Final answer:
A language L is decidable if a Turing machine can determine membership of strings in that language by halting in state "y" for yes or state "n" for no, signifying whether a string is in the language L or not. This comes from computational theory and highlights the capacity for problems to be algorithmically solvable.
Step-by-step explanation:
A language L is decidable if there is a Turing machine (computer program) that halts on all inputs, and for strings in L, halts in state "y", and for other strings, halts in state "n." This definition stems from the field of computational theory, which falls under theoretical computer science. Decidable languages are also known as recursively enumerable languages, and the concept is essential in understanding which problems can be algorithmically solved.
The Turing machine is a fundamental model of computation that can simulate any computer algorithm, regardless of complexity. Thus, when a language is decidable, it implies that there is an algorithm (i.e., a computer program) that can determine with certainty whether a given string belongs to that language or not. The binary states "y" for yes and "n" for no clearly delineate which strings are part of the language in question.
It is the precision and logical structuring within mathematics that enables this kind of clear definition. Unlike natural languages, which have exceptions and irregularities that defy logic, mathematical concepts like integers, addition, multiplication, and more complex functions, follow consistent rules that would likely be recognized even by an alien species. This universality of mathematics as a language makes it a powerful tool for expressing and exploring computational problems and their solutions.