Related papers: Beyond Turing Machines
According to the Church-Turing Thesis (CTT), effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider claim true,…
The Turing machine (TM) and the Church thesis have formalized the concept of computable number, this allowed to display non-computable numbers. This paper defines the concept of number "approachable" by a TM and shows that some (if not all)…
Cellular automata are arrays of finite state machines that can exist in a finite number of states. These machines update their states simultaneously based on specific local rules that govern their interactions. This framework provides a…
Metastability is a spurious mode of operation in digital signals, where an electrical signal fails to settle into a stable state within a specified time, leading to uncertainty and potentially failing downstream hardware. A system that…
A physical system is determined by a finite set of initial conditions and "laws" represented by equations. The system is computable if we can solve the equations in all instances using a "finite body of mathematical knowledge". In this…
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…
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…
Computation models such as circuits describe sequences of computation steps that are carried out one after the other. In other words, algorithm design is traditionally subject to the restriction imposed by a fixed causal order. We address a…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…
Can computers overcome human capabilities? This is a paradoxical and controversial question, particularly because there are many hidden assumptions. This article focuses on that issue putting on evidence some misconception related with…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
The Turing Machine is the paradigmatic case of computing machines, but there are others such as analogical, connectionist, quantum and diverse forms of unconventional computing, each based on a particular intuition of the phenomenon of…
To date, work on formalizing connectionist computation in a way that is at least Turing-complete has focused on recurrent architectures and developed equivalences to Turing machines or similar super-Turing models, which are of more…
Computing is a high-level process of a physical system. Recent interest in non-standard computing systems, including quantum and biological computers, has brought this physical basis of computing to the forefront. There has been, however,…
The world has seen the emergence of machines based on pretrained models, transformers, also known as generative artificial intelligences for their ability to produce various types of content, including text, images, audio, and synthetic…
We propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum…
We present a generalization of standard Turing machines based on allowing unusual tapes. We present a set of reasonable constraints on tape geometry and classify all tapes conforming to these constraints. Surprisingly, this generalization…
Approaching limitations of digital computing technologies have spurred research in neuromorphic and other unconventional approaches to computing. Here we argue that if we want to systematically engineer computing systems that are based on…
This paper presents a theory of systemic undecidability, reframing incomputability as a structural property of systems rather than a localized feature of specific functions or problems. We define a notion of causal embedding and prove a…
In this paper we present an introduction to the area of computability in dynamical systems. This is a fairly new field which has received quite some attention in recent years. One of the central questions in this area is if relevant…