Hierarchies of Provably Recursive Functions

The Power of Priority Channel Systems

“…The verification of infinite-state systems, and WSTSs in particular, often turns out to require astronomic computational resources expressed as subrecursive functions [LW70,FW98] of the input size. We show in this section how to bound the complexity of the algorithms presented in Section 2.2 and classify the Reachability and Inevitability problems for PCSs using fast-growing complexity classes [Sch13].…”

“…The latter hierarchy of function classes (F α ) α is well-established [LW70,FW98]. The class F α is the set of functions computable in time F c α , a finite iterate of F α .…”

“…The class F α is the set of functions computable in time F c α , a finite iterate of F α . In particular, F 2 is the set of elementary functions, α<ω F α the set of primitive-recursive functions, while α<ε 0 F α is exactly the set of ordinal-recursive (aka "provably recursive") functions [FW98]. The complexity classes (F α ) α are more recent [Sch13]: F α is the set of decision problems that can be solved in time F α • p for some p in β<α F β .…”

Revisiting Ackermann-Hardness for Lossy Counter Machines and Reset Petri Nets

“…The Fast-Growing Hierarchy [10] turns the class of all primitive-recursive functions into a strict cumulative hierarchy built from a sequence (F k ) k=0,1,2,... of number-theoretic functions. The functions F k : N → N are defined by induction over k ∈ N:…”

Things that can and things that can’t be done in PRA

“…(0) are just n-ary number theoretic functions). 2 We have the axioms and rules of many-sorted classical predicate logic as well as symbols and defining equations for all primitive recursive functionals of type level ≤ 2 in the sense of Kleene [7] (i.e. ordinary primitive recursion uniformly in function parameters, for details see e.g.…”