Of all the classes shown above, only two strict containments are known:. Unlike for NP, however, the polynomial-time machine doesn't need to detect fraudulent advice strings; it is not a verifier. If it contains NP, then the polynomial hierarchy collapses to the second level. On the other hand, it also contains some impractical problems, including some undecidable problems such as the unary version of any undecidable problem. In , Jin-Yi Cai and D. Polynomial-time algorithms are closed under composition.
Intuitively, this says that if one writes a function that is polynomial-time assuming that function calls are constant-time, and if those called functions themselves require polynomial time, then the entire algorithm takes polynomial time.
One consequence of this is that P is low for itself. This is also one of the main reasons that P is considered to be a machine-independent class; any machine 'feature', such as random access, that can be simulated in polynomial time can simply be composed with the main polynomial-time algorithm to reduce it to a polynomial-time algorithm on a more basic machine.
Languages in P are also closed under reversal, intersection, union, concatenation, Kleene closure, inverse homomorphism, and complementation.
Some problems are known to be solvable in polynomial-time, but no concrete algorithm is known for solving them. Flow Indicator symbol. Flow Meter symbol. Flow Nozzle Meter symbol. Flow Recorder symbol. Flow Transmitter symbol. Flowmeter symbol. Flume Meter symbol. Indicator 01 symbol. Level Alarm symbol. Level Controller symbol. Level Gauge symbol. Level Indicator symbol. Level Meter symbol. Level Recorder symbol.
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