We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals. Every Pi^1_1 set, for example, is decidable by such machines, and the semi-decidable sets form a portion of the Delta^1_2 sets. Our oracle concept leads to a notion of relative computability for reals and sets of reals and a rich degree structure, stratified by two natural jump operators.
Infinite time Turing machines are extended in several ways to allow for iterated oracle calls. The expressive power of these machines is discussed and in some cases determined.
We propose a notion of autoreducibility for infinite time computability and explore it and its connection with a notion of randomness for infinite time machines.
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 the final loop, in the classes 3x10, 4x6, 5x4, and 13x2.
Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.
We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bmΔ^1_2$ in a satisfying way.
Clift and Murfet (2019) introduced a naive Bayesian smooth relaxation of Turing machines motivated by work in differential linear logic; this was subsequently used to endow spaces of program codes of bounded length with a smooth manifold structure via the staged-pseudo universal Turing machine introduced by Clift, Murfet and Wallbridge (2021). In this paper, we give a general construction for simulating n-tape Turing machines on a single tape Turing machine such that the (naive Bayesian) uncertainty is propagated in an equivalent manner. This result suggests a stronger kind of equivalence betw…
We give several different encodings of the step function of a Turing machine in intuitionistic linear logic, and calculate the denotations of these encodings in the Sweedler semantics.
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 least for functions f:R-->N, the same class of computable functions. Nevertheless, there are infinite time computable functions f:R-->R that are not one-tape computable, and so the two models of supertask computation are not equivalent. Surprisingly, the class of one-tape computable functions is not closed under composition; but closing it under composition y…
We consider how changes in transfinite machine architecture can sometimes alter substantially their capabilities. We approach the subject by answering three open problems touching on: firstly differing halting time considerations for machines with multiple as opposed to single heads, secondly space requirements, and lastly limit rules. We: 1) use admissibility theory, $Σ_{2}$-codes and $Π_{3}$-reflection properties in the constructible hierarchy to classify the halting times of ITTMs with multiple independent heads; the same for Ordinal Turing Machines which have $\tmop{On}$ length tapes; 2) d…
Schindler recently addressed two versions of the question P $\stackrel{?}{=}$ NP for Turing machines running in transfinite ordinal time. These versions differ in their definition of input length. The corresponding complexity classes are labelled P, NP and ${\rm P}^+,{\rm NP}^+$. Schindler showed that P $\neq$ NP and ${\rm P}^+ \neq {\rm NP}^+$. We show that P $=$ NP $\cap$ co-NP and NP $\neq$ co-NP, whereas ${\rm P}^+ \subset$ NP $\cap$ co-NP and ${\rm NP}^+ \neq $ co-NP$^+$.
We calculate denotations under the Sweedler semantics of the Ehrhard-Regnier derivatives of various encodings of Turing machines into linear logic. We show that these derivatives calculate the rate of change of probabilities naturally arising in the Sweedler semantics of linear logic proofs. The resulting theory is applied to the problem of synthesising Turing machines by gradient descent.
If we establish that the counterexample function for P=NP, if total, overtakes all total recursive functions when extended over all Turing machines, then what happens to the same counterexample function when defined over the so-called Baker-Gill-Solovay (BGS) set of poly machines? We state and prove here a lemma that tries to answer this query.
Extending results of Schindler [math.LO/0106087] and Hamkins and Welch [math.LO/0212046], we establish in the context of infinite time Turing machines that P is properly contained in NP intersect coNP. Furthermore, NP intersect coNP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P+ = (NP+ intersect coNP+) = (NP intersect coNP), though P++ is properly contained in NP++ intersect coNP++. Within any contiguous block of infinite clockable ordinals, we show that P_alpha is not equal to NP_alpha intersect coNP_alpha, but if beta begins a gap in the cloc…
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 ($IAM$s) of this which will turn out to be equivalent in strength to the Ordinal Turing Machines defined by P. Koepke.
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.
