Vector Addition Systems Research Articles

Monotone Procedure Summarization via Vector Addition Systems and Inductive Potentials

This paper presents a technique for summarizing recursive procedures operating on integer variables. The motivation of our work is to create more predictable program analyzers, and in particular to formally guarantee compositionality and monotonicity of procedure summarization. To summarize a procedure, we compute its best abstraction as a vector addition system with resets (VASR) and exactly summarize the executions of this VASR over the context-free language of syntactic paths through the procedure. We improve upon this technique by refining the language of syntactic paths using (automatically synthesized) linear potential functions that bound the number of recursive calls within valid executions of the input program. We implemented our summarization technique in an automated program verification tool; our experimental evaluation demonstrates that our technique computes more precise summaries than existing abstract interpreters and that our tool’s verification capabilities are comparable with state-of-the-art software model checkers.

Reachability in Continuous Pushdown VASS

Pushdown Vector Addition Systems with States (PVASS) consist of finitely many control states, a pushdown stack, and a set of counters that can be incremented and decremented, but not tested for zero. Whether the reachability problem is decidable for PVASS is a long-standing open problem. We consider continuous PVASS , which are PVASS with a continuous semantics. This means, the counter values are rational numbers and whenever a vector is added to the current counter values, this vector is first scaled with an arbitrarily chosen rational factor between zero and one. We show that reachability in continuous PVASS is NEXPTIME-complete. Our result is unusually robust: Reachability can be decided in NEXPTIME even if all numbers are specified in binary. On the other hand, NEXPTIME-hardness already holds for coverability, in fixed dimension, for bounded stack, and even if all numbers are specified in unary.

Linear equations for unordered data vectors in $[D]^k\to{}Z^d$

Following a recently considered generalisation of linear equations to unordered-data vectors and to ordered-data vectors, we perform a further generalisation to data vectors that are functions from k-element subsets of the unordered-data set to vectors of integer numbers. These generalised equations naturally appear in the analysis of vector addition systems (or Petri nets) extended so that each token carries a set of unordered data. We show that nonnegative-integer solvability of linear equations is in nondeterministic exponential time while integer solvability is in polynomial time.

Algorithmic analysis of termination and counter complexity in vector addition systems with states

We give an overview of recently established results about the effective asymptotic analysis of termination and counter complexity of VASS computations. In contrast to "classical" problems such as reachability, boundedness, liveness, coverability, etc., that are EXPSPACE-hard, the decision problems related to VASS asymptotic analysis tend to have low complexity and many important variants are even decidable in polynomial time. We also present selected concepts and techniques used to achieve these results.

The Reachability Problem for Two-Dimensional Vector Addition Systems with States

We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This, in turn, enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.

Strategic reasoning with a bounded number of resources: The quest for tractability

The resource-bounded alternating-time temporal logic RB±ATL combines strategic reasoning with reasoning about resources. Its model-checking problem is known to be 2exptime-complete (the same as its proper extension RB±ATL⁎) and fragments have been identified to lower the complexity.In this work, we consider the variant RB±ATL+ that allows for Boolean combinations of path formulae starting with single temporal operators, but restricted to a single resource, providing an interesting trade-off between temporal expressivity and resource analysis. We show that the model-checking problem for RB±ATL+ restricted to a single agent and a single resource is Δ2P-complete, hence the same as for the standard branching-time temporal logic CTL+. In this case reasoning about resources comes at no extra computational cost. When a fixed finite set of linear-time temporal operators is considered, the model-checking problem drops to ptime, which includes the special case of RB±ATL restricted to a single agent and a single resource. Furthermore, we show that, with an arbitrary number of agents and a fixed number of resources, the model-checking problem for RB±ATL+ can be solved in exptime using a sophisticated Turing reduction to the parity game problem for alternating vector addition systems with states (AVASS).

The Complexity of Reachability in Affine Vector Addition Systems with States

Vector addition systems with states (VASS) are widely used for the formal verification of concurrent systems. Given their tremendous computational complexity, practical approaches have relied on techniques such as reachability relaxations, e.g., allowing for negative intermediate counter values. It is natural to question their feasibility for VASS enriched with primitives that typically translate into undecidability. Spurred by this concern, we pinpoint the complexity of integer relaxations with respect to arbitrary classes of affine operations. More specifically, we provide a trichotomy on the complexity of integer reachability in VASS extended with affine operations (affine VASS). Namely, we show that it is NP-complete for VASS with resets, PSPACE-complete for VASS with (pseudo-)transfers and VASS with (pseudo-)copies, and undecidable for any other class. We further present a dichotomy for standard reachability in affine VASS: it is decidable for VASS with permutations, and undecidable for any other class. This yields a complete and unified complexity landscape of reachability in affine VASS. We also consider the reachability problem parameterized by a fixed affine VASS, rather than a class, and we show that the complexity landscape is arbitrary in this setting.

