The definitions I've found were highly technical and using terms I've never seen before. Say, I have a certain irrational number e that I get get closer and closer to with a computer and I want to determine if there is a particular digit d appearing after a certain position. Or what about the halting problem (give a program and ask 'will the computer ever halt?')? Are these problems semi-computable problems or noncomputable?
What's an intuitive distinction between semi-computable problems and noncomputable problems/functions?
computability;terminology
Intuitively, a set of numbers is computable or decidable or recursive if there is an algorithm deciding membership for the set. A set of numbers is semidecidable or recursively enumerable (r.e.) if it has verifiable proofs of membership. For example, the set of programs which halt is semidecidable, since given a program $A$ and a time bound $t$, one can check that $A$ halts after at most $t$ steps. But this set is not computable, since without the time bound, we don't know how long to wait for. (That's an intuitive explanation; you can google for a proof that the halting problem is not decidable.)There are several equivalent definitions of semidecidable sets:A set $S$ is semidecidable if there exists an algorithm $A$ which always terminates with either YES or NO as an answer, such that $x \in S$ iff there is a witness $w$ for which $A(x,w)$ answers YES.A set $S$ is semidecidable if there exists an algorithm $A$ such that $x \in S$ iff $A(x)$ halts.A set $S$ is semidecidable if there is an algorithm that enumerates all members of $S$.See if you can show that all these definitions are equivalent. You might want to look up the technique of dovetailing.What about the digits of $e$? We need to phrase the problem of computing the digits of $e$ in our framework. The set of digits of $e$ is the set of pairs $(k,d)$, where $d$ is the $k$th decimal digit of $e=2.71828\ldots$, i.e. $\{(0,2),(1,7),(2,1),(3,8),(4,2),(5,8),\ldots\}$. Since there is an algorithm computing the $k$th digit of $e$ (e.g. using the Taylor series $e = \sum_{k=0}^\infty 1/k!$), the set of digits of $e$ (under this encoding) is computable.Edit: Following the OP's comment, suppose we're interested in the set of positions $n$ such that the decimal expansion of $e$ contains the digit $2$ somewhere past the $n$th digit. This set is decidable, for the following reason: either the digit $2$ appears infinitely often, or its last appearance is at digit $N$. In the first case, the set in question contains all natural numbers, in the second case it is $\{0,\ldots,N-1\}$.The same trick works for any fixed real number. Consider, however, the set consisting of pairs $(A,n)$ such that $A$ is an algorithm that enumerates the decimal expansion of some real number, which contains the digit $2$ somewhere past the $n$th digit. This set is not computable, only semidecidable, since one can reduce the halting problem to it and vice versa (how?).