Non-computable Functions Research Articles

Generating candidate busy beaver machines (or how to build the zany zoo)

The busy beaver problem is a well-known example of a non-computable function. In order to determine a particular value of this function, it is necessary to generate and classify a large number of Turing machines. The advent of cloud computing provides unparallelled opportunities to explore this problem, but also to develop infrastructures that support distributed investigations and relevant computational evidence for all results obtained, such as a list of all machines of a given size together with their classifications. This list can then be used for variants of the problem or related ones such as the placid platypus problem. We investigate how to generate classes of machines to be considered for the busy beaver problem, which involves separating the generation of machines from their analysis. We discuss the relationship between quadruple and quintuple variants of Turing machines, and show that the latter are more general than the former. We give some formal results to justify our strategy for minimising the number of machines generated, and define a process reflecting this strategy for generating machines. We describe our implementation, and the results of generating various classes of machines with up to 5 states or up to 5 symbols, all of which (together with our code) are available on the author's website.

Uncertainty on Discrete-Event System Simulation

Uncertainty Propagation methods are well-established when used in modeling and simulation formalisms like differential equations. Nevertheless, until now there are no methods for Discrete-Dynamic Systems. Uncertainty-Aware Discrete-Event System Specification (UA-DEVS) is a formalism for modeling Discrete-Event Dynamic Systems that include uncertainty quantification in messages, states, and event times. UA-DEVS models provide a theoretical framework to describe the models’ uncertainty and their properties. As UA-DEVS models can include continuous variables and non-computable functions, their simulation could be non-computable. For this reason, we also introduce Interval-Approximated Discrete-Event System Specification (IA-DEVS), a formalism that approximates UA-DEVS models using a set of order and bounding functions to obtain a computable model. The computable model approximation produces a tree of all trajectories that can be traversed from the original model and some erroneous ones introduced by the approximation process. We also introduce abstract simulation algorithms for IA-DEVS, present a case study of UA-DEVS, its IA-DEVS approximation and, its simulation results using the algorithms defined.

Algebra of Countably Functions and Theorems of Completeness

An algebraic approach to the theory of countable functions is given. Compositions (superpositions) of functions are used instead of recursions. Arithmetic and analytic algorithms are defined. All closed sets are founded. Mathematically precise definitions of logic algorithms with quantifiers of existence and universality are given. Logic algorithm for fast-growing function is built as example. Classification of functions is given. There are non-computable functions. These functions are fictitious (useless) and their set is continual. The set of computable functions is countable. Incompleteness of disjunction and negation, conjunction and negation, of Pierce, Sheffer and diagonal Webb functions is proved. The completeness of the set of one-place functions and any all-valued essential function (Slupecki theorem) is proved for computable functions. Existence of generators of all computable functions is proved too.

THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

AbstractIn a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

Diagonally non-computable functions and fireworks

A set C of reals is said to be negligible if there is no probabilistic algorithm which generates a member of C with positive probability. Various classes have been proven to be negligible, for example the Turing upper-cone of a non-computable real, the class of coherent completions of Peano Arithmetic or the class of reals of minimal Turing degree. One class of particular interest in the study of negligibility is the class of diagonally non-computable (DNC) functions, proven by Kučera to be non-negligible in a strong sense: every Martin-Löf random real computes a DNC function. Ambos-Spies et al. showed that the converse does not hold: there are DNC functions which compute no Martin-Löf random real. In this paper, we show that the set of such DNC functions is in fact non-negligible using a technique we call ‘fireworks argument’. We also use this technique to prove further results on DNC functions.

Busy beaver machines and the observant otter heuristic (or how to tame dreadful dragons)

The busy beaver is a well-known specific example of a non-computable function. Whilst many aspects of this problem have been investigated, it is not always easy to find thorough and convincing evidence for the claims made about the maximality of particular machines, and the phenomenal size of some of the numbers involved means that it is not obvious that the problem can be feasibly addressed at all. In this paper we address both of these issues. We discuss a framework in which the busy beaver problem and similar problems may be addressed, and the appropriate processes for providing evidence of claims made. We also show how a simple heuristic, which we call the observant otter, can be used to evaluate machines with an extremely large number of execution steps required to terminate. We also show empirical results for an implementation of this heuristic which show how this heuristic is effective for all known ‘monster’ machines.

Problems in number theory from busy beaver competition

By introducing the busy beaver competition of Turing machines, in 1962, Rado defined noncomputable functions on positive integers. The study of these functions and variants leads to many mathematical challenges. This article takes up the following one: How can a small Turing machine manage to produce very big numbers? It provides the following answer: mostly by simulating Collatz-like functions, that are generalizations of the famous 3x+1 function. These functions, like the 3x+1 function, lead to new unsolved problems in number theory.

Beyond Physics? On the Prospects of Finding a Meaningful Oracle

Certain enterprises at the fringes of science, such as intelligent design creationism, claim to identify phenomena that go beyond not just our present physics but any possible physical explanation. Asking what it would take for such a claim to succeed, we introduce a version of physicalism that formulates the proposition that all available data sets are best explained by combinations of “chance and necessity”—algorithmic rules and randomness. Physicalism would then be violated by the existence of oracles that produce certain kinds of noncomputable functions. Examining how a candidate for such an oracle would be evaluated leads to questions that do not admit an easy resolution. Since we lack any plausible candidate for any such oracle, however, chance-and-necessity physicalism appears very likely to be correct.

