91Ó°ÊÓ

$P$Under union, concatenation, and complement, the class P is closed.

$P$seems to be a category of languages that can be deduced in polynomial time on a randomized single-tap Turing computer.

That is,$P=U_{k}TIME\left(n^k\right)$

$P$ is closed under the conditions of union, concatenation, and complement

Assume two languages$P_1∈P and P_2∈P$

The Turing machine $Mthataccepts{P}_1∪P_2$works as follows:

$M=\text{'}\text{'}Oninputw:$

1. Check if $w∈P_1$

2. if not then check if $w∈P_2$

3. Accept $W$it and only if$P_1 or P_2$accepts.

4. If one or both rejects, this same input $W$ is rejected.

The entire time is polynomial because every membership check takes polynomial time.

Complement:

Assume a language $P_1∈P $The Turing machine $Mthataccepts{P}_1$works as follows:

$M=\text{'}\text{'}Oninputw:$

1. Check if $w∈P_1$

2. if not then check if $w∈P_2$

3. Accept $W$it and only if $P_1 or P_2$accepts.

Clearly the overall time is polynomial.

Therefore, the class $P$ is closed under union, concatenation, and complement.