Glossary
Decidability
The property of a problem indicating whether an algorithm can be written to solve it, specifically if it can always produce a correct yes/no answer for all possible inputs.
Example:
Understanding decidability helps computer scientists know if a problem, like perfectly predicting the stock market's exact movements for the next year, is even theoretically solvable by an algorithm.
Decidable Problem
A decision problem for which an algorithm can be written that always produces a correct output (yes/no) for all possible inputs.
Example:
Determining if a given word is spelled correctly according to a dictionary is a decidable problem because an algorithm can always correctly check and provide a 'yes' or 'no' answer.
Halting Problem
The classic undecidable problem that asks whether an algorithm can determine if any given program, with a specific input, will eventually stop running (halt) or run forever.
Example:
If you tried to write a program that could analyze any other program and tell you if it would ever finish its execution, you'd be attempting to solve the halting problem, which Alan Turing proved impossible.
Limits of Computation
The inherent boundaries to what algorithms and computers can achieve, demonstrating that not all problems are solvable by computational means.
Example:
The existence of the Halting Problem illustrates the fundamental limits of computation, showing that even the most powerful supercomputers cannot solve every conceivable problem.
Undecidable Problem
A decision problem for which no algorithm can be written that will always produce a correct yes or no answer for all possible inputs.
Example:
The challenge of creating a single algorithm that can perfectly predict if any given AI will achieve consciousness is an undecidable problem, as no universal method exists.