A class of Recursive Permutations which is Primitive Recursive complete

The fixed point problem of a simple reversible language

“…Finally, concerning Control operators, the instruction set of a CM typically contains conditional jumps, something which, to us, is not at all immediately available in SRL. By the way, were we able to encode some sort of if-then-else in SRL, we would prove its equivalence with the class of Reversible Primitive Permutations in [19,20], by incontrovertibly making SRL the simplest primitive recursive complete reversible programming language.…”

“…The inverse of an SRL-program is obtained by transforming inc x, dec x, P 0 ; P 1 and for r(P) in dec x, inc x, P 1 −1 ; P 0 −1 and for r(P −1 ), respectively. For more information on SRL and its extensions, as well as results related with that language, consult for instance [6] and [20].…”

“…Among the languages whose programs represent total reversible functions [6,19,20,10], SRL and ESRL are the simplest ones, but they are far from being trivial. No language which is expressive enough to contain all the computable bijections can be eectively formalized.…”

“…On the other side, the last two authors characterize classes of total invertible functions according to formalisms and ideas typical of Recursion Theory [19,20,10]. The long term goal is to prove that some analogous of Kleene Normal Form Theorem [22] holds for Turing complete classes of recursive reversible functions, like the one in [10].…”

“…In this work we focus on total reversible functions only. Eventually, we are interested to the relation between SRL and some of the classes of total reversible functions in [19,20], the simplest one being RPP [20]. RPP is the class of Reversible Primitive Recursive functions.…”