The geometric dimension g of a Vector Addition System with States (VASS) is the dimension of the vector space generated by cycles in the VASS; this parameter refines the standard dimension d, the number of counters. Recently, it was discovered that the fastest-known algorithm for solving the reachability problem for VASS has the same complexity in terms of g as in terms of d. This suggests that the geometric dimension may in fact be a more adequate parameter for measuring the complexity of VASS reachability problems. We initiate a more systematic study of the geometric dimension. We discuss differences between two parameters: the geometric dimension and the SCC dimension. Our main technical result states that classical results about the coverability and boundedness problems can be improved from dimension d to geometric dimension g. Namely, coverability is witnessed by runs of length n^{2^𝒪(g)} instead of n^{2^𝒪(d)}, and unboundedness can be witnessed by runs of length n^{2^𝒪(g log g)} instead of n^{2^𝒪(d log d)}, where n is the size of the instance. We also study integer reachability and simultaneous unboundedness in VASS parameterised by the geometric dimension.

We consider a variant of VASS extended with integer counters, denoted VASS+ℤ. These are automata equipped with ℕ- and ℤ-counters; the ℕ-counters are required to remain nonnegative and the ℤ-counters do not have this restriction. We study the complexity of the reachability problem for VASS+ℤ when the number of ℕ-counters is fixed. We show that reachability is NP-complete in 1-VASS+ℤ (i.e. when there is only one ℕ-counter) regardless of unary or binary encoding. For d ≥ 2, using a KLMST-based algorithm, we prove that reachability in d-VASS+ℤ lies in the complexity class ℱ_{d+2}. Our upper bound improves on the naively obtained Ackermannian complexity by simulating the ℤ-counters with ℕ-counters. To complement our upper bounds, we show that extending VASS with integer counters significantly lowers the number of ℕ-counters needed to exhibit hardness. We prove that reachability in unary 2-VASS+ℤ is PSpace-hard; without ℤ-counters this lower bound is only known in dimension 5. We also prove that reachability in unary 3-VASS+ℤ is Tower-hard. Without ℤ-counters, reachability in 3-VASS has elementary complexity and Tower-hardness is only known in dimension 8.

The aim of this paper is to deliver broad understanding of a class of languages of boundedly-ambiguous VASSs, that is k-ambiguous VASSs for some natural k. These are languages of Vector Addition Systems with States with the acceptance condition defined by the set of accepting states such that each accepted word has at most k accepting runs. We develop tools for proving that a given language is not accepted by any k-ambiguous VASS. Using them we show a few negative results: lack of some closure properties of languages of k-ambiguous VASSs and undecidability of the k-ambiguity problem, namely the question whether a given VASS language is a language of some k-ambiguous VASS. In fact we show an even more general undecidability result stating that for any class containing all regular languages and only k-ambiguous VASS languages for some k ∈ ℕ it is undecidable whether a language of a given 1-dimensional VASS belongs to this class. Finally, we show that the regularity problem is decidable for k-ambiguous VASSs.

The reachability problem in 3-dimensional vector addition systems with states (3-VASS) is known to be PSpace-hard, and to belong to Tower. We significantly narrow down the complexity gap by proving the problem to be solvable in doubly-exponential space. The result follows from a new upper bound on the length of the shortest path: if there is a path between two configurations of a 3-VASS then there is also one of at most triply-exponential length. We show it by introducing a novel technique of approximating the reachability sets of 2-VASS by small semi-linear sets.

Vector addition systems with states (VASS), also known as Petri nets, are a popular model of concurrent systems. Many problems from many areas reduce to the reachability problem for VASS, which consists of deciding whether a target configuration of a VASS is reachable from a given initial configuration. In this paper, we obtain an Ackermannian (primitive-recursive in fixed dimension) upper bound for the reachability problem in VASS with nested zero tests. Furthermore, we provide a uniform approach which also allows to decide most related problems, for example semilinearity and separability, in the same complexity. For some of these problems like semilinearity the complexity was unknown even for plain VASS.

We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASS) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed conditions). In general, the problem of language equivalence is undecidable even for one-dimensional VASS, thus to get decidability we investigate restricted subclasses. On the one hand, we show that the problem of language inclusion of a VASS in k-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand, we prove that the language equivalence problem is already Ackermann-hard for deterministic VASS. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems.

