A Superpolynomial Lower Bound for the Size of Non-Deterministic Complement of an Unambiguous Automaton

Unambiguous non-deterministic finite automata (UFA) are non-deterministic automata (over finite words) such that there is at most one accepting run over each input. Such automata are known to be potentially exponentially more succinct than deterministic automata, and non-deterministic automata can be exponentially more succinct than them. In this paper we establish a superpolynomial lower bound for the state complexity of the translation of an UFA to a non-deterministic automaton for the complement language. This disproves the formerly conjectured polynomial upper bound for this translation. This lower bound only involves a one letter alphabet, and makes use of the random graph methods. The same proof also shows that the translation of sweeping automata to non-deterministic automata is superpolynomial

Geometry of VAS reachability sets

Vector Addition Systems (VAS) or equivalently petri-nets are a popular model for representing concurrent systems. Many important decidability results about VAS were obtained by considering geometric properties of their reachability sets, i.e. the set of configurations reachable from some initial configuration $c_0$. For example, in 2012 Jerome Leroux proved that if a configuration $c_t$ is not reachable, then there exists a semilinear inductive invariant separating the reachability set from $c_t$. This gave an alternative proof of decidability of the reachability problem. The paper introduced the class of petri-sets, proved that reachability sets are petri-sets, and that petri-sets have this property. In a follow-up paper in 2013, Jerome Leroux again used the class of petri-sets to prove that if a reachability set is semilinear, then a representation of it can be computed. In this paper, we utilize the class of petri-sets to answer the opposite type of question: Even if the reachability set is non-semilinear, what form can it have? We give another proof that semilinearity of the reachability set is decidable, which was first shown by Hauschildt in 1990. We prove that reachability sets can be partitioned into nicely shaped sets we call almost-hybridlinear, and how to utilize this to decide semilinearity.Comment: 21 pages, 5 figure

Regular Model Checking Upside-Down: An Invariant-Based Approach

Regular model checking is a technique for the verification of infinite-state systems whose configurations can be represented as finite words over a suitable alphabet. It applies to systems whose set of initial configurations is regular, and whose transition relation is captured by a length-preserving transducer. To verify safety properties, regular model checking iteratively computes automata recognizing increasingly larger regular sets of reachable configurations, and checks if they contain unsafe configurations. Since this procedure often does not terminate, acceleration, abstraction, and widening techniques have been developed to compute a regular superset of the reachable configurations. In this paper we develop a complementary procedure. Instead of approaching the set of reachable configurations from below, we start with the set of all configurations and approach it from above. We use that the set of reachable configurations is equal to the intersection of all inductive invariants of the system. Since this intersection is non-regular in general, we introduce b-bounded invariants, defined as those representable by CNF-formulas with at most b clauses. We prove that, for every b ? 0, the intersection of all b-bounded inductive invariants is regular, and we construct an automaton recognizing it. We show that whether this automaton accepts some unsafe configuration is in EXPSPACE for every b ? 0, and PSPACE-complete for b = 1. Finally, we study how large must b be to prove safety properties of a number of benchmarks

Flatness and Complexity of Immediate Observation Petri Nets

In a previous paper we introduced immediate observation (IO) Petri nets, a class of interest in the study of population protocols and enzymatic chemical networks. In the first part of this paper we show that IO nets are globally flat, and so their safety properties can be checked by efficient symbolic model checking tools using acceleration techniques, like FAST. In the second part we study Branching IO nets (BIO nets), whose transitions can create tokens. BIO nets extend both IO nets and communication-free nets, also called BPP nets, a widely studied class. We show that, while BIO nets are no longer globally flat, and their sets of reachable markings may be non-semilinear, they are still locally flat. As a consequence, the coverability and reachability problem for BIO nets, and even a certain set-parameterized version of them, are in PSPACE. This makes BIO nets the first natural net class with non-semilinear reachability relation for which the reachability problem is provably simpler than for general Petri nets