Equivalence kernels of sequential functions and sequential observation synthesis

TL;DR

This paper investigates the decidability of whether a rational equivalence relation can be realized as the equivalence kernel of a sequential transducer, with implications for modeling imperfect information in games.

Contribution

It establishes the decidability of synthesizing sequential transducers from rational equivalence relations and proves undecidability in certain cases.

Findings

Deciding if a rational equivalence relation is a transducer kernel is decidable.

It is undecidable to determine this when the relation is given as a transducer.

The results impact modeling imperfect information in game theory.

Abstract

We show that one can decide if a rational equivalence relation can be given as the equivalence kernel of a sequential letter-to-letter transduction. This problem comes from the setting of games with imperfect information. In [1, p. 6] the authors propose to model imperfect information by a rational equivalence relation and leave open the problem of deciding if one can synthesize a sequential letter-to-letter transducer (Mealy machine) which maps equivalent histories to the same sequence of observations. We also show that knowing if an equivalence relation can be given as the equivalence kernel of a sequential transducer is undecidable, even if the relation is given as a letter-to-letter transducer.