Vector Addition Systems with States (VASS), equiv-alent to Petri nets, are a well-established model of concurrency. A d-VASS can be seen as directed graph whose edges are labelled by d-dimensional integer vectors. While following a path, the values of$d$nonnegative integer counters are updated according to the integer labels. The central algorithmic challenge in VASS is the reachability problem: is there a run from a given starting node and counter values to a given target node and counter values? When the input is encoded in binary, reachability is computationally intractable: even in dimension one, it is NP-hard. In this paper, we comprehensively characterise the tractability border of the problem when the input is encoded in unary. For our main result, we prove that reachability is NP-hard in unary encoded 3-VASS, even when structure is heavily restricted to be a simple linear-path scheme. This improves upon a recent result of Czerwiński and Orlikowski [LICS 2022], in both the number of counters and expressiveness of the considered model, as well as answers open questions of Englert, Lazić, and Totzke [LICS 2016] and Leroux [PETRI NETS 2021]. The underlying graph structure of a simple linear path scheme (SLPS) is just a path with self-loops at each node. We also study the exceedingly weak model of computation that is SPLS with counter updates in { -1, 0, + 1 }. Here, we show that reachability is NP-hard when the dimension is bounded by O(a(k)), where$a$is the inverse Ackermann function and$k$bounds the size of the SLPS. We complement our result by presenting a polynomial-time algorithm that decides reachability in 2-SLPS when the initial and target configurations are specified in binary. To achieve this, we show that reachability in such instances is well-structured: all loops, except perhaps for a constant number, are taken either polynomially many times or almost maximally. This extends the main result of Englert, Lazić, and Totzke [LICS 2016] who showed the problem is in NL when the initial and target configurations are specified in unary.

In this paper we propose two new subclasses of Petri nets with resets, for which the reachability and coverability problems become tractable. Namely, we add an acyclicity condition that only applies to the consumptions and productions, not the resets. The first class is acyclic Petri nets with resets, and we show that coverability is PSPACE-complete for them. This contrasts the known Ackermann-hardness for coverability in (not necessarily acyclic) Petri nets with resets. We prove that the reachability problem remains undecidable for acyclic Petri nets with resets. The second class concerns workflow nets, a practically motivated and natural subclass of Petri nets. Here, we show that both coverability and reachability in acyclic workflow nets with resets are PSPACE-complete. Without the acyclicity condition, reachability and coverability in workflow nets with resets are known to be equally hard as for Petri nets with resets, that being Ackermann-hard and undecidable, respectively.

This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary. Let $n$ denote the maximum of the number of states of the input finite automata considered in the corresponding results. The following main results are obtained: (1) Given two unary NFAs recognising $L$ and $H$, respectively, one can decide whether $L \subseteq H$ as well as whether $L = H$ in time $2^{O((n \log n)^{1/3})}$. The previous upper bound on time was $2^{O((n \log n)^{1/2})}$ as given by Chrobak (1986), and this bound was not significantly improved since then. (2) Given two unary UFAs (unambiguous finite automata) recognising $L$ and $H$, respectively, one can determine a UFA recognising $L \cup H$ and a UFA recognising complement of $L$, where these output UFAs have the number of states bounded by a quasipolynomial in $n$. However, in the worst case, a UFA for recognising concatenation of languages recognised by two $n$-state UFAs, uses $2^{Θ((n \log^2 n)^{1/3})}$ states. (3) Given a unary language $L$, if $L$ contains the word of length $k$, then let $L(k)=1$ else let $L(k)=0$. Let $ω_L$ be the $ω$-word $L(0)L(1)\ldots$ and let $\cal L$ be a fixed $ω$-regular language. The last section studies how difficult it is to decide, given an $n$-state UFA or NFA

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We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASS) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed conditions). In general, the problem of language equivalence is undecidable even for one-dimensional VASS, thus to get decidability we investigate restricted subclasses. On the one hand, we show that the problem of language inclusion of a VASS in k-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand, we prove that the language equivalence problem is already Ackermann-hard for deterministic VASS. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems.

We consider linear cost-register automata (equivalent to weighted automata) over the semiring of nonnegative rationals, which generalise probabilistic automata. The two problems of boundedness and zero isolation ask whether there is a sequence of words that converge to infinity and to zero, respectively. In the general model both problems are undecidable so we focus on the copyless linear restriction. There, we show that the boundedness problem is decidable.

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.

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 complexity classes.

We investigate computational complexity of the reachability problem for vector addition systems (or, equivalently, Petri nets), the central algorithmic problem in verification of concurrent systems. Concerning its complexity, after 40 years of stagnation, a non-elementary lower bound has been shown recently: the problem needs a tower of exponentials of time or space, where the height of tower is linear in the input size. We improve on this lower bound, by increasing the height of tower from linear to exponential. As a side-effect, we obtain better lower bounds for vector addition systems of fixed dimension.

We present a data structure that in a dynamic graph of treedepth at most d, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time , which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time f(d) for some non-elementary (i.e. tower-exponential) function f. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth d from doubly-exponential in d to dO(d). As applications, we design new fully dynamic parameterized data structures for detecting long paths and cycles in general graphs. More precisely, for a fixed parameter k and a dynamic graph G, modified over time by edge insertions and deletions, our data structures maintain answers to the following queries: Does G contain a simple path on k vertices? Does G contain a simple cycle on at least k vertices? In the first case, the data structure achieves amortized update time . In the second case, the amortized update time is . In both cases we assume access to a dictionary on the edges of G.

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant $C$ such that for all positive integers $a,b$ and a graph $G$, if the treedepth of $G$ is at least $Cab$, then the treewidth of $G$ is at least $a$ or $G$ contains a subcubic (i.e., of maximum degree at most 3) tree of treedepth at least $b$ as a subgraph. As a direct corollary, we obtain that every graph of treedepth $\Omega(k^3)$ either is of treewidth at least $k$, contains a subdivision of full binary tree of depth $k$, or contains a path of length $2^k$. This improves the bound of $\Omega(k^5 \log^2 k)$ of Kawarabayashi and Rossman [Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 234--246]. We also show an application of our techniques for approximation algorithms of treedepth: given a graph $G$ of treedepth $k$ and treewidth $t$, one can in polynomial time compute a treedepth decomposition of $G$ of width $\mathcal{O}(kt \log^{3/2} t)$. This improves upon a bound of $\mathcal{O}(kt^2 \log t)$ stemming from a tradeoff between known results. The main technical ingredient in our result is a proof that every tree of treedepth $d$ contains a subcubic subtree of treedepth at least $d \cdot \log_3 ((1+\sqrt{5})/2)$.

The reachability problem is a central decision problem in verification of vector addition systems with states (VASS). In spite of recent progress, the complexity of the reachability problem remains unsettled, and it is closely related to the lengths of shortest VASS runs that witness reachability. We obtain three main results for VASS of fixed dimension. For the first two, we assume that the integers in the input are given in unary, and that the control graph of the given VASS is flat (i.e., without nested cycles). We obtain a family of VASS in dimension 3 whose shortest runs are exponential, and we show that the reachability problem is NP-hard in dimension 7. These results resolve negatively questions that had been posed by the works of Blondin et al. in LICS 2015 and Englert et al. in LICS 2016, and contribute a first construction that distinguishes 3-dimensional flat VASS from 2-dimensional ones. Our third result, by means of a novel family of products of integer fractions, shows that 4-dimensional VASS can have doubly exponentially long shortest runs. The smallest dimension for which this was previously known is 14.

We study languages of unambiguous VASS, that is, Vector Addition Systems with States, whose transitions read letters from a finite alphabet, and whose acceptance condition is defined by a set of final states (i.e., the coverability language). We show that the problem of universality for unambiguous VASS is ExpSpace-complete, in sheer contrast to Ackermann-completeness for arbitrary VASS, even in dimension 1. When the dimension d ∈ ℕ is fixed, the universality problem is PSpace-complete if d ≥ 2, and coNP-hard for 1-dimensional VASSes (also known as One Counter Nets).

We study the problem of regular separability of languages of vector addition systems with states (VASS). It asks whether for two given VASS languages K and L, there exists a regular language R that includes K and is disjoint from L. While decidability of the problem in full generality remains an open question, there are several subclasses for which decidability has been shown: It is decidable for (i) one-dimensional VASS, (ii) VASS coverability languages, (iii) languages of integer VASS, and (iv) commutative VASS languages.

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Several distinct techniques have been proposed to design quasi-polynomial algorithms for solving parity games since the breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan (2017): play summaries, progress measures and register games. We argue that all those techniques can be viewed as instances of the separation approach to solving parity games, a key technical component of which is constructing (explicitly or implicitly) an automaton that separates languages of words encoding plays that are (decisively) won by either of the two players. Our main technical result is a quasi-polynomial lower bound on the size of such separating automata that nearly matches the current best upper bounds. This forms a barrier that all existing approaches must overcome in the ongoing quest for a polynomial-time algorithm for solving parity games. The key and fundamental concept that we introduce and study is a universal ordered tree. The technical highlights are a quasi-polynomial lower bound on the size of universal ordered trees and a proof that every separating safety automaton has a universal tree hidden in its state space.

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.

We show that any one-counter automaton with $n$ states, if its language is non-empty, accepts some word of length at most $O(n^2)$. This closes the gap between the previously known upper bound of $O(n^3)$ and lower bound of $\Omega(n^2)$. More generally, we prove a tight upper bound on the length of shortest paths between arbitrary configurations in one-counter transition systems (weaker bounds have previously appeared in the literature). Comment: 28 pages, 2 figures

We investigate the languages recognized by well-structured transition systems (WSTS) with upward and downward compatibility. Our first result shows that, under very mild assumptions, every two disjoint WSTS languages are regular separable: There is a regular language containing one of them and being disjoint from the other. As a consequence, if a language as well as its complement are both recognized by WSTS, then they are necessarily regular. In particular, no subclass of WSTS languages beyond the regular languages is closed under complement. Our second result shows that for Petri nets, the complexity of the backwards coverability algorithm yields a bound on the size of the regular separator. We complement it by a lower bound construction.

A vector addition system (VAS) with an initial and a final marking and transition labels induces a language. In part because the reachability problem in VAS remains far from being well-understood, it is difficult to devise decision procedures for such languages. This is especially true for checking properties that state the existence of infinitely many words of a particular shape. Informally, we call these unboundedness properties. We present a simple set of axioms for predicates that can express unboundedness properties. Our main result is that such a predicate is decidable for VAS languages as soon as it is decidable for regular languages. Among other results, this allows us to show decidability of (i) separability by bounded regular languages, (ii) unboundedness of occurring factors from a language K with mild conditions on K, and (iii) universality of the set of factors.

Many of today’s graph query languages are based on graph pattern matching. We investigate optimization of tree-shaped patterns that have transitive closure operators. Such patterns not only appear in the context of graph databases but also were originally studied for querying tree-structured data, where they can perform child, descendant, node label, and wildcard tests. The minimization problem aims at reducing the number of nodes in patterns and goes back to the early 2000s. We provide an example showing that, in contrast to earlier claims, tree patterns cannot be minimized by deleting nodes only. The example resolves the M = ? NR problem, which asks if a tree pattern is minimal if and only if it is nonredundant. The example can be adapted to prove that minimization is Σ P 2 -complete, which resolves another question that was open since the early research on the problem. The latter result shows that, unless NP = Π P 2 , more general approaches for minimizing tree patterns are also bound to fail in general.

We investigate a subclass of languages recognized by vector addition systems, namely languages of nondeterministic Parikh automata. While the regularity problem (is the language of a given automaton regular?) is undecidable for this model, we surprisingly show decidability of the regular separability problem: given two Parikh automata, is there a regular language that contains one of them and is disjoint from the other? We supplement this result by proving undecidability of the same problem already for languages of visibly one counter automata.

The regular separability problem asks, for two given languages, if there exists a regular language including one of them but disjoint from the other. Our main result is decidability, and PSPACE-completeness, of the regular separability problem for languages of one counter automata without zero tests (also known as one counter nets). This contrasts with undecidability of the regularity problem for one counter nets, and with undecidability of the regular separability problem for one counter automata, which is our second result.