In this paper, we analyze the problem of finding the minimum dimension $n$ such that a closed-form analytic map/ordinary differential equation can simulate a Turing machine over $\mathbb{R}^{n}$ in a way that is robust to perturbations. We show that one-dimensional closed-form analytic maps are sufficient to robustly simulate Turing machines; but the minimum dimension for the closed-form analytic ordinary differential equations to robustly simulate Turing machines is two, under some reasonable assumptions. We also show that any Turing machine can be simulated by a two-dimensional $C^{\infty}$ …
This note introduces a generalization to the setting of infinite-time computation of the busy beaver problem from classical computability theory, and proves some results concerning the growth rate of an associated function. In our view, these results indicate that the generalization is both natural and promising.
We consider graph Turing machines, a model of parallel computation on a graph, in which each vertex is only capable of performing one of a finite number of operations. This model of computation is a natural generalization of several well-studied notions of computation, including ordinary Turing machines, cellular automata, and parallel graph dynamical systems. We analyze the power of computations that can take place in this model, both in terms of the degrees of computability of the functions that can be computed, and the time and space resources needed to carry out these computations. We furt…
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 combinatoric structures of the classical Turing machines. We will show the following. The internal deterministic Turing machines can do in $*$-polynomial time what a classical deterministic Turing machine can do in an arbitrary finite amount of time. Given an element of $<M;x>\in HALT$ (more precisely, the $*$-embedding of $HALT$), there is an internal deterministic…
We introduce a model of infinitary computation which enhances the infinite time Turing machine model slightly but in a natural way by giving the machines the capability of detecting cardinal stages of computation. The computational strength with respect to ITTMs is determined to be precisely that of the strong halting problem and the nature of the new characteristic ordinals (clockable, writable, etc.) is explored.
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 does not lead to yet another equivalent formulation of the notion of computable function. Rather, it gives an alternative definition of the recursively enumerable Turing degrees that does not rely on oracles. The definitions give rise to a number of questions about computable paths inside Cayley graphs of finitely generated groups, and several of these questi…
We answer two questions posed by Castro and Cucker, giving the exact complexities of two decision problems about cardinalities of omega-languages of Turing machines. Firstly, it is $D_2(Σ_1^1)$-complete to determine whether the omega-language of a given Turing machine is countably infinite, where $D_2(Σ_1^1)$ is the class of 2-differences of $Σ_1^1$-sets. Secondly, it is $Σ_1^1$-complete to determine whether the omega-language of a given Turing machine is uncountable.
An ω-language is a set of infinite words over a finite alphabet X. We consider the class of recursive ω-languages, i.e. the class of ω-languages accepted by Turing machines with a Büchi acceptance condition, which is also the class Σ11 of (effective) analytic subsets of Xω for some finite alphabet X. We investigate here the notion of ambiguity for recursive ω-languages with regard to acceptance by Büchi Turing machines. We first present in detail essentials on the literature on ω-languages accepted by Turing Machines. Then we give a complete and broad view on the notion of ambiguity and unambi…
Wolfram [2, p. 707] and Cook [1, p. 3] claim to prove that a (2,5) Turing machine (2 states, 5 symbols) is universal, via a universal cellular automaton known as Rule 110. The first part of this paper points out a critical gap in their argument. The second part bridges the gap, thereby giving what appears to be the first proof of universality.
Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation that can compute more than the Turing machine and addresses their implications. In this report, I survey much of the work that has been done on hypercomputation, explaining how such non-classical models fit into the classical theory of computation and comparing their relative powers. I also examine the physical requirements for such machines to be constructib…
In this paper, we have compared r.e. sets based on their enumeration orders with Turing machines. Accordingly, we have defined novel concept uniformity for Turing machines and r.e. sets and have studied some relationships between uniformity and both one-reducibility and Turing reducibility. Furthermore, we have defined type-2 uniformity concept and studied r.e. sets and Turing machines based on this concept. In the end, we have introduced a new structure called Turing Output Binary Search Tree that helps us lighten some ideas.
