Class Of Computable Functions Research Articles

On the Generic Undecidability of the Halting Problem for Normalized Turing Machines

Hamkins and Miasnikov presented in (Notre Dame J. Formal Logic 47(4), 515---524, 2006) a generic algorithm deciding the classical Halting Problem for Turing machines with one-way tape on a set of asymptotic probability one (on a so-called generic set). Rybalov (Theor. Comput. Sci. 377, 268---270, 2007) showed that there is no generic algorithm deciding this problem on strongly generic sets of inputs (some subclass of generic sets). In this paper we prove that there is no generic algorithm deciding the Halting Problem for normalized Turing machines on generic sets of inputs. Normalized Turing machines have programs with the following natural restriction: internal states with large indices are not allowed at the beginning of the program. This condition does not reduce the class of computable functions because for every Turing machine there exists a normalized Turing machine which computes the same function. Our proof holds for machines with one-way and two-way tape. It also implies that the Hamkins-Miasnikov algorithm is not generic for normalized Turing machines.

Mutual Dimension

We define the lower and upper mutual dimensions mdim ( x : y ) and Mdim ( x : y ) between any two points x and y in Euclidean space. Intuitively, these are the lower and upper densities of the algorithmic information shared by x and y . We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that if f : R m → R n is computable and Lipschitz, then the inequalities mdim ( f ( x ): y ) ≤ mdim ( x : y ) and Mdim ( f ( x ): y ) ≤ Mdim ( x : y ) hold for all x ∈ R m and y ∈ R t . We use this inequality and related inequalities that we prove in like fashion to establish conditions under which various classes of computable functions on Euclidean space preserve or otherwise transform mutual dimensions between points.

Modelling of Complex Systems: Systems as Dataflow Machines

We develop a unified functional formalism for modelling complex systems, that is to say systems that are composed of a number of heterogeneous components, including typically software and physical devices. Our approach relies on non-standard analysis that allows us to model continuous time in a discrete way. S ystems are defined as generalized Turing machines with temporized input, internal and output mechanisms. Behaviors of systems are represented by transfer functions. A transfer function is said to be implementable if it is associated with a system. This notion leads us to define a new class - which is natural in our framework - of computable functions on (usual) real numbers. We show that our definitions are robust: on one hand, the class of implementable transfer functions is closed under composition; on the other hand, the class of computable functions in our meaning includes analytical functions whose coefficients are computable in the usual way, and is closed under addition, multiplication, differentiation and integration. Our class of computable functions also includes solutions of dynamical and Hamiltonian systems defined by computable functions. Hence, our notion of system appears to take suitably into account physical systems.

Inferring answers to queries

One focus of inductive inference is to infer a program for a function f from observations or queries about f. We propose a new line of research which examines the question of inferring the answers to queries. For a given class of computable functions, we consider the learning (in the limit) of properties of these functions that can be captured by queries formulated in a logical language L. We study the inference types that arise in this context. Of particular interest is a comparison between the learning of properties and the learning of programs. Our results suggest that these two types of learning are incomparable. In addition, our techniques can be used to prove a general lemma about query inference [W. Gasarch, C. Smith, Learning via queries, J. ACM 39 (1992) 649–676]. We show that I ⊂ J ⇒ Q I ( L ) ⊂ Q J ( L ) for many standard inference types I , J and many query languages L. Hence any separation that holds between these inference types also holds between the corresponding query inference types. One interesting consequence is that [ 24 , 49 ] QEX 0 ( [ Succ , < ] 2 ) − [ 2 , 4 ] QEX 0 ( [ Succ , < ] 2 ) ≠ ∅ .

Infinite Time Turing Machines With Only One Tape

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 partial 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 infinitary computation are not equivalent. Surprisingly, the class of one-tape computable functions is not closed under composition; but closing it under composition yields the full class of all infinite time computable functions. Finally, every ordinal that is clockable by an infinite time Turing machine is clockable by a one-tape machine, except certain isolated ordinals that end gaps in the clockable ordinals.

Maximal Machine Learnable Classes

A class of computable functions ismaximaliff it can be incrementally learned by some inductive inference machine (IIM), but no infinitely larger class of computable functions can be so learned. Rolf Wiehagen posed the question whether there exist such maximal classes. This question and many interesting variants are answered herein in the negative. Viewed positively, each IIM can be infinitely improved upon! Also discussed are the problems of algorithmically finding the improvements proved to exist.

Approximation methods in inductive inference

In many areas of scientific inquiry, the phenomena under investigation are viewed as functions on the real numbers. Since observational precision is limited, it makes sense to view these phenomena as bounded functions on the rationals. One may translate the basic notions of recursion theory into this framework by first interpreting a partial recursive function as a function on Q. The standard notions of inductive inference carry over as well, with no change in the theory. When considering the class of computable functions on Q, there are a number of natural ways in which to define the distance between two functions. We utilize standard metrics to explore notions of approximate inference — our inference machines will attempt to guess values which converge to the correct answer in these metrics. We show that the new inference notions, NV ∞, EX ∞, and BC ∞, infer more classes of functions than their standard counterparts, NV, EX, and BC. Furthermore, we give precise inclusions between the new inference notions and those in the standard inference hierarchy. We also explore weaker notions of approximate inference, leading to inference hierarchies analogous to the EX n and BC n hierarchies. Oracle inductive inference is also considered, and we give sufficient conditions under which approximate inference from a generic oracle G is equivalent to approximate inference with only finitely many queries to G.

The Mathematical Work of S.C.Kleene

§1. The origins of recursion theory. In dedicating a book to Steve Kleene, I referred to him as the person who made recursion theory into a theory. Recursion theory was begun by Kleene's teacher at Princeton, Alonzo Church, who first defined the class of recursive functions; first maintained that this class was the class of computable functions (a claim which has come to be known as Church's Thesis); and first used this fact to solve negatively some classical problems on the existence of algorithms. However, it was Kleene who, in his thesis and in his subsequent attempts to convince himself of Church's Thesis, developed a general theory of the behavior of the recursive functions. He continued to develop this theory and extend it to new situations throughout his mathematical career. Indeed, all of the research which he did had a close relationship to recursive functions.Church's Thesis arose in an accidental way. In his investigations of a system of logic which he had invented, Church became interested in a class of functions which he called the λ-definable functions. Initially, Church knew that the successor function and the addition function were λ-definable, but not much else. During 1932, Kleene gradually showed1 that this class of functions was quite extensive; and these results became an important part of his thesis 1935a (completed in June of 1933).

Church's thesis without tears

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

Augmented loop languages and classes of computable functions

A classification of all the computable functions is given in terms of subrecursive programming languages. These classes are those which arise from the relation “primitive recursive in”. By distinguishing between honest and dishonest classes the classification is related to the computational complexity of the functions classified and the classification has a wide degree of measure invariance. The structure of the honest and dishonest classes under inclusion is explored. It is shown that any countable partial ordering can be embedded in the dishonest classes, and that the dishonest classes are dense in the honest classes. Every honest class is minimal over some dishonest class, but there are dishonest classes with no honest class minimal over them.

A Note on the Intersection of Complexity Classes of Functions

The classes of computable functions defined by a bound on the computation time are shown not to be closed under infinite descending intersection.