Algorithms and Programs

$\operatorname {dec}(x)=\lfloor x-1\rfloor$, where $\lfloor 0-1\rfloor =0$ and $\lfloor (x+1)-1\rfloor =x$;

$\operatorname {sub}(x, y)=\lfloor x-y\rfloor$, where $\lfloor x-y\rfloor =x-y$ if $x \geqslant y$, and $\lfloor x-y\rfloor =0$ otherwise;

$\operatorname {fact}(x)=x!$ — factorial;

$\operatorname {div}(x, y)=\lfloor x / y\rfloor$ — integer quotient;

$\operatorname {sqrt}(x)=\lfloor \sqrt{x}\rfloor$;

$\operatorname {pow}(x, y)=x^{y}$ — exponentiation;

$\log (y, x)=\left\lfloor \log_{y} x\right\rfloor$;

$\bmod (x, y)$ — the remainder of dividing $x$ by $y$;

$\tau (x)$ — the number of distinct divisors of $x$, $\tau (0)=0$.

$f_{1}(x, y)= \begin{cases} \lfloor (x+y) / 2\rfloor & \text{ if } x<y, \\ x^{2} y^{3} & \text{ otherwise; }\end{cases}$

$f_{2}(x, y)= \begin{cases} 3^{y} & \text{ if } \log_{2}(x+1) \geqslant y, \\ \left|x-y\right| & \text{ otherwise; }\end{cases}$

$f_{3}(x, y)= \begin{cases} \lfloor x y / 2\rfloor & \text{ if } \log_{2} x \leqslant y+2, \\ x^{\lfloor y / 2\rfloor } & \text{ otherwise; }\end{cases}$

$f_{4}(x, y)= \begin{cases} \left\lfloor \log_{3}(x+y+1)\right\rfloor & \text{ if } 2 x \leqslant y^{3}+y, \\ \lfloor \sqrt{x+y}\rfloor & \text{ otherwise; }\end{cases}$

$f_{5}(x, y)= \begin{cases} 1, & \text{ if } x \text{ is a prime number, } \\ 0 & \text{ otherwise. }\end{cases}$

What is the maximum number of functions of $n \leqslant m$ variables that П can compute? How many different functions in total can П compute? Functions having a different number of arguments are considered distinct a priori.

Construct a program $\Pi_{m, n}$ that computes the maximum number of different functions of $n \leqslant m$ variables.

Construct a program $\Pi_{m},\left|\boldsymbol {V}_{\Pi_{m}}\right|=m$, that computes, for each $n \leqslant m$, the maximum number of different functions of $n$ variables.