# Strict sequential P-completeness

## Abstract

In this paper we present a new notion of what it means for a problem in *P* to be inherently sequential. Informally, a problem *L* is *strictly sequential P-complete* if when the best known *sequential* algorithm for *L* has polynomial speedup by parallelization, this implies that *all* problems in *P* have a polynomial speedup in the parallel setting. The motivation for defining this class of problems is to try and capture the problems in *P* that axe truly inherently sequential. Our work extends the results of Condon who exhibited problems such that if a polynomial speedup of their best known *parallel* algorithms could be achieved, then all problems in *P* would have polynomial speedup. We demonstrate one such natural problem, namely the *Multiplex-select Circuit Problem* (MCP). MCP has one of the highest degrees of sequentiality of any problem yet defined. On the way to proving MCP is strictly sequential *P*-complete, we define an interesting model, the *register stack machine*, that appears to be of independent interest for exploring pure sequentiality.

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