1. Diagonalization: The Halting problem is defined as follows: Given a TuringMachineM = 〈Q, {0, 1},Γ, δ, q0,2, F 〉 and a string w ∈ Σ∗, determine whether M halts on w. In class, we proved that the Halting problem is undecidable using two different techniques. The first technique was constructive, where we constructed a series of Turing Machines, which led to a contradiction. The second techniqu...

We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that is exactly as random as the halting probability of a universal machine equipped with an oracle for the second jump of the halti...

This short Letter elucidates a fundamental relationship between self-replication of living systems and the halting problem in computation theory.

In computability and complexity theory, all problems are reduced to language recognition, where a language is defined as a set of strings over some alphabet, typically {0, 1}. For example, L1 = {10, 11, 101, · · · } is the language that contains all prime numbers, represented in binary. Then the problem of determining whether or not a number x is a prime is the same as determining whether or no...

The halting of universal quantum computers is shown to be incompatible with the constraint of unitarity of the dynamics. The prospects of using quantum dynamics to process information have opened up new research horizon. While most of the activities is in the theoretical study and experimental realisation of quantum networks [1], the principles –and the halting problem, in particular– of a univ...

Within Valiant's model of learning as formalized by Kearns, we show that computable total predicates for two formally uncomputable problems (the classical Halting Problem, and the Halting Problem relative to a speci ed oracle) are formally learnable from examples, to arbitrarily high accuracy with arbitrarily high con dence, under any probability distribution. The Halting Problem relative to th...

We consider a class of stochastic games, where each state is identified with a player. At any moment during play, one of the players is called active. The active player can terminate the game, or he can announce any player, who then becomes the active player. There is a non-negative payoff for each player upon termination of the game, which depends only on the player who decided to terminate. W...

Rendezvous with detection differs from the usual rendezvous problem in that two mobile agents not only accomplish rendezvous whenever this is possible, but can also detect the impossibility of rendezvous (e.g., due to symmetrical initial positions of the agents) in which case they are able to halt. We study the problem of rendezvous with and without detection of two anonymous mobile agents in a...

A new MS Excel application has been developed which seeks to reduce the risks associated with the development, operation and auditing of Excel spreadsheets. FormulaDataSleuth provides a means of checking spreadsheet formulas and data as they are developed or used, enabling the users to identify actual or potential errors quickly and thereby halt their propagation. In this paper, we will describ...