What is modal logic

In this paper we develop a new algorithm for deciding the winner in parity games, and hence also for the modal μ-calculus model checking. The design and analysis of the algorithm is based on a notion of game progress measures: they are witnesses for winning strategies in parity games. We characterize game progress measures as pre-fixed points of certain monotone operators on a complete lattice. As a result we get the existence of the least game progress measures and a straightforward way to compute them. The worst-case running time of our algorithm matches the best worst-case running time bounds known so far for the problem, achieved by the algorithms due to Browne et al., and Seidl. Our algorithm has better space complexity: it works in small polynomial space; the other two algorithms have exponential worst-case space complexity.

Small progress measures for solving parity games

Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and the currently best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from LICS 2019. We establish a non-elementary lower bound, i.e. that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi and other areas, that are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the currently best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. At the heart of our proof is a novel gadget so called the factorial amplifier that, assuming availability of counters that are zero testable and bounded by k, guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k. We also develop a novel construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R. Repeatedly composing the factorial amplifier with itself by means of the construction then enables us to compute in linear time Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we in fact establish hardness for h-exponential space already for Petri nets with h + 13 counters.

/pdf/the-reachability-problem-for-petri-nets-is-not-elementary-8u04h5hbz0.pdf

The reachability problem for Petri nets is not elementary

Parikh’s Theorem states that semilinear sets are effectively equivalent with the Parikh images of regular languages and those of context-free languages. In this paper, we study the complexity of Parikh’s Theorem over any fixed alphabet size d. We prove various normal form the oremsin the case of NFAs and CFGs. In particular, the normalform theorems ensure that a union of linear sets with dgenerators suffice to express such Parikh images, which in the case of NFAs can further be computed in polynomial time. We then apply apply our results to derive: (1) optimal complexity for decision problems concerning Parikh images(e.g. membership, universality, equivalence, and disjointness), (2) a new polynomial fragment of integer programming, (3) an answer to an open question about PAC-learnability of semilinear sets, and (4) an optimal algorithm for verifying LTL over discrete-timed reversal-bounded counter systems.

/pdf/parikh-images-of-grammars-complexity-and-applications-2bxhqfpwt3.pdf

Parikh Images of Grammars: Complexity and Applications

More than 30 years after their inception, the decidability proofs for reach ability in vector addition systems (VAS) still retain much of their mystery. These proofs rely crucially on a decomposition of runs successively refined by Mayr, Kosaraju, and Lambert, which appears rather magical, and for which no complexity upper bound is known. We first offer a justification for this decomposition technique, by showing that it computes the ideal decomposition of the set of runs, using the natural embedding relation between runs as well quasi ordering. In a second part, we apply recent results on the complexity of termination thanks to well quasi orders and well orders to obtain a cubic Ackermann upper bound for the decomposition algorithms, thus providing the first known upper bounds for general VAS reach ability.

Demystifying Reachability in Vector Addition Systems

Known to be decidable since 1981, there still remains a huge gap between the best known lower and upper bounds for the reach ability problem for vector addition systems with states (VASS). Here the problem is shown PSPACE-complete in the two-dimensional case, vastly improving on the doubly exponential time bound established in 1986 by Howell, Rosier, Huynh and Yen. Cover ability and bounded ness for two-dimensional VASS are also shown PSPACE-complete, and reach ability in two-dimensional VASS and in integer VASS under unary encoding are considered.

Reachability in Two-Dimensional Vector Addition Systems with States Is PSPACE-Complete

We investigate the decidability and complexity of various model checking problems over one-counter automata. More specifically, we consider succinct one-counter automata, in which additive updates are encoded in binary, as well as parametric one-counter automata, in which additive updates may be given as unspecified parameters. We fully determine the complexity of model checking these automata against CTL, LTL, and modal µ-calculus specifications.

/pdf/model-checking-succinct-and-parametric-one-counter-automata-znigc2q0yx.pdf

Model checking succinct and parametric one-counter automata

One-counter processes are pushdown systems over a singleton stack alphabet (plus a stack-bottom symbol). We study the complexity of two closely related verification problems over one-counter processes: model checking with the temporal logic EF, where formulas are given as directed acyclic graphs, and weak bisimilarity checking against finite systems. We show that both problems are $\P^\NP$-complete. This is achieved by establishing a close correspondence with the membership problem for a natural fragment of Presburger Arithmetic, which we show to be$\P^\NP$-complete. This fragment is also a suitable representation for the global versions of the problems. We also show that there already exists a fixed EF formula(resp. a fixed finite system) such that model checking (resp. weak bisimulation) over one-counter processes is hard for $\P^{\NP[\log]}$. However, the complexity drops to $\P$ if the one-counter process is fixed.

/pdf/on-the-computational-complexity-of-verifying-one-counter-1bx1tut2kq.pdf

On the Computational Complexity of Verifying One-Counter Processes

We prove that language equivalence of deterministic one-counter automata is NL-complete. This improves the superpolynomial time complexity upper bound shown by Valiant and Paterson in 1975. Our main contribution is to prove that two deterministic one-counter automata are inequivalent if and only if they can be distinguished by a word of length polynomial in the size of the two input automata.

/pdf/equivalence-of-deterministic-one-counter-automata-is-nl-1kvsf4jpq6.pdf

Equivalence of deterministic one-counter automata is NL-complete

This paper introduces a class of register machines whose registers can be updated by polynomial functions when a transition is taken, and the domain of the registers can be constrained by linear constraints. This model strictly generalises a variety of known formalisms such as various classes of Vector Addition Systems with States. Our main result is that reachability in our class is PSPACE-complete when restricted to one register. We moreover give a classification of the complexity of reachability according to the type of polynomials allowed and the geometry induced by the range-constraining formula.

/pdf/reachability-in-register-machines-with-polynomial-updates-2lgyss7ke3.pdf

Stefan Göller

Papers

Reachability in Two-Dimensional Vector Addition Systems with States Is PSPACE-Complete

Model checking succinct and parametric one-counter automata

On the Computational Complexity of Verifying One-Counter Processes

Equivalence of deterministic one-counter automata is NL-complete

Reachability in Register Machines with Polynomial Updates