Bit-size analysis of transition matrices under binary prefix coding in the MPCP-to-PFA reduction

Determine the bit complexity (e.g., denominator sizes of rational entries) of the transition matrices that arise when the copying rules in the Modified Post Correspondence Problem to probabilistic finite automaton reduction are replaced by a binary prefix code using only two copying pairs, and quantify the resulting bounds in the fixed‑matrix undecidability constructions.

Background

To reduce the number of MPCP pairs, the paper suggests encoding the tape alphabet with a binary prefix code so that the |Γ|+1 symbol-copying pairs can be replaced by just two pairs, potentially lowering the number of fixed matrices needed in the PFA constructions.

While this optimization saves pairs, the author notes that the resulting effect on the bit size (denominator exponents) of the PFA transition matrices has not been analyzed, leaving quantitative bounds on matrix entries open.