In this work, dedicated to Shafi Goldwasser, we consider a relaxation of the notion of pseudodeterministic algorithms, which was put forward by Gat and Goldwasser ({\em ECCC}, TR11--136, 2011).

Pseudodeterministic algorithms are randomized algorithms that solve search problems by almost always providing the same canonical solution (per each input).
Multi-pseudodeterministic algorithms relax the former notion by allowing the algorithms to output one of a bounded number of canonical solutions (per each input).
We show that efficient multi-seudodeterministic algorithms can solve natural problems that are not solveable by efficient pseudodeterministic algorithms, present a composition theorem regarding multi-pseudodeterministic algorithms,
and relate them to other known notions.

Adding technical details and some discussions.

In this work, dedicated to Shafi Goldwasser, we consider a relaxation of the notion of pseudodeterministic algorithms, which was put forward by Gat and Goldwasser ({\em ECCC}, TR11--136, 2011).

Pseudodeterministic algorithms are randomized algorithms that solve search problems by almost always providing the same canonical solution (per each input).
Multi-pseudodeterministic algorithms relax the former notion by allowing the algorithms to output one of a bounded number of canonical solutions (per each input).
We show that efficient multi-seudodeterministic algorithms can solve natural problems that are not solveable by efficient pseudodeterministic algorithms, present a composition theorem regarding multi-pseudodeterministic algorithms,
and relate them to other known notions.