Abstract We study the parameterized complexity of the problem to decide whether a given natural number n satisfies a given $\Delta _0$ -formula $\varphi (x)$ ; the parameter is the size of $\varphi $ . This parameterization focusses attention on instances where n is large compared to the size of $\varphi $ . We show unconditionally that this problem does not belong to the parameterized analogue of $\mathsf {AC}^0$ . From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameterized halting problem. Some of these upper bounds follow assuming that $I\Delta _0$ proves the MRDP theorem in a certain weak sense.

The Turing Machine Halting Problem is a major problem in computer theory, Russell’s Paradox is the root of the Third Mathematical Crisis, and the Gödel Incompleteness Theorem is a major discovery in modern logic. The three have had a profound impact on the development of science and have attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned; in particular, Wittgenstein regards it as a certain logical paradox, and Russell’s Paradox has not yet been settled. This paper makes a detailed analysis of the three based on the view of dialectical infinity. The author notes that the Principle of Comprehension based on the view of actual infinity is the root of Russell’s Paradox. The Turing Machine Halting Problem shows that it is impossible to make an actual-infinite ultimate judgment of the constantly generated infinite world, but the philosophical significance of the Gödel Incompleteness Theorem is that our understanding of the world is essentially potentially infinite. At the end of the article, the author raises several questions about the proof of the Gödel Incompleteness Theorem, finds out the specific paradox form in the proof, points out the high consistency of its proof method and Russell’s Paradox, which strongly supports Wittgenstein’s view. The author points out that the philosophical basis of the proof of the Gödel Incompleteness Theorem is the idea of actual infinity, the proof of the theorem is based on a logically invalid circular formula, the contradiction of the proof originates from the Gödel formula itself, and cannot be attributed to the incompleteness of the system, so the proof is wrong. Therefore, the conclusion of this paper is that the world is constantly developing and changing, and our human understanding of the world is essentially a potential infinite, that is, the world is Aristotelian, not Platonic.

Heisenberg deduced the famous uncertainty principle which shows that there exist several conjugated quantities that can never be measured precisely at the same time. Further, Copenhagen interpretation gives a new perspective about quantum mechanics, which claims that particles do not have properties like position or momentum until people measure it. Hence, the famous EPR paradox was proposed to question the realism and locality of Quantum mechanics, which leads to the Hidden Variable explanation. However, this theorem was proved wrong by John Bell in 1964 with Bell Inequality. In addition, Hidden Variable Interpretation was developed by De Broglie and Bohm, they came up with the Bohmian Mechanics, which can be considered as Non-local Hidden Variable theorem. This interpretation gives physical meaning to waves. Unlike Copenhagen Interpretation, this theorem claims that particles do have a determined position. Some people may argue that it can be proved contradicted by using the Turing method (self-reference). Because if the algorithm represented by physics law described Hidden Mechanics to determine particles state, then the future is determined and predictable. Therefore, another algorithm can be established based on that, which will lead to Liar Paradox. This article will briefly introduce the uncertainty principle and some interpretations about quantum mechanics. Furthermore, this article will combine some ideas in Alan Turings Halting Problem to the universe of non-local hidden variables as a thought experiment, which involve self-reference, to give an interesting result of locality and determinism.

The Halting Problem, the most (in)famous undecidable problem, has important applications in theoretical and applied computer science and beyond, hence the interest in its approximate solutions. Experimental results reported on various models of compu

As an example of an algorithmically undecidable problem, most textbooks list the impossibility to check whether a given program halts on given data.A usual proof of this result is based on the assumption that the hypothetical halt-checker works for all programs.To show that a halt-checker is impossible, we design an auxiliary program for which the existence of such a halt-checker leads to a contradiction.However, this auxiliary program is usually very artificial.So, a natural question arises: what if we only require that the halt-checker work for reasonable programs?In this paper, we show that even with such a restriction, haltcheckers are not possible -and thus, we make a proof of halting problem more convincing for students.

No abstract available.

In this paper a detailed proofwill be given of the solvability of the halting and reachability problem for binary 2-tag systems.

In this paper a detailed proofwill be given of the solvability of the halting and reachability problem for binary 2-tag systems.

Weighted graphs: A tool for studying the halting problem and time complexity in term rewriting systems and logic programming

Monadic table counter schemas (MTCS) are defined as extensions of recursive monadic schemas by incorporating a depth-of-recursion counter. The family of languages generated by free MTCS under Herbrand interpretation is shown to be the family of ETOL languages. It is proven that the halting and divergence problems are decidable for free MTCS and that the freedom problem is decidable. Most of these results are obtained using results on regular control sequences from L system theory.

Modular machines were introduced in [1] and [2], where they were used to give simple proofs of various unsolvability results in group theory. Here we discuss the degrees of the halting, word, and confluence problems for modular machines, both for their own sake and in the hope that the results may be useful in group theory (see [4] for an application of a related result to group theory).In the course of the analysis, I found it convenient to compare degrees of these problems for a Turing machine T and for a Turing machine T1 obtained from T by enlarging the alphabet but retaining the same quintuples (or quadruples). The results were surprising. The degree for a problem of T1 depends not just on the corresponding degree for T, but also on the degrees of the corresponding problems when T is restricted to a semi-infinite tape (both semi-infinite to the right and semi-infinite to the left). For the halting and confluence problems, the Turing degrees of the problems for these three machines can be any r.e. degrees. In particular the halting problem of T can be solvable, while that of T1 has any r.e. degree.A machine M (in the general sense) consists of a countable set of configurations (together with a numbering, which we usually take for granted), a recursive subset of configurations called the terminal configurations, and a recursive function, written C ⇒ C′, on the set of configurations. If, for some n ≥ 0, we have C = C0 ⇒ C1 ⇒ … ⇒ Cn = C′, we write C → C′. We say M halts from C if C → C′ for some terminal C′.

Spectra and halting problems

The degree representations of the general halting, word, and confluence problems for Markov algorithms are investigated. Each of these problems is shown to represent every r.e. (recursively enumerable) many-one degree but not every r.e. one-one degree of unsolvability. In the course of proving this we also show that every total recursive function is computed by a Markov algorithm which always halts and that the immortality problem for the class of Markov algorithms is Σ2o-complete.

Strong Computability and Variants of the Uniform Halting Problem