Related papers: Turing Machines on Graphs and Inescapable Groups
We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length $\omega$ to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite…
Infinite time Turing machines with only one tape are in many respects fully as powerful as their multi-tape cousins. In particular, the two models of machine give rise to the same class of decidable sets, the same degree structure and, at…
We define a generalization of the Turing machine that computes on general sets. Our main theorem states that the class of generalized Turing machine computable functions and the class of Set Recursive functions coincide.
Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has…
Indexed monoidal algebras are introduced as an equivalent structure for self-dual compact closed categories, and a coherence theorem is proved for the category of such algebras. Turing automata and Turing graph machines are defined by…
This paper presents an algebraic theory of instruction sequences with instructions for Turing tapes as basic instructions, the behaviours produced by the instruction sequences concerned under execution, and the interaction between such…
We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only…
Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…
The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the…
The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of…
We propose a new generalisation of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language in the…
Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…
As an example of the concept of rulial space, we explore the case of simple Turing machines. We construct the rulial multiway graph which represents the behavior of all possible Turing machines with a certain class of rules. This graph…
We characterize the set of planar locally finite Cayley graphs, and give a finite representation of these graphs by a special kind of finite state automata called labeling schemes. As a result, we are able to enumerate and describe all…
The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a…
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…
A variant of Turing machines is introduced where the tape is replaced by a single tree which can be manipulated in a style akin to purely functional programming. This yields two benefits: first, the extra structure on the tape can be…
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…
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…
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…