are all NP problems made up of P problems? that is, can NP problems be thought of as an accumulation of P problems? or can NP problems be divided up into a series of P problems?



are all NP problems made up of P problems?
The answer is yes. Suppose that $A$ is any NP problem, so there is a polynomial time algorithm $p$ such that $a\in A$ just in case there is some $b$ (of size at most $q(|a|)$, where $q$ is a fixed polynomial) such that $p$ accepts the pair $(a,b)$. Let $A\_b$ be the set of $a$ such that $b$ has size at most $q(|a|)$ and $p$ accepts $(a,b)$. So each $A\_b$ is in the class $P$, and $A$ is clearly the union $$A=\bigcup\_b A\_b.$$ So every NP problem is an accumulation of P problems in a natural way.


