The Turing Halting Problem is a fundamental question in computer science that asks whether we can determine if any given program will eventually halt or run forever. This problem was first formulated by Alan Turing in 1936 and has profound implications for the limits of computation.
A Turing Machine is a theoretical model of computation that consists of an infinite tape divided into cells, each containing a symbol. A read-write head can move left or right along the tape, reading and writing symbols. The machine has a finite set of internal states and follows transition rules that determine its next action based on the current state and symbol being read.
The Halting Problem asks whether we can create a universal program H that takes any program P and input I as arguments, and determines whether P will halt when run with input I. If such a detector existed, it would output YES if the program halts, and NO if it runs forever. But can such a detector actually exist?
Turing proved the Halting Problem is undecidable using proof by contradiction. First, we assume that a halting detector H exists. Then we construct a special program D that uses H to create a paradox. When we run D on itself, we get a logical contradiction that proves H cannot exist. This elegant proof shows there are fundamental limits to what computers can decide.