Related papers: Turing Machines and Understanding Computational Co…
The preceding paper constructed tangle machines as diagrammatic models, and illustrated their utility with a number of examples. The information content of a tangle machine is contained in characteristic quantities associated to equivalence…
Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not being fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example…
This is yet another version of the course notes in chao-dyn/9407003. Here we change the universal Turing machine that is used to measure program-size complexity so that the constants in our information-theoretic incompleteness theorems are…
The study of computability has its origin in Hilbert's conference of 1900, where an adjacent question, to the ones he asked, is to give a precise description of the notion of algorithm. In the search for a good definition arose three…
Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important…
A coherent mathematical overview of computation and its generalisations is described. This conceptual framework is sufficient to comfortably host a wide range of contemporary thinking on embodied computation and its models.
Hypercomputation or super-Turing computation is a ``computation'' that transcends the limit imposed by Turing's model of computability. The field still faces some basic questions, technical (can we mathematically and/or physically build a…
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…
Generative AI techniques have opened the path for new generations of machines in diverse domains. These machines have various capabilities for example, they can produce images, generate answers or stories, and write codes based on the…
This paper describes the formal verification of two Turing machines using the program verifier Dafny. Both machines are deciders, so we prove total correctness. They are typical first examples of Turing machines used in any course of…
The essay consists of three parts. In the first part, it is explained how theory of algorithms and computations evaluates the contemporary situation with computers and global networks. In the second part, it is demonstrated what new…
We give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios. The aim is to provide simple but clear definitions, and to highlight…
Artificial computing machinery transforms representations through an objective process, to be interpreted subjectively by humans, so the machine and the interpreter are different entities, but in the putative natural computing both…
The paper offers a mathematical formalization of the Turing test. This formalization makes it possible to establish the conditions under which some Turing machine will pass the Turing test and the conditions under which every Turing machine…
Recently Quantum Computation has generated a lot of interest due to the discovery of a quantum algorithm which can factor large numbers in polynomial time. The usefulness of a quantum com puter is limited by the effect of errors. Simulation…
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…
In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of boundary is given either as a list of…
In this chapter, I review the main methods and techniques of complex systems science. As a first step, I distinguish among the broad patterns which recur across complex systems, the topics complex systems science commonly studies, the tools…
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…
The Turing Machine has two implicit properties that depend on its underlying notion of computing: the format is fully determinate and computations are information preserving. Distributed representations lack these properties and cannot be…