[desertrider12]

L_R is not decidable. Deciding if x##y is in L requires simulating y until it produces |x| digits of output. x being anything and y = “while(true) {}” is a counterexample. Another way to look at it is that for any fixed x, "y produces x as its first |x| digits of output" is a nontrivial semantic property of y, so Rice's theorem shows that the property is undecidable.

[shunyaekam]

R is basicially a function from rationals to algorithms computing irrationals.

L_R is decidable by a master algorithm running every y of x#y, e.g. 3.14#<y_pi> runs y_pi(|3.14|), so y runs up to 2 decimals.

The master algorithm runs y(|x|) and then compares the output to x.

It halts correctly for every x#y unless you can provide a counter-example?