Related papers: Symmetry for transfinite computability
What can one do with a given tunable quantum device? We provide complete symmetry criteria deciding whether some effective target interaction(s) can be simulated by a set of given interactions. Symmetries lead to a better understanding of…
Turing computability is the standard computability paradigm which captures the computational power of digital computers. To understand whether one can create physically realistic devices which have super-Turing power, one needs to…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
This is the first of a series of papers in which we study deep computations (ultracomputations) and deep iterates, formalizing the ideas of "asymptotic limit" of computations and compositional iterates, respectively. In this first paper of…
Deutsch, Feynman, and Manin viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
In the paper we present results to develop an irreducible theory of complex systems in terms of self-organization processes of prime integer relations. Based on the integers and controlled by arithmetic only the self-organization processes…
Manin, Feynman, and Deutsch have viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum logic circuits and quantum Turing machines has shown how these machines can simulate an…
Causality serves as an abstract notion of time for concurrent systems. A computation is causal, or simply valid, if each observation of a computation event is preceded by the observation of its causes. The present work establishes that this…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
Fixed point iterations are known to generate chaos, for some values in their parameter range. It is an established fact that Turing Machines are fixed point iterations. However, as these Machines operate in integer space, the standard…
Purpose: This study extends the structural theory of finite commutative ternary $\Gamma$-semirings into a computational and categorical framework for explicit classification and constructive reasoning. Methods: Constraint-driven enumeration…
We consider the question whether there is an infinitary analogue of the Church-Turing-thesis. To this end, we argue that there is an intuitive notion of transfinite computability and build a canonical model, called Idealized Agent Machines…
Predictions for measurement outcomes in physical theories are usually computed by combining two distinct notions: a state, describing the physical system, and an observable, describing the measurement which is performed. In quantum theory,…
Symmetry plays a crucial role in understanding the properties of mathematical structures and optimization problems. Recent work has explored this phenomenon in the context of neural networks, where the loss function is invariant under…
Dynamical Systems theory generally deals with fixed point iterations of continuous functions. Computation by Turing machine although is a fixed point iteration but is not continuous. This specific category of fixed point iterations can only…
A concept of randomness for infinite time register machines (ITRMs) is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies computability and that an analogue of…
We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions…
The computational cost of micromechanics for heterogeneous materials can be reduced in certain cases where symmetric boundary conditions are applicable. We derived an eighth symmetric formulation of the Generalized Method of Cells for…
We initiate the computability-theoretic study of ringed spaces and schemes. In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures…