A Characterisation of Multiply Recursive Functions with Higman’s Lemma

KBOs, ordinals, subrecursive hierarchies and all that

“…This result may suggest that a fast-growing hierarchy provides the link between order type of a reduction order and the complexity of a rewrite system. In particular Touzet conjectured: "the Hardy hierarchy is the right tool for connecting derivation length and order type" [38]. This conjecture is partly emphasised by the observation that the below presented proof of Lepper's result exploits the order type of kbo and the connection of the Ackermann function to the Hardy hierarchy.…”

Turchin's Relation and Subsequence Relation in Loop Approximation

“…We prove that while applied only to sequences generated by prefix grammars the relation admits bad sequences not more than exponential over a grammar size. If we apply the relation to a direct product of sequences generated by prefix grammars we receive the multiple recursive upper bound found by H. Touzet [14]. Also we show how to make a refinement of the subsequence relation that solves the empty word problem for languages generated by alphabetic prefix grammars and inherits some useful features of Turchin's relation, which was also used in program transformation (in particular, in the supercompiler SCP4 [8]).…”

“…Recall that in the case of pairs over the upper bound on arbitrary sequences with restricted word length growth is multiple recursive [14]. We show that the upper bound is exact even if the sequences are built by a direct product of two prefix grammars.…”

“…Example 6. Consider the following rewrite grammar (it not necessarily rewrites only prefixes; the grammar is similar to the one described in [14]).…”

“…On the other hand, well-binariness of the homeomorphic embedding is shown to be nonprovable in the Peano arithmetic with the first-order induction scheme [12], and this fact aroused interest of logicians and computer scientists with background in mathematical logic (a thorough study of the proof-theoretical strength of the fact is in [16]). Studies of the homeomorphic embedding as a termination criterion for term rewriting systems ( [11,14]) are located in the middle between these poles of pure theory and practice.…”