorithm to find the Moore machine with the minimum number of states equivalent to a given one.
A weakly consecutive sequence (WCS) is a permutation $\sigma$ of $\{1, \ldots, k\}$ such that if an integer $d$ divides $\sigma(i)$, then $d$ also divides $\sigma(i \pm d)$ insofar as these are defined. The structure of weakly consecutive…
Current quantum computer technology is sufficient to realize weak measurements and the corresponding concept of weak values. We demonstrate how the weak value anomaly can be tested, along with consistency and simultaneity of weak values,…
We give new Turing machines that simulate the iteration of the Collatz 3x+1 function. First, a never halting Turing machine with 3 states and 4 symbols, improving the known 3x5 and 4x4 Turing machines. Second, Turing machines that halt on…
In this paper, we construct a weakly universal cellular automaton with two states only on the tiling {11,3}. The cellular automaton is rotation invariant and it is a true planar one.
An example shows that weak decoherence is more restrictive than the minimal logical decoherence structure that allows probabilities to be used consistently for quantum histories. The probabilities in the sum rules that define minimal…
We construct reversible Boolean circuits efficiently simulating reversible Turing machines. Both the circuits and the simulation proof are rather simple. Then we give a fairly straightforward generalization of the circuits and the…
Conventional quantum mechanics describes a pre- and post-selected system in terms of virtual (Feynman) paths via which the final state can be reached. In the absence of probabilities, a weak measurement (WM) determines the probability…
Ordered binary decision diagrams (OBDDs) are a fundamental data structure for the manipulation of Boolean functions, with strong applications to finite-state symbolic model checking. OBDDs allow for efficient algorithms using top-down…
Weak measurement is a standard measuring procedure with two changes: it is performed on pre- and post-selected quantum systems and the coupling to the measuring device is weakened. The outcomes of weak measurements, ``weak values'' are very…
Multiway Turing machines (also known as nondeterministic Turing machines or NDTMs) with explicit, simple rules are studied. Even very simple rules are found to generate complex behavior, characterized by complex multiway graphs, that can be…
For any fixed $k$, a remarkably simple single-tape Turing machine can simulate $k$ independent counters in real time. Informally, a counter is a storage unit that maintains a single integer (initially 0), incrementing it, decrementing it,…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
Since Cocke and Minsky proved 2-tag systems universal, they have been extensively used to prove the universality of numerous computational models. Unfortunately, all known algorithms give universal 2-tag systems that have a large number of…
In quantum theory, a weak value is a complex number with a somewhat technical definition: it is a ratio whose numerator is the matrix element of a self-adjoint operator and whose denominator is the inner product of a corresponding pair of…
We provide partial implementations of von Neumann's universal constructor and universal copier, starting out with three types of simple building blocks using minimal assumptions. Using the same principles, we also construct Turing machines.…
We study finite-state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite…
A Turing machine with faults, failures and recovery (TMF) is described. TMF is (weakly) non-deterministic Turing machine consisting of five semi-infinite tapes (Master Tape, Synchro Tape, Backup Tape, Backup Synchro Tape, User Tape) and…
The power of real-time Turing machines using sublinear space is investigated. In contrast to a claim appearing in the literature, such machines can accept non-regular languages, even if working in deterministic mode. While maintaining a…