Joining non-low C.E. sets with diagonally non-computable functions

Joining non-low C.E. sets with diagonally non-computable functions

Diagonally Non-Computable Functions and Bi-Immunity

AbstractWe prove that every diagonally noncomputable function computes a set A which is bi-immune, meaning that neither A nor its complement has an infinite computably enumerable subset.

Noncomputable functions in the Blum-Shub-Smale model

Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d + 1. We add a number of further results on relative computability of these sets and their unions. Then we show that the halting problem for BSS-computation is not decidable below any countable oracle set, and give a more specific condition, related to the cardinalities of the sets, necessary for relative BSS-computability. Most of our results involve the technique of using as input a tuple of real numbers which is algebraically independent over both the parameters and the oracle of the machine.

Simple section interchange and properties of non-computable functions

We here extend our earlier work on the theory of partially defined computer instructions and guards to cover partially defined computer expressions and programs. The notion of the relevant region of an expression is generalized to conditional relevant regions, and we specify upper bounds on these for expressions built up from simpler ones. We then proceed to specify upper bounds on the input and output regions of programs containing goto statements, even though the state transformations of such programs are not necessarily computable. This is then combined with our earlier commutativity results to obtain a general condition under which two simple sections of such a program commute with each other, and therefore may be interchanged, or possibly done in parallel.

Absolutely non-computable predicates and functions in analysis

In the representation approach (TTE) to computable analysis, the representations of an algebraic or topological structure for which the basic predicates and functions become computable are of particular interest. There are, however, many predicates (like equality of real numbers) and functions that areabsolutely non-computable, that is, not computable for any representation. Many of these results can be deduced from a simple lemma. In this article we prove this lemma for multi-representations and apply it to a number of examples. As applications, we show that various predicates and functions on computable measure spaces are absolutely non-computable. Since all the arguments are topological, we prove that the predicates are not relatively open and the functions are not relatively continuous for any multi-representation.

Quantum Hypercomputation—Hype or Computation?

A recent attempt to compute a (recursion-theoretic) noncomputable function using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion-theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from.

Betti numbers of finitely presented groups and very rapidly growing functions

Define the length of a finite presentation of a group G as the sum of lengths of all relators plus the number of generators. How large can the k th Betti number b k ( G ) = rank H k ( G ) be providing that G has length ≤ N and b k ( G ) is finite? We prove that for every k ≥ 3 the maximum b k ( N ) of the k th Betti numbers of all such groups is an extremely rapidly growing function of N . It grows faster that all functions previously encountered in mathematics (outside of logic) including non-computable functions (at least those that are known to us). More formally, b k grows as the third busy beaver function that measures the maximal productivity of Turing machines with ≤ N states that use the oracle for the halting problem of Turing machines using the oracle for the halting problem of usual Turing machines. We also describe the fastest possible growth of a sequence of finite Betti numbers of a finitely presented group. In particular, it cannot grow as fast as the third busy beaver function but can grow faster than the second busy beaver function that measures the maximal productivity of Turing machines using an oracle for the halting problem for usual Turing machines. We describe a natural problem about Betti numbers of finitely presented groups such that its answer is expressed by a function that grows as the fifth busy beaver function. Also, we outline a construction of a finitely presented group all of whose homology groups are either Z or trivial such that its Betti numbers form a random binary sequence.

A new conceptual framework for analog computation

In this paper we show how to explore the classical theory of computability using the tools of Analysis: A differential scheme is substituted for the classical recurrence scheme and a limit operator is substituted for the classical minimization. We show that most relevant problems of computability over the non-negative integers can be dealt with over the reals: elementary functions are computable, Turing machines can be simulated, the hierarchy of non-computable functions can be represented (the classical halting problem being solvable at some level). The most typical concepts in Analysis become natural in this framework. The most relevant question is posed: Can we solve open problems of classical computability and computational complexity using, as Popper says, the toolbox of Analysis?

Quantum Hypercomputability?

A recent proposal to solve the halting problem with the quantum adiabatic algorithm is criticized and found wanting. Contrary to other physical hypercomputers, where one believes that a physical process "computes" a (recursive-theoretic) non-computable function simply because one believes the physical theory that presumably governs or describes such process, believing the theory (i.e., quantum mechanics) in the case of the quantum adiabatic "hypercomputer" is tantamount to acknowledging that the hypercomputer cannot perform its task.

A new Gödelian argument for hypercomputing minds based on the busy beaver problem

Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using as a springboard Gödel’s little-known assertion that the human mind has a power “converging to infinity”, and as an anchoring problem Rado’s [T. Rado, On non-computable functions, Bell System Technical Journal 41 (1963) 877–884] Turing-uncomputable “busy beaver” (or Σ) function, we present in this short paper a new argument that, in fact, human persons can hypercompute. The argument is intended to be formidable, not conclusive: it brings Gödel’s intuition to a greater level of precision, and places it within a sensible case against computationalism.

Some thoughts on hypercomputation

We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory within the setting of formal mathematical theories.

Mechanizing coinduction and corecursion in higher-order logic

A theory of recursive and corecursive deflnitions has been developed in higher-order logic (HOL) and mechanized using Isabelle. Least flxedpoints express inductive data types such as strict lists; greatest flxedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresses recursive functions over inductive data types; corecursion expresses functions that yield elements of coinductive data types. The theory rests on a traditional formalization of inflnite trees. The theory is intended for use in speciflcation and veriflcation. It supports reasoning about a wide range of computable functions, but it does not formalize their operational semantics and can express noncomputable functions also. The theory is illustrated using flnite and inflnite lists. Corecursion expresses functions over inflnite lists; coinduction reasons about such functions.