Given two families of sets F and G, the F-separability problem for G asks whether for two given sets U, V in G there exists a set S in F, such that U is included in S and V is disjoint with S. We consider two families of sets F: modular sets S which are subsets of N^d, defined as unions of equivalence classes modulo some natural number n in N, and unary sets, which extend modular sets by requiring equality below a threshold n, and equivalence modulo n above n. Our main result is decidability of modular- and unary-separability for the class G of reachability sets of Vector Addition Systems, Petri Nets, Vector Addition Systems with States, and for sections thereof.

We study a class of integrity constraints for tree-structured data modelled as data trees, whose nodes have a label from a finite alphabet and store a data value from an infinite data domain. The constraints require each tuple of nodes selected by a conjunctive query (using navigational axes and labels) to satisfy a positive combination of equalities and a positive combination of inequalities over the stored data values. Such constraints are instances of the general framework of XML-to-relational constraints proposed recently by Niewerth and Schwentick. They cover some common classes of constraints, including W3C XML Schema key and unique constraints, as well as domain restrictions and denial constraints, but cannot express inclusion constraints, such as reference keys. Our main result is that consistency of such integrity constraints with respect to a given schema (modelled as a tree automaton) is decidable. An easy extension gives decidability for the entailment problem. Equivalently, we show that validity and containment of unions of conjunctive queries using navigational axes, labels, data equalities and inequalities is decidable, as long as none of the conjunctive queries uses both equalities and inequalities; without this restriction, both problems are known to be undecidable. In the context of XML data exchange, our result can be used to establish decidability for a consistency problem for XML schema mappings. All the decision procedures are doubly exponential, with matching lower bounds. The complexity may be lowered to singly exponential, when conjunctive queries are replaced by tree patterns, and the number of data comparisons is bounded.

We investigate minimization of tree pattern queries that use the child relation, descendant relation, node labels, and wildcards. We prove that minimization for such tree patterns is Sigma2P-complete and thus solve a problem first attacked by Flesca, Furfaro, and Masciari in 2003. We first provide an example that shows that tree patterns cannot be minimized by deleting nodes. This example shows that the M-NR conjecture, which states that minimality of tree patterns is equivalent to their nonredundancy, is false. We then show how the example can be turned into a gadget that allows us to prove Sigma2P-completeness.

Branching bisimilarity of nor med Basic Process Algebra (BPA) processes was shown to be decidable by Yuxi Fu (ICALP 2013) but his proof has not provided any upper complexity bound. We present a simpler approach based on relative prime decompositions that leads to a nondeterministic exponential-time algorithm, this is "close" to the known exponential-time lower bound. We also derive that semantic finiteness (the question if a given nor med BPA process is branching bisimilar with some finite-state process) belongs to NExpTime as well.

Tree pattern queries are being investigated in database theory for more than a decade. They are a fundamental and flexible query mechanism and have been considered in the context of querying tree structured as well as graph structured data. We revisit their containment, validity, and satisfiability problem, both with and without schema information. We present a comprehensive overview of what is known about the complexity of containment and develop new techniques which allow us to obtain tractability- and hardness results for cases that have been open since the early work on tree pattern containment. For the tree pattern queries we consider in this paper, it is known that the containment problem does not depend on whether patterns are evaluated on trees or on graphs. This means that our results also shed new light on tree pattern queries on graphs.

We consider sequences of vectors from ℕ d . Each coordinate of a vector can be reset or incremented by 1 with respect to the same coordinate of the preceding vector. We give an example of non-dominating sequence, like in Dickson’s Lemma, of length 2 2 θ( n) , what matches the previously known upper bound.

We investigate normed commutative context-free processes (Basic Parallel Processes). We show that branching bisimilarity admits the bounded response property : in the Bisimulation Game, Duplicator always has a response leading to a process of size linearly bounded with respect to the Spoiler’s process. The linear bound is effective, which leads to decidability of branching bisimilarity. For weak bisimilarity, we are able merely to show existence of some linear bound, which is not sufficient for decidability. We conjecture however that the same effective bound holds for weak bisimilarity as well. We suppose that further elaboration of novel techniques developed in this paper may be sufficient to demonstrate decidability.

Bisimulation equivalence is decidable in polynomial time over normed graphs generated by a context-free grammar. We present a new algorithm, working in time $O(n^5)$, thus improving the previously known complexity $O(n^8 * polylog(n))$. It also improves the previously known complexity $O(n^6 * polylog(n))$ of the equality problem for simple grammars.