# Folklore

## P vs. BPP ★★★

Author(s): Folklore

**Conjecture**Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a deterministic Turing machine in polynomial time? That is, does P = BPP?

Keywords: BPP; circuit complexity; pseudorandom generators

## Shuffle-Exchange Conjecture ★★★

Author(s): Beneš; Folklore; Stone

Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as ( times), where is the *shuffle permutation* defined by , and is the *exchange group* consisting of all permutations in preserving the first letters in the words.

**Problem (SE)**Evaluate .

**Conjecture (SE)**, for all .

Keywords:

## Shuffle-Exchange Conjecture (graph-theoretic form) ★★★

Author(s): Beneš; Folklore; Stone

Given integers , the *2-stage Shuffle-Exchange graph/network*, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).

Given integers , the *-stage Shuffle-Exchange graph/network*, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).

Let be the smallest integer such that the graph is rearrangeable.

**Problem**Find .

**Conjecture**.

Keywords:

## P vs. PSPACE ★★★

Author(s): Folklore

**Problem**Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?

Keywords: P; PSPACE; separation; unconditional