Decision problems for HNN groups and vector addition systems

A group G is S-universal if every countable group can be embedded in a quotient of G.In this paper we give conditions under which HNN groups are S-universal.These are then applied to showing the S-universality of certain classes of Kleinian groups (discontinuous subgroups of PSL^C)).Finally some connections are mentioned between Sg-universality and the theory of ends.Introduction.A group G is So-universal if every countable group can be embedded in a quotient of G. Thus all nonabelian free groups are Suniversal [4], while it is known that all nontrivial free products except the infinite dihedral group [10], and large classes of amalgamated free products are also , [11].Sacerdote and Schupp have extended these to certain HNN and one-relator groups [9].In this paper we give a series of results which complement those of Sacerdote and Schupp but are somewhat less technical and more directly applicable to geometrically and arithmetically defined groups [2], [3].This will be applied in 3 to showing the S?-universality of a class of Kleinian groups.Finally we make some observations on the relationship between SG;-universality and the theory of ends.1. Recall that an HNN group is a group of the form G = </" ...,t",K; rel K, t^tr1 = <P,(A)>where {L,} are subgroups of K, and {<p,} are isomorphisms L, - <p,(jL,) < K. K is called the base, {L, <p(L)} associated subgroups and </" . .., i") the free part.Groups of this type were used by Higman, Neumann and Neumann to show the S-universality of free groups of rank 2 [4].The base K can be embedded isomorphically in G, and a subgroup structure theory for HNN groups, paralleling that for amalgamated free products, has been developedby Karrass and Solitar [5].Now G has a free quotient of rank n if the free part has rank n, and since clearly a group with an S-universal quotient is itself S-universal we getLemma.An HNN group with free part of rank > 1 is SQ-universal.

Seminal results establish that the coverability problem for Vector Addition Systems with States (VASS) is in EXPSPACE (Rackoff, ’78) and is EXPSPACE -hard already under unary encodings (Lipton, ’76). More precisely, Rosier and Yen later utilise Rackoff’s bounding technique to show that if coverability holds then there is a run of length at most \(n^{2^{\mathcal {O}(d \log (d))}}\) , where d is the dimension and n is the size of the given unary VASS. Earlier, Lipton showed that there exist instances of coverability in d -dimensional unary VASS that are only witnessed by runs of length at least \(n^{2^{\Omega (d)}}\) . Our first result closes this gap. We improve the upper bound by removing the twice-exponentiated \(\log (d)\) factor, thus matching Lipton’s lower bound. This closes the corresponding gap for the exact space required to decide coverability. This also yields a deterministic \(n^{2^{\mathcal {O}(d)}}\) -time algorithm for coverability. Our second result is a matching lower bound, that there does not exist a deterministic \(n^{2^{o(d)}}\) -time algorithm, conditioned upon the exponential time hypothesis. When analysing coverability, a standard proof technique is to consider VASS with bounded counters. Bounded VASS make for an interesting and popular model due to strong connections with timed automata. Withal, we study a natural setting where the counter bound is linear in the size of the VASS. Here the trivial exhaustive search algorithm runs in \(\mathcal {O}(n^{d+1})\) time. We give evidence to this being near-optimal. We prove that in dimension one this trivial algorithm is conditionally optimal, by showing that \(n^{2-o(1)}\) time is required under the k -cycle hypothesis. In general, for any fixed dimension d ≥ 4, we show that \(n^{d-2-o(1)}\) time is required under the 3-uniform hyperclique hypothesis.

Vector addition systems are an important model in theoretical computer science and have been used in a variety of areas. In this paper, we consider vector addition systems with states over a parameterized initial configuration. For these systems, we are interested in the standard notion of computational time complexity, i.e., we want to understand the length of the longest trace for a fixed vector addition system with states depending on the size of the initial configuration. We show that the asymptotic complexity of a given vector addition system with states is either varTheta (N^k) for some computable integer k, where N is the size of the initial configuration, or at least exponential. We further show that k can be computed in polynomial time in the size of the considered vector addition system. Finally, we show that 1 le k le 2^n, where n is the dimension of the considered vector addition system.

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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.

