The fixed point problem of a simple reversible language

SRL is a total programming language with distinctive features: (i) every program that mentions n registers defines a bijection Zn→Zn, and (ii) the generation of the SRL-program that computes the inverse of that bijection can be automatic. Containing SRL a very essential set of commands, it is suitable for studying strengths and weaknesses of reversible computations. We deal with the fixed points of SRL-programs. Given any SRL-program P, we are interested in the problem of deciding if a tuple of initial register values of P exists which remains unaltered after its execution. We show that the existence of fixed points in SRL is undecidable and complete in Σ10. We show that such problem remains undecidable even when the number of registers mentioned by P is limited to 12. Moreover, if we restrict to the linear programs of SRL, i.e. to those programs where different registers control nested loops, then the problem is already undecidable for the class of SRL-programs that mention no more than 3712 registers. Last, we show that, except for trivial cases, finding if the number of fixed points has a given cardinality is also undecidable.