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

Inclusion problems for one-counter systems

We study the decidability and complexity of verification problems for infinite-state systems. A fundamental question in formal verification is if the behaviour of one process is reproducible by another. This inclusion problem can be studied for various models of computation and behavioural preorders. It is generally intractable or even undecidable already for very limited computational models. The aim of this work is to clarify the status of the decidability and complexity of some well-known inclusion problems for suitably restricted computational models. In particular, we address the problems of checking strong and weak simulation and trace inclusion for processes definable by one-counter automata (OCA), that consist of a finite control and a single counter ranging over the non-negative integers. We take special interest of the subclass of one-counter nets (OCNs), that cannot fully test the counter for zero and which is subsumed both by pushdown automata and Petri nets / vector addition systems. Our new results include the PSPACE-completeness of strong and weak simulation, and the undecidability of trace inclusion for OCNs. Moreover, we consider semantic preorders between OCA/OCN and finite systems and close some gaps regarding their complexity. Finally, we study deterministic processes, for which simulation and trace inclusion coincide

Controlling a Random Population is EXPTIME-hard

Bertrand et al. [1] (LMCS 2019) describe two-player zero-sum games in which one player tries to achieve a reachability objective in $n$ games (on the same finite arena) simultaneously by broadcasting actions, and where the opponent has full control of resolving non-deterministic choices. They show EXPTIME completeness for the question if such games can be won for every number $n$ of games. We consider the almost-sure variant in which the opponent randomizes their actions, and where the player tries to achieve the reachability objective eventually with probability one. The lower bound construction in [1] does not directly carry over to this randomized setting. In this note we show EXPTIME hardness for the almost-sure problem by reduction from Countdown Games

History-deterministic Vector Addition Systems

We consider history-determinism, a restricted form of non-determinism, for Vector Addition Systems with States (VASS) when used as acceptors to recognise languages of finite words. History-determinism requires that the non-deterministic choices can be resolved on-the-fly; based on the past and without jeopardising acceptance of any possible continuation of the input word. Our results show that the history-deterministic (HD) VASS sit strictly between deterministic and non-deterministic VASS regardless of the number of counters. We compare the relative expressiveness of HD systems, and closure-properties of the induced language classes, with coverability and reachability semantics, and with and without $\varepsilon$-labelled transitions. Whereas in dimension 1, inclusion and regularity remain decidable, from dimension two onwards, HD-VASS with suitable resolver strategies, are essentially able to simulate 2-counter Minsky machines, leading to several undecidability results: It is undecidable whether a VASS is history-deterministic, or if a language equivalent history-deterministic VASS exists. Checking language inclusion between history-deterministic 2-VASS is also undecidable.Comment: This is the full version of a paper published in CONCUR 202

Optimally Resilient Strategies in Pushdown Safety Games

Infinite-duration games with disturbances extend the classical framework of infinite-duration games, which captures the reactive synthesis problem, with a discrete measure of resilience against non-antagonistic external influence. This concerns events where the observed system behavior differs from the intended one prescribed by the controller. For games played on finite arenas it is known that computing optimally resilient strategies only incurs a polynomial overhead over solving classical games. This paper studies safety games with disturbances played on infinite arenas induced by pushdown systems. We show how to compute optimally resilient strategies in triply-exponential time. For the subclass of safety games played on one-counter configuration graphs, we show that determining the degree of resilience of the initial configuration is PSPACE-complete and that optimally resilient strategies can be computed in doubly-exponential time

Parity Games on Temporal Graphs

Temporal graphs are a popular modelling mechanism for dynamic complex systems that extend ordinary graphs with discrete time. Simply put, time progresses one unit per step and the availability of edges can change with time. We consider the complexity of solving $\omega$-regular games played on temporal graphs where the edge availability is ultimately periodic and fixed a priori. We show that solving parity games on temporal graphs is decidable in PSPACE, only assuming the edge predicate itself is in PSPACE. A matching lower bound already holds for what we call punctual reachability games on static graphs, where one player wants to reach the target at a given, binary encoded, point in time. We further study syntactic restrictions that imply more efficient procedures. In particular, if the edge predicate is in $P$ and is monotonically increasing for one player and decreasing for the other, then the complexity of solving games is only polynomially increased compared to static graphs

Reachability in two-dimensional unary vector addition systems with states is NL-complete

Blondin et al. showed at LICS 2015 that two-dimensional vector addition systems with states have reachability witnesses of length exponential in the number of states and polynomial in the norm of vectors. The resulting guess-and-verify algorithm is optimal (PSPACE), but only if the input vectors are given in binary. We answer positively the main question left open by their work, namely establish that reachability witnesses of pseudo-polynomial length always exist. Hence, when the input vectors are given in unary, the improved guess-and-verify algorithm requires only logarithmic space

Timed Basic Parallel Processes

Timed basic parallel processes (TBPP) extend communication-free Petri nets (aka. BPP or commutative context-free grammars) by a global notion of time. TBPP can be seen as an extension of timed automata (TA) with context-free branching rules, and as such may be used to model networks of independent timed automata with process creation. We show that the coverability and reachability problems (with unary encoded target multiplicities) are PSPACE-complete and EXPTIME-complete, respectively. For the special case of 1-clock TBPP, both are NP-complete and hence not more complex than for untimed BPP. This contrasts with known super-Ackermannian-completeness and undecidability results for general timed Petri nets. As a result of independent interest, and basis for our NP upper bounds, we show that the reachability relation of 1-clock TA can be expressed by a formula of polynomial size in the existential fragment of linear arithmetic, which improves on recent results from the literature