Reentrant locking is a recursive locking mechanism which allows a thread in a multi-threaded program to acquire the reentrant lock multiple times. The thread must release this lock an equal number of times before another thread can acquire this lock. We consider the control state reachability problem for recursive multi-threaded programs synchronizing via a finite number of reentrant locks. Such programs can be abstracted as multi-pushdown systems with a finite number of counters. The pushdown stacks model the call stacks of the threads and the counters model the reentrant locks. The control state reachability problem is already undecidable for non-reentrant locks. As a consequence, for non-reentrant locks, under-approximation techniques which restrict the search space have gained traction. One popular technique is to limit the number of context switches. Our main result is that the problem of checking whether a control state is reachable within a bounded number of context switches is decidable for recursive multi-threaded programs synchronizing via a finite number of reentrant locks if we restrict the lock-usage to contextual locking: a release of an instance of reentrant lock can only occur if the instance was acquired before in the same procedure and each instance of a reentrant lock acquired in a procedure call must be released before the procedure returns. The decidability is obtained by a reduction to the reachability problem of Vector Addition Systems with States (VASS).KeywordsGlobal StateIterative StepProcedure CallShared StateContext SwitchThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Does the trace language of a given vector addition system (VAS) intersect with a given context-free language? This question lies at the heart of several verification questions involving recursive programs with integer parameters. In particular, it is equivalent to the coverability problem for VAS that operate on a pushdown stack. We show decidability in dimension one, based on an analysis of a new model called grammar-controlled vector addition systems.

Rackoff’s small witness property for the coverability problem is the standard means to prove tight upper bounds in vector addition systems (VAS) and many extensions. We show how to derive the same bounds directly on the computations of the VAS instantiation of the generic backward coverability algorithm. This relies on a dual view of the algorithm using ideal decompositions of downwards-closed sets, which exhibits a key structural invariant in the VAS case. The same reasoning readily generalises to several VAS extensions.

A graph product is the fundamental group of a graph of groups Amongst the simplest examples are HNN groups and free products with amalgamation.The conjugacy problem is solvable for recursively presented graph products with cyclic edge groups over finite graphs if the vertex groups have solvable conjugacy problem and the sets of cyclic generators in them are semicritical.For graph products over infinite graphs these conditions are insufficient: a further condition ensures that graph products over infinite graphs of bounded path length have solvable conjugacy problem.These results generalise the known ones for HNN groups and free products with amalgamation.

We introduce a linear extension of the rotor-routing model in directed graphs, akin to the sandpile model and vector addition systems, together with new rotor mechanisms that extend standard cyclic rotors. In this framework, rotor-routing is interpreted as the simultaneous movement of particles accross two coupled graphs, involving both vertex and arc-based particles. The standard, combinatorial rotor-routing of positive particles (legal routing) based on rotor-configurations, then becomes a special case of a linear equivalence. We give comprehensive reachability results characterizing legal routings among linear equivalences, expanding on previous results, and settle the algorithmic complexities associated with these problems.

Multi-pushdown systems are a standard model for concurrent recursive programs, but they have an undecidable reachability problem. Therefore, there have been several proposals to underapproximate their sets of runs so that reachability in this underapproximation becomes decidable. One such underapproximation that covers a relatively high portion of runs is scope boundedness. In such a run, after each push to stack i, the corresponding pop operation must come within a bounded number of visits to stack i. In this work, we generalize this approach to a large class of infinite-state systems. For this, we consider the model of valence systems, which consist of a finite-state control and an infinite-state storage mechanism that is specified by a finite undirected graph. This framework captures pushdowns, vector addition systems, integer vector addition systems, and combinations thereof. For this framework, we propose a notion of scope boundedness that coincides with the classical notion when the storage mechanism happens to be a multi-pushdown. We show that with this notion, reachability can be decided in PSPACE for every storage mechanism in the framework. Moreover, we describe the full complexity landscape of this problem across all storage mechanisms, both in the case of (i) the scope bound being given as input and (ii) for fixed scope bounds. Finally, we provide an almost complete description of the complexity landscape if even a description of the storage mechanism is part of the input.

This paper studies the boundedness and termination problems for vector addition systems equipped with one stack. We introduce an algorithm, inspired by the Karp & Miller algorithm, that solves both problems for the larger class of well-structured pushdown systems. We show that the worst-case running time of this algorithm is hyper-Ackermannian for pushdown vector addition systems. For the upper bound, we introduce the notion of bad nested words over a well-quasi-ordered set, and we provide a general scheme of induction for bounding their lengths. We derive from this scheme a hyper-Ackermannian upper bound for the length of bad nested words over vectors of natural numbers. For the lower bound, we exhibit a family of pushdown vector addition systems with finite but large reachability sets (hyper-Ackermannian).

Vector addition systems, initially introduced by Karp and Miller, have been very successfully used for the description of complex concurrent behaviour. These models are essentially the same as Petri nets. Our purpose is to develop a quantitative “measurement” oriented theory of such systems in order to complement the existing qualitative theory. This paper is a first step in that direction.

article Free Access Share on The Complexity of the Finite Containment Problem for Petri Nets Authors: Ernst W. Mayr Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MassachusettsView Profile , Albert R. Meyer Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MassachusettsView Profile Authors Info & Claims Journal of the ACMVolume 28Issue 3pp 561–576https://doi.org/10.1145/322261.322271Published:01 July 1981Publication History 68citation651DownloadsMetricsTotal Citations68Total Downloads651Last 12 Months25Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

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).