In 2015, it was shown that reachability for arbitrary directed graphs can be updated by first-order formulas after inserting or deleting single edges. Later, in 2018, this was extended for changes of size $\frac{\log n}{\log \log n}$, where $n$ is the size of the graph. Changes of polylogarithmic size can be handled when also majority quantifiers may be used.
In this paper we extend these results by showing that, for changes of polylogarithmic size, first-order update formulas suffice for maintaining (1) undirected reachability, and (2) directed reachability under insertions. For classes of directed graphs for which efficient parallel algorithms can compute non-zero circulation weights, reachability can be maintained with update formulas that may use "modulo 2" quantifiers under changes of polylogarithmic size. Examples for these classes include the class of planar graphs and graphs with bounded treewidth. The latter is shown here.
As the logics we consider cannot maintain reachability under changes of larger sizes, our results are optimal with respect to the size of the changes.
This paper studies parametric Markov decision processes (pMDPs), an extension to Markov decision processes (MDPs) where transitions probabilities are described by polynomials over a finite set of parameters. Fixing values for all parameters yields MDPs. In particular, this paper studies the complexity of finding values for these parameters such that the induced MDP satisfies some reachability constraints. We discuss different variants depending on the comparison operator in the constraints and the domain of the parameter values. We improve all known lower bounds for this problem, and notably p…
This paper proves the NP-completeness of the reachability problem for the class of flat counter machines with difference bounds and, more generally, octagonal relations, labeling the transitions on the loops. The proof is based on the fact that the sequence of powers $\{R^i\}_{i=1}^\infty$ of such relations can be encoded as a periodic sequence of matrices, and that both the prefix and the period of this sequence are $2^{\mathcal{O}(\bin{R})}$ in the size of the binary encoding $\bin{R}$ of a relation $R$. This result allows to characterize the complexity of the reachability problem for one of…
The dynamic complexity of the reachability query is studied in the dynamic complexity framework of Patnaik and Immerman, restricted to quantifier-free update formulas. It is shown that, with this restriction, the reachability query cannot be dynamically maintained, neither with binary auxiliary relations nor with unary auxiliary functions, and that ternary auxiliary relations are more powerful with respect to graph queries than binary auxiliary relations. Further inexpressibility results are given for the reachability query in a different setting as well as for a syntactical restriction of qua…
We propose a semi-decision procedure for checking generalized reachability properties, on generalized Petri nets, that is based on the Property Directed Reachability (PDR) method. We actually define three different versions, that vary depending on the method used for abstracting possible witnesses, and that are able to handle problems of increasing difficulty. We have implemented our methods in a model-checker called SMPT and give empirical evidences that our approach can handle problems that are difficult or impossible to check with current state of the art tools.
Determining the complexity of the reachability problem for vector addition systems with states (VASS) is a long-standing open problem in computer science. Long known to be decidable, the problem to this day lacks any complexity upper bound whatsoever. In this paper, reachability for two-dimensional VASS is shown PSPACE-complete. This improves on a previously known doubly exponential time bound established by Howell, Rosier, Huynh and Yen in 1986. The coverability and boundedness problems are also noted to be PSPACE-complete. In addition, some complexity results are given for the reachability p…
This article presents the complexity of reachability decision problems for parametric Markov decision processes (pMDPs), an extension to Markov decision processes (MDPs) where transitions probabilities are described by polynomials over a finite set of parameters. In particular, we study the complexity of finding values for these parameters such that the induced MDP satisfies some maximal or minimal reachability probability constraints. We discuss different variants depending on the comparison operator in the constraints and the domain of the parameter values. We improve all known lower bounds …
Vector Addition Systems and equivalent Petri nets are a well established models of concurrency. The central algorithmic problem for Vector Addition Systems with a long research history is the reachability problem asking whether there exists a run from one given configuration to another. We settle its complexity to be Ackermann-complete thus closing the problem open for 45 years. In particular we prove that the problem is $\mathcal{F}_k$-hard for Vector Addition Systems with States in dimension $6k$, where $\mathcal{F}_k$ is the $k$-th complexity class from the hierarchy of fast-growing complex…
We provide an Ackermannian complexity lower bound for the reachability problem for checking programs, a model equivalent to Petri nets. Moreover in fixed dimension $2d+4$, we show that the problem is $\mathbb{F}_d$-hard. As a direct corollary, the reachability problem in dimension 10 is not elementary.
Recently it was shown that the transitive closure of a directed graph can be updated using first-order formulas after insertions and deletions of single edges in the dynamic descriptive complexity framework by Dong, Su, and Topor, and Patnaik and Immerman. In other words, Reachability is in DynFO. In this article we extend the framework to changes of multiple edges at a time, and study the Reachability and Distance queries under these changes. We show that the former problem can be maintained in DynFO$(+, \times)$ under changes affecting O($\frac{\log n}{\log \log n}$) nodes, for graphs with $…
A complete characterization of the complexity of the reachability problem for vector addition system has been open for a long time. The problem is shown to be Tower complete.
