For the additive real BSS machines using only constants 0 and 1 and order tests we consider the corresponding Turing reducibility and characterize some semidecidable decision problems over the reals. In order to refine, step-by-step, a linear hierarchy of Turing degrees with respect to this model, we define several halting problems for classes of additive machines with different abilities and c...

The basic question: what can and cannot be computed? ˆ Hilbert’s 10th problem, 1900. (CANNOT) ˆ Error testing (a.k.a. the Halting Problem). (CANNOT) ˆ Godel’s Theorem. (CANNOT) Tools: ˆ Well specified models. ˆ Mathematical arguments (convincing and unassailable) — analogous to geometry in high school. Important goal of the course: ˆ Have you better able to recognize when an argument is mathema...

Up to now, many properties of macro instructions of SCMFSA are described by the parahalting concepts. However, many practical programs are not always halting while they are halting for initialization states. For this reason, we propose initialization halting concepts. That a program is initialization halting (called ”InitHalting” for short) means it is halting for initialization states.In order...

We propose railway line planning optimization models that determines the frequency of each train so that it can meet passenger origindestination demands while minimizing the related costs. Most line planning models assume that all trains on a supply network stop at all intermediate stations. However to minimize passenger travelling time and provide train service to as many stations as possible,...

This contribution is the third of four parts. Based on the gain-number concept, canonical forms will be presented of the ambiguity search spaces of the geometry-based model, the time-averaged model, and the geometry-free model. These forms reveal the intrinsic geometry of the search spaces and allow one to study their size, shape, and orientation as function of data precision, sampling rate, sa...

No algorithm can of course solve the Halting Problem, that is, decide within finite time always correctly whether a given program halts on a certain given input. It might however be able to give correct answers for ‘most’ instances and thus solve it at least approximately. Whether and how well such approximations are feasible highly depends on the underlying encodings and in particular the Göde...

In a historical overview, Cristian S. Calude, Elena Calude, and Solomon Marcus identify eight stages in the development of the concept of a mathematical proof in support of an ambitious conjecture: we can express classical mathematical concepts adequately only in a mathematical language in which both truth and provability are essentially unverifiable. In this essay we show, first, that the conc...

A simple reversible Turing machine with four states, three symbols and no halting configuration is constructed that has no periodic orbit, simplifying a construction by Blondel, Cassaigne and Nichitiu and positively answering a conjecture by Kari and Ollinger. The constructed machine has other interesting properties: it is symmetric both for space and time and has a topologically minimal associ...

After a brief description of the Church-Turing Thesis, we suggest that, according to the latest results on classical recursive probabilistic solution of the Halting Problem, such thesis is asymptotically false.