Complexity Hierarchies Beyond Elementary

We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a non-elementary complexity, which occur naturally in logic, combinatorics, formal languages, verification, etc., with complexities ranging from simple towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep updating the catalogue of problems from Section 6 in future revision

The complexity of coverability in ν-Petri nets

We show that the coverability problem in ν-Petri nets is complete for ‘double Ackermann’ time, thus closing an open complexity gap between an Ackermann lower bound and a hyper-Ackermann upper bound. The coverability problem captures the verification of safety properties in this nominal extension of Petri nets with name management and fresh name creation. Our completeness result establishes ν-Petri nets as a model of intermediate power among the formalisms of nets enriched with data, and relies on new algorithmic insights brought by the use of well-quasi-order ideals

What makes petri nets harder to verify : stack or data?, Concurrency, security, and puzzles : Festschrift for A.W. Roscoe on the occasion of his 60th birthday

We show how the yardstick construction of Stockmeyer, also developed as counter bootstrapping by Lipton, can be adapted and extended to obtain new lower bounds for the coverability problem for two prominent classes of systems based on Petri nets: Ackermann-hardness for unordered data Petri nets, and Tower-hardness for pushdown vector addition systems

Coverability Trees for Petri Nets with Unordered Data

We study an extension of classical Petri nets where tokens carry values from a countable data domain, that can be tested for equality upon firing transitions. These Unordered Data Petri Nets (UDPN) are well-structured and therefore allow generic decision procedures for several verification problems including coverability and boundedness. We show how to construct a finite representation of the coverability set in terms of its ideal decomposition. This not only provides an alternative method to decide coverability and boundedness, but is also an important step towards deciding the reachability problem. This also allows to answer more precise questions about the reachability set, for instance whether there is a bound on the number of tokens on a given place (place boundedness), or if such a bound exists for the number of different data values carried by tokens (place width boundedness). We provide matching Hyper-Ackermann bounds on the size of cover-ability trees and on the running time of the induced decision procedures

The Parametric Complexity of Lossy Counter Machines

The reachability problem in lossy counter machines is the best-known ACKERMANN-complete problem and has been used to establish most of the ACKERMANN-hardness statements in the literature. This hides however a complexity gap when the number of counters is fixed. We close this gap and prove F_d-completeness for machines with d counters, which provides the first known uncontrived problems complete for the fast-growing complexity classes at levels 3 < d < omega. We develop for this an approach through antichain factorisations of bad sequences and analysing the length of controlled antichains