can u solve this halting paradox?

Open Journal of Mathematics and Physics, 2020

We discuss the dubbed term "logical loop" and it's implication regarding provability in undecidable theorems.

2023

The work offers a unique perspective on the Turing machine halting problem, attributing it to arithmetic logic irreversibility and memory erasure, which introduce computational uncertainty by causing loss of information during computation. It discusses the concept of arithmetic logical entropy as a measure of uncertainty and indicator of information loss, along with characteristics of irreversibility, Landauer's principle, and memory erasure. The connection between Turing machines and general recursive functions, explored through λ1 calculus and the Turing/Church thesis, highlights the unpredictability of certain recursive functions due to the lack of information in their definitions. The text emphasizes that algorithms cannot generate more information than provided in the input, leading to indecision in determining whether a program will halt or not, especially when semantic characteristics are unknown. It also mentions the Turing oracle machine, suggesting that it introduces external information to compensate for the lack of internal information, thus influencing the computation's outcome.

2016

In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy Beaver function which we will call Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a “paradox” of computability a la Skolem: a function is computable when “seen from the outside” the subsystem, but uncomputable when “seen from within” the same subsystem. Finally, we will raise the possibility of defining universal submachines, and a hierarchy of negative Turing degrees.

Engineering and the Ultimate: An Interdisciplinary Investigation of Order and Design in Nature and Craft, 2014

Calculating the complexity of software projects is important to software engineering as it helps in estimating the likely locations of bugs as well as the number of resources required to modify certain program areas. Cyclomatic complexity is one of the pri- mary estimators of software complexity which operates by counted branch points in software code. However, cyclomatic complexity assumes that all branch points are equally complex. Some types of branch points require more creativity and foresight to understand and program correctly than others. Specifically, when knowledge of the behavior of a loop or recursion requires solving a problem similar to the halting problem, that loop has intrinsically more complexity than other types of loops or con- ditions. Halting-problem-like problems can be detected by looking for loops whose termination conditions are not intrinsically bound in the looping construct. These types of loops are counted to find the program complexity. This metric is orthogonal to cyclomatic complexity (which remains useful) rather than as a substitute for it.

Lecture Notes in Computer Science, 1993

This paper proposes a codi cation of the halting problem of any Turing machine in the form of only one right{linear binary Horn clause as follows : p(t) p(tt) : where t (resp. tt) is any (resp. linear) term. Recursivity is well{known to be a crucial and fundamental concept in programming theory. This result proves that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes. Some direct consequences are presented here. For instance, there exists an explicitly constructible right{linear binary Horn clause for which no decision algorithm, given a goal, always decides in a nite number of steps whether or not the resolution using this clause is nite. The halting problem of derivations w.r.t. one binary Horn clause had been shown decidable if the goal is ground SS88] or if the goal is linear Dev88, Dev90, DLD90]. The undecidability in the non{linear case is an unexpected extension. The proof of the main result is based on the unpredictable iterations of periodically linear functions de ned by J.H. Conway within number theory. Let us note that these new undecidability results are proved w.r.t. any type of resolution (bottom{up or top{down, depth{ rst or breadth{ rst, uni cation with or without occur{check).

INTERNATIONAL JOURNAL OF RESEARCH IN MATHEMATICS AND COMPUTATION (IJRMC), 2024

Here we define, mathematically, a program : ⟶ {0,1} ℵ. Where is a set of all programmable words, we consider as the domain, and {0,1} ℵ is the co-domain is the set of all finite or infinite strings of 0 & 1. (*Ref.1) In this paper, we propose a function *, which we call the stop function and we propose another function h, which we call the halt function. Our objective of the paper is to show their existence in a completely mathematical form of the, well known halting problem and its solution using simple functional compositions. Our next approach is to study the structure of the domain of programmable functions i.e. and its topology with respect to the topology of {0,1} ℵ. Followed by defining the finite string topology and the product topology on {0,1} ℵ and study the continuous functions from to {0,1} ℵ. Our main intention is to show that a programmable function will terminate for a specific input if and only if the function is continuous at that specific input (point on).

2005

Tests of branch splitting and branch-splitting independence in Allais

Fundamenta Informaticae, 1997

The abstract approach proposed here encompasses both the detection of some periodic loops during the execution of Prolog programs and the detection of some periodic loops during recursive computations (an attempt to look at the loop detection problem for Prolog from an abstract point of view has been done previously in a paper by the same author published in 1993). ). 1 Of course, a trivial partial decision procedure computable in the same sense exists, namely comparing each trajectory member with all preceding ones. This procedure, however, is not practically usable in most cases of interest. We have not included references to in most of our previous papers due to not knowing about that publication during a long period of time.

CR: New Centennial Review, 2021

CR: The New Centennial Review, Volume 21, Number 1, Spring 2021, pp. 11-35 (Article)

The Journal of Logic Programming, 1994