PL (complexity)

PL, or probabilistic L, is the class of languages recognizable by a polynomial time logarithmic space randomized machine with probability > ​¹⁄₂ (this is called unbounded error). Equivalently, as shown below, PL is the class of languages recognized by unbounded time unbounded error logspace randomized machine.

An example of PL complete problem (under logspace reduction) is finding whether the determinant of a matrix (with integral coefficients) is positive. Given a matrix M and a number n, testing with ${\displaystyle |M|>n}$ is also PL complete. By contrast, testing whether matrix permanent is positive is PP complete.

PL^PL=PL in the sense that for every f in PL, PL is unchanged if it is extended to allow ${\displaystyle x\to f(A,x)}$ as a subroutine, where A is the input string.

PL contains NL and BPL and is contained in NC₂.

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