Linear Ordering on Graphs, Anti-Founded Sets and Polynomial Time Computability

It is proved deenability in FO+IFP of a global linear ordering on vertices of strongly extensional (SE) nitely-branching graphs. In the case of nite SE graphs this also holds for FO+LFP. This gives capturing results for PTIME computability on the latter class of graphs by FO+LFP and FO+IFP, and also on the corresponding anti-founded universe HFA of hereditarily-nite sets by a language of a bounded set theory BSTA. Oracle PTIME computability over HFA is also captured by an appropriate extension of the language by predicate variables and a bounded 2-recursion schema. It is also characterized the type of corresponding linear ordering on the universe HFA and on its natural extension HFA 1 consisting of hereditarily-nite anti-founded sets with possibly innnite (unlike HFA) transitive closures.