What are HP and MP in this context? From Kozen's Automata and Computability, 3ed, lecture 32 p. 328:


What are HP and MP in this context?  I tried looking around and this text says:



How did the halting problem and membership problem become regular expression?  I am so loss here.
Both HP and MP are decision problems, namely sets of instances that you can think of as descriptive strings. For example,

HP is a collection of pairs $M, x$ where $M$ is the description of a Turing Machine (think of a listing) and $x$ is an input to that machine. An instance is such a pair and a pair is in the set HP if the machine described by $M$ eventually halts when started on input $x$,

and, similarly

MP is a collection of pairs $M, x$ as above and an instance of MP is in the set MP if the machine described by $M$ eventually halts in an accept state when started on input $x$.



In simple terms, HP is about programs and inputs and answers "Does this program halt on this input?" and MP answers "Does this program accept this input?" 

The octothorpe (the # symbol) is just a separator so we know where $M$ ends and $x$ starts.