Vector addition system with states is an ubiquitous model of computation with extensive applications in computer science. The reachability problem for vector addition systems is central since many other problems reduce to that question. The problem is decidable and it was recently proved that the dimension of the vector addition system is an important parameter of the complexity. In fixed dimensions larger than two, the complexity is not known (with huge complexity gaps). In dimension two, the reachability problem was shown to be PSPACE-complete by Blondin et al. in 2015. We consider an extens…
Technische Universität Darmstadt（达姆施塔特工业大学）；Université libre de Bruxelles（布鲁塞尔自由大学）；Departement d’Informatique（计算机系）
We study the never-worse relation (NWR) for Markov decision processes with an infinite-horizon reachability objective. A state q is never worse than a state p if the maximal probability of reaching the target set of states from p is at most the same value from q, regard- less of the probabilities labelling the transitions. Extremal-probability states, end components, and essential states are all special cases of the equivalence relation induced by the NWR. Using the NWR, states in the same equivalence class can be collapsed. Then, actions leading to sub- optimal states can be removed. We show …
In this paper we propose augmented interval Markov chains (AIMCs): a generalisation of the familiar interval Markov chains (IMCs) where uncertain transition probabilities are in addition allowed to depend on one another. This new model preserves the flexibility afforded by IMCs for describing stochastic systems where the parameters are unclear, for example due to measurement error, but also allows us to specify transitions with probabilities known to be identical, thereby lending further expressivity. The focus of this paper is reachability in AIMCs. We study the qualitative, exact quantitativ…
We consider finite-state concurrent stochastic games, played by k>=2 players for an infinite number of rounds, where in every round, each player simultaneously and independently of the other players chooses an action, whereafter the successor state is determined by a probability distribution given by the current state and the chosen actions. We consider reachability objectives that given a target set of states require that some state in the target is visited, and the dual safety objectives that given a target set require that only states in the target set are visited. We are interested in the …
We study pushdown systems where control states, stack alphabet, and transition relation, instead of being finite, are first-order definable in a fixed countably-infinite structure. We show that the reachability analysis can be addressed with the well-known saturation technique for the wide class of oligomorphic structures. Moreover, for the more restrictive homogeneous structures, we are able to give concrete complexity upper bounds. We show ample applicability of our technique by presenting several concrete examples of homogeneous structures, subsuming, with optimal complexity, known results …
Patnaik and Immerman introduced the dynamic complexity class DynFO of database queries that can be maintained by first-order dynamic programs with the help of auxiliary relations under insertions and deletions of edges (Patnaik and Immerman 1997). This article confirms their conjecture that the Reachability query is in DynFO. As a byproduct it is shown that the rank of a matrix with small values can be maintained in DynFO(+,x). It is further shown that the (size of the) maximum matching of a graph can be maintained in non-uniform DynFO, another extension of DynFO, with non-uniform initialisati…
In a dynamic parametric process every subprocess may spawn arbitrarily many, identical child processes, that may communicate either over global variables, or over local variables that are shared with their parent. We show that reachability for dynamic parametric processes is decidable under mild assumptions. These assumptions are e.g. met if individual processes are realized by pushdown systems, or even higher-order pushdown systems. We also provide algorithms for subclasses of pushdown dynamic parametric processes, with complexity ranging between NP and DEXPTIME.
We study the complexity of the reachability problem for Vector Addition Systems with States (VASSes) in fixed dimensions. We provide four lower bounds improving the currently known state-of-the-art: 1) \np-hardness for unary flat $4$-VASSes (VASSes in dimension 4), 2) \pspace-hardness for unary $5$-VASSes, 3) \expspace-hardness for binary $6$-VASSes and 4) \tower-hardness for unary $8$-VASSes.
This paper studies dynamic complexity under definable change operations in the DynFO framework by Patnaik and Immerman. It is shown that for changes definable by parameter-free first-order formulas, all (uniform) $AC^1$ queries can be maintained by first-order dynamic programs. Furthermore, many maintenance results for single-tuple changes are extended to more powerful change operations: (1) The reachability query for undirected graphs is first-order maintainable under single tuple changes and first-order defined insertions, likewise the reachability query for directed acyclic graphs under qua…
Petri nets are a classical model of concurrency widely used and studied in formal verification with many applications in modeling and analyzing hardware and software, data bases, and reactive systems. The reachability problem is central since many other problems reduce to reachability questions. The reachability problem is known to be decidable but its complexity is extremely high (non primitive recursive). In 2011, a variant of the reachability problem, called the mutual reachability problem, that consists in deciding if two configurations are mutually reachable was proved to be exponential-s…
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…
In this paper, we investigate the rational synthesis problem for concurrent game structure for a variety of objectives ranging from reachability to Muller condition. We propose a new algorithm that establishes the decidability of the non cooperative rational synthesis problem that relies solely on game theoretic technique as opposed to previous approaches that are logic based. Thanks to this approach, we construct a zero-sum turn-based game that can be adapted to each one of the afore mentioned objectives thus obtain new complexity results. In particular, we show that reachability, safety, Büc…
Answer Set Programming (ASP) is a problem modeling and solving framework for several problems in KR with growing industrial applications. Also for studies of computational complexity and deeper insights into the hardness and its sources, ASP has been attracting researchers for many years. These studies resulted in fruitful characterizations in terms of complexity classes, fine-grained insights in form of dichotomy-style results, as well as detailed parameterized complexity landscapes. Recently, this lead to a novel result establishing that for the measure treewidth, which captures structural d…
Dynamic networks of concurrent pushdown systems (DCPS) are a theoretical model for multi-threaded recursive programs with shared global state and dynamical creation of threads. The (global) state reachability problem for DCPS is undecidable in general, but Atig et al. (2009) showed that it becomes decidable, and is in 2EXPSPACE, when each thread is restricted to a fixed number of context switches. The best known lower bound for the problem is EXPSPACE-hard and this lower bound follows already when each thread is a finite-state machine and runs atomically to completion (i.e., does not switch co…
