Multiply-Recursive Upper Bounds with Higman’s Lemma

Abstract

We develop a new analysis for the length of controlled bad sequences in well-quasi-orderings based on Higman's Lemma. This leads to tight multiply-recursive upper bounds that readily apply to several verification algorithms for well-structured systems.

Cutting through Regular Post Embedding Problems

“…This proof is simpler and more direct than the proof (for PEP only) based on blockers [2]. [12]. We prove two "cutting lemmas" giving sufficient conditions for "cutting" a solution…”

“…However, complexity upper-bound can be derived if the complexity of the sequences (or more precisely of the process that generates bad sequences) is taken into account. The interested reader can consult [12] for more details. Here we only need the simplest version of these results, i.e., the statement that the maximal length of bad sequences is computable.…”

The Power of Priority Channel Systems

“…The heart of the upper bound proof is a specialized Length Function Theorem for (Σ * d,Γ , p,Γ ), obtained in Section 4.2 by instrumenting the proof of Theorem 3.6 and applying the generic Length Function Theorem from [SS11]. This allows us to derive F ε 0 upper bounds and new combinatorial algorithms for PCS verification in Section 4.3.…”

“…Controlled Sequences. We employ to this end the framework and results of [SS11]. The first observation is that bad sequences over (Σ * d , p ) can be of arbitrary length: for every N > 0, the sequence 1, 0…”

“…As shown in [SS11], for a given g and n, bad (g, n)-controlled sequences over (A, ≤ A , |.| A ) have a maximal length denoted L A,g (n).…”