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

The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin’s omega number, and is the most well known example of a real which is random in the sense of Martin-Löf. Although omega numbers depend on the underlying universal T...

A classical result says that the halting problem is undecidable: there is no algorithm that, given a computation, says whether it terminates or not. A related result says that for some computations the termination statement is undecidable in Gödel’s sense (neither provable nor refutable). Still in many cases the termination question is not that hard. Maybe, the difficult cases are rare exceptio...

Classical models of computation traditionally resort to halting schemes in order to enquire about the state of a computation. In such schemes, a computational process is responsible for signaling an end of a calculation by setting a halt bit, which needs to be systematically checked by an observer. The capacity of quantum computational models to operate on a superposition of states requires an ...

It is shown that it is undecidable in general whether a graph rewriting system (in the \double pushout approach") is terminating. The proof is by a reduction of the Post Correspondence Problem. It is also argued that there is no straightforward reduction of the halting problem for Turing machines or of the termination problem for string rewriting systems to the present problem.

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the conve...

Interactive debugging requires that the programmer be able to halt a program at interesting points in its execution. This paper presents an algorithm for halting a distributed program in a consistent state, and presents a definition of distributed breakpoints with an algorithm for implementing the detection of these breakpoints. The Halting Algorithm extends Chandy and Lamport’s algorithm for r...

It is shown that it is undecidable in general whether a graph rewriting system (in the \double pushout approach") is terminating. The proof is by a reduction of the Post Correspondence Problem. It is also argued that there is no straightforward reduction of the halting problem for Turing machines or of the termination problem for string rewriting systems to the present problem.

Let TM(k, l) be the set of one-tape Turing machines with k states and l symbols. It is known that the halting problem is decidable for machines in TM(2, 3) and TM(3, 2). We prove that the decidability of machines in TM(2, 4) and TM(3, 3) will be difficult to settle, by giving machines in these sets for which the halting problem depends on an open problem in number theory. A machine in TM(5, 2) ...