Cutting through Regular Post Embedding Problems

Abstract

Abstract. The Regular Post Embedding Problem extended with partial (co)directness is shown decidable. This extends to universal and/or counting versions. It is also shown that combining directness and codirectness in Post Embedding problems leads to undecidability.

Complexity Hierarchies beyond Elementary

“…There is a wealth of variants and applications, see e.g. [17,50,49]. We give here a slightly different viewpoint, taken from [9,49], that uses regular relations (i.e.…”

Unidirectional Channel Systems Can Be Tested

“…Since they are upward-closed, they eventually stabilize by the well-quasi-ordering property: (see [17,25]), our reductions prove that reachability for UCST[Z, N] is at level F ω ω in the extended Grzegorczyck hierarchy, and at level F ω m−1 , where m = |M|, when we restrict to systems with a fixed-sized alphabet of messages.…”

“…As an example, let us informally describe the simplest one and show how to solve a PEP partial dir instance with a UCST[Z r 1 ] system. We recall from [25] that PEP partial dir is the question whether there is a σ ∈ R such that u(σ) v(σ) and furthermore u(σ ) v(σ ) for all prefixes of σ that belong to R (thus PEP partial dir and PEP partial codir are equivalent problems and one switches from one to the other by taking the mirror images of u, v, R, R ).…”

“…The proof goes through a series of reductions that leave us with UCS's extended by only emptiness tests on a single side of a single channel (called "Z l 1 tests"). This minimal extension can then be reformulated as PEP partial codir , or "PEP with partial codirectness", a nontrivial extension of PEP that was recently proved decidable [25].…”

The Parametric Ordinal-Recursive Complexity of Post Embedding Problems

“…In fact, this motivation has been present from their inception in (Chambart and Schnoebelen, 2007): find a "master" decision problem complete for F ω ω , the class of hyper-Ackermannian problems, solvable with non multiply-recursive complexity, but no less-much like SAT is often taken as the canonical NPTime-complete problem, or the Post Correspondence Problem for Σ 0 1 . This has also prompted a wealth of research into variants and related questions Schnoebelen, 2008b, 2010a,b;Barceló et al, 2012;Karandikar and Schnoebelen, 2012).…”