Autosolvability of halting problem instances for instruction sequences

Long multiplication by instruction sequences with backward jump instructions

For each function on bit strings, its restriction to bit strings of any given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers , forward jump instructions, and a termination instruction. Backward jump instructions are not necessary for this, but instruction sequences can be significantly shorter with them. We...

Asymptotic Proportion of Hard Instances of the Halting Problem

Although the halting problem is undecidable, imperfect testers that fail on some instances are possible. Such instances are called hard for the tester. One variant of imperfect testers replies “I don’t know” on hard instances, another variant fails to halt, and yet another replies incorrectly “yes” or “no”. Also the halting problem has three variants. A tester may test halting on the empty inpu...

The Asymptotic Behaviour of the Proportion of Hard Instances of the Halting Problem (Extended Version)

On Approximating Real-World Halting Problems

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...

Some improvements in fuzzy turing machines

In this paper, we improve some previous definitions of fuzzy-type Turing machines to obtain degrees of accepting and rejecting in a computational manner. We apply a BFS-based search method and some level’s upper bounds to propose a computational process in calculating degrees of accepting and rejecting. Next, we introduce the class of Extended Fuzzy Turing Machines equipped with indeterminacy s...