The Halting Theorem establishes that there is no machine H that can decide in all cases if a machine n halts on input m. The conjecture of this paper is that nevertheless there exist a machine H such that it can identify all the cases it is unable to decide. This become possible if the Recursion Theorem is reinterpreted as mutual necessitation rather than equivalence.

When Alan Turing laid the foundation for computation in 1936, he wanted to show what computation can do, and what it cannot do. For the latter, he invented a problem that we now call the “Halting Problem”. In modern terms, it is as follows. In a general-purpose programming language, write a program that reads a text (character string) p representing a program in that same language, and reads an...

We present a formalization of the halting problem in Agda, a language based on Martin-Löf’s intuitionistic type theory. The key features are: – We give a constructive proof of the halting problem. The “constructive halting problem” is a natural reformulation of the classic variant. – A new abstract model of computation is introduced, in type theory. – The undecidability of the halting problem i...

The undecidability of the halting problem is a well-known research result of theoretical computer science, dating back to Turing’s work in 1936. Nevertheless, it is commonly known that the halting problem on finite-state computer systems is decidable. Thus, any undecidability proof given for the halting problem must imply that it does not apply to finite-state computer systems. The aim of this ...

Acoustic scattering problems are considered when the material parameters (density and speed of sound) are functions of position within a bounded region. An integro-differential equation for the pressure in this region is obtained. It is proved that solving this equation is equivalent to solving the scattering problem. Problems of this kind are often solved by regarding the effects of the inhomo...

Limit computable functions can be characterized by Turing jumps on the input side or limits on the output side. As a monad of this pair of adjoint operations we obtain a problem that characterizes the low functions and dually to this another problem that characterizes the functions that are computable relative to the halting problem. Correspondingly, these two classes are the largest classes of...

We investigate which computable equivalence structures are isomorphic relative to the Halting problem.