Affine Extensions of Integer Vector Addition Systems with States

We study the reachability problem for affine $\mathbb{Z}$-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine $\mathbb{Z}$-VASS with the finite-monoid property (afmp-$\mathbb{Z}$-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-$\mathbb{Z}$-VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-$\mathbb{Z}$-VASS reduces to reachability in a $\mathbb{Z}$-VASS whose control-states grow linearly in the size of the matrix monoid. Our construction shows that reachability relations of afmp-$\mathbb{Z}$-VASS are semilinear, and in particular enables us to show that reachability in $\mathbb{Z}$-VASS with transfers and $\mathbb{Z}$-VASS with copies is PSPACE-complete. We then focus on the reachability problem for affine $\mathbb{Z}$-VASS with monogenic monoids: (possibly infinite) matrix monoids generated by a single matrix. We show that, in a particular case, the reachability problem is decidable for this class, disproving a conjecture about affine $\mathbb{Z}$-VASS with infinite matrix monoids we raised in a preliminary version of this paper. We complement this result by presenting an affine $\mathbb{Z}$-VASS with monogenic matrix monoid and undecidable reachability relation.

Context-bounded verification of liveness properties for multithreaded shared-memory programs

We study context-bounded verification of liveness properties of multi-threaded, shared-memory programs, where each thread can spawn additional threads. Our main result shows that context-bounded fair termination is decidable for the model; context-bounded implies that each spawned thread can be context switched a fixed constant number of times. Our proof is technical, since fair termination requires reasoning about the composition of unboundedly many threads each with unboundedly large stacks. In fact, techniques for related problems, which depend crucially on replacing the pushdown threads with finite-state threads, are not applicable. Instead, we introduce an extension of vector addition systems with states (VASS), called VASS with balloons (VASSB), as an intermediate model; it is an infinite-state model of independent interest. A VASSB allows tokens that are themselves markings (balloons). We show that context bounded fair termination reduces to fair termination for VASSB. We show the latter problem is decidable by showing a series of reductions: from fair termination to configuration reachability for VASSB and thence to the reachability problem for VASS. For a lower bound, fair termination is known to be non-elementary already in the special case where threads run to completion (no context switches). We also show that the simpler problem of context-bounded termination is 2EXPSPACE-complete, matching the complexity bound---and indeed the techniques---for safety verification. Additionally, we show the related problem of fair starvation, which checks if some thread can be starved along a fair run, is also decidable in the context-bounded case. The decidability employs an intricate reduction from fair starvation to fair termination. Like fair termination, this problem is also non-elementary.

The Reachability Problem for Petri Nets Is Not Elementary

Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modeling 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, currently, the best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from Symposium on Logic in Computer Science 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, which are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the current best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. We develop a construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R . At the heart of our proof is then a novel gadget, the so-called 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 . Repeatedly composing the factorial amplifier with itself by means of the former construction 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, already establish hardness for h -exponential space for Petri nets with h + 13 counters.

A lower bound for the coverability problem in acyclic pushdown VAS

We investigate the coverability problem for a one-dimensional restriction of pushdown vector addition systems with states. We improve the lower complexity bound to PSpace, even in the acyclic case.

A formal framework for spiking neural P systems

Spiking neural P systems are a class of distributed parallel computing models, inspired by the way in which neurons process information and communicate with each other by means of spikes. In 2007, Freund and Verlan developed a formal framework for P systems to capture most of the essential features of P systems and to define their functioning in a formal way. In this work, we present an extension of the formal framework related to spiking neural P systems by considering the applicability of each rule to be controlled by specific conditions on the current contents of the cells. The main objective of this extension is to also capture spiking neural P systems in the formal framework. Another goal of our extension is to incorporate the notions of input and output. Finally, we also show that in the case of spiking neural P systems, the rules have a rather simple form and in that way spiking neural P systems correspond to vector addition systems where the application of rules is controlled by semi-linear sets.

The ideal view on Rackoff's coverability technique

Well-structured transition systems form a large class of infinite-state systems, for which safety verification is decidable thanks to a generic backward coverability algorithm. However, for several classes of systems, the generic upper bounds one can extract from the algorithm are far from optimal. In particular, in the case of vector addition systems (VAS) and several of their extensions, the known tight upper bounds were rather derived thanks to ad-hoc arguments based on Rackoff's small witness property.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. This reasoning offers a uniform setting for all well-structured transition systems, including branching ones, and we further apply it to several VAS extensions, deriving optimal upper bounds.

Co-Finiteness and Co-Emptiness of Reachability Sets in Vector Addition Systems with States

The coverability and boundedness problems are well-known exponential-space complete problems for vector addition systems with states (or Petri nets). The boundedness problem asks if the reachability set (for a given initial configuration) is finite. Here we consider a dual problem, the co-finiteness problem that asks if the complement of the reachability set is finite; by restricting the question we get the co-emptiness (or universality) problem that asks if all configurations are reachable.

Verification of Hierarchical Artifact Systems

Data-driven workflows, of which IBM’s Business Artifacts are a prime exponent, have been successfully deployed in practice, adopted in industrial standards, and have spawned a rich body of research in academia, focused primarily on static analysis. The present work represents a significant advance on the problem of artifact verification by considering a much richer and more realistic model than in previous work, incorporating core elements of IBM’s successful Guard-Stage-Milestone model. In particular, the model features task hierarchy, concurrency, and richer artifact data. It also allows database key and foreign key dependencies, as well as arithmetic constraints. The results show decidability of verification and establish its complexity, making use of novel techniques including a hierarchy of Vector Addition Systems and a variant of quantifier elimination tailored to our context.

Binary Reachability of Timed-register Pushdown Automata and Branching Vector Addition Systems

Timed-register pushdown automata constitute a very expressive class of automata, whose transitions may involve state, input, and top-of-stack timed registers with unbounded differences. They strictly subsume pushdown timed automata of Bouajjani et al., dense-timed pushdown automata of Abdulla et al., and orbit-finite timed-register pushdown automata of Clemente and Lasota. We give an effective logical characterisation of the reachability relation of timed-register pushdown automata. As a corollary, we obtain a doubly exponential time procedure for the non-emptiness problem. We show that the complexity reduces to singly exponential under the assumption of monotonic time. The proofs involve a novel model of one-dimensional integer branching vector addition systems with states. As a result interesting on its own, we show that reachability sets of the latter model are semilinear and computable in exponential time.

ML, Visibly Pushdown Class Memory Automata, and Extended Branching Vector Addition Systems with States

We prove that the observational equivalence problem for a finitary fragment of the programming langauge ML is recursively equivalent to the reachability problem for extended branching vector addition systems with states (EBVASS). This result has two natural and independent parts. We first prove that the observational equivalence problem is equivalent to the emptiness problem for a new class of class memory automata equipped with a visibly pushdown stack, called Visibly Pushdown Class Memory Automata (VPCMA). Our proof uses the fully abstract game semantics of the language. We then prove that the VPCMA emptiness problem is equivalent to the reachability problem for EBVASS. The results of this article complete our programme to give an automata classification of the ML types with respect to the observational equivalence problem for closed terms.

On the complexity of resource-bounded logics

We revisit decidability results for resource-bounded logics and use decision problems on vector addition systems with states (VASS) in order to establish complexity characterisations of (decidable) model checking problems. We show that the model checking problem for the logic RB±ATL is 2exptime-complete by using recent results on alternating VASS (and in exptime when the number of resources is bounded). Moreover, we establish that the model checking problem for RBTL is expspace-complete. The problem is decidable and of the same complexity for RBTL⁎, proving a new decidability result as a by-product of the approach. When the number of resources is bounded, the problem is in pspace. We also establish that the model checking problem for RB±ATL⁎, the extension of RB±ATL with arbitrary path formulae, is decidable by a reduction to parity games for single-sided VASS (a variant of alternating VASS). Furthermore, we are able to synthesise values for resource parameters. Hence, the paper establishes formal correspondences between model checking problems for resource-bounded logics advocated in the AI literature and decision problems on alternating VASS, paving the way for more applications and cross-fertilizations.

Chordal Networks of Polynomial Ideals

We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. Chordal networks provide a computationally convenient decomposition into simpler (triangular) polynomial sets, while preserving the underlying graphical structure. We show that many interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Chordal networks can be computed for arbitrary polynomial systems using a refinement of the chordal elimination algorithm from [Cifuentes-Parrilo-2016]. Furthermore, they can be effectively used to obtain several properties of the variety, such as its dimension, cardinality, and equidimensional components, as well as an efficient probabilistic test for radical ideal membership. We apply our methods to examples from algebraic statistics and vector addition systems; for these instances, algorithms based on chordal networks outperform existing techniques by orders of magnitude.

Reasoning about Data Repetitions with Counter Systems

We study linear-time temporal logics interpreted over data words with multiple attributes. We restrict the atomic formulas to equalities of attribute values in successive positions and to repetitions of attribute values in the future or past. We demonstrate correspondences between satisfiability problems for logics and reachability-like decision problems for counter systems. We show that allowing/disallowing atomic formulas expressing repetitions of values in the past corresponds to the reachability/coverability problem in Petri nets. This gives us 2EXPSPACE upper bounds for several satisfiability problems. We prove matching lower bounds by reduction from a reachability problem for a newly introduced class of counter systems. This new class is a succinct version of vector addition systems with states in which counters are accessed via pointers, a potentially useful feature in other contexts. We strengthen further the correspondences between data logics and counter systems by characterizing the complexity of fragments, extensions and variants of the logic. For instance, we precisely characterize the relationship between the number of attributes allowed in the logic and the number of counters needed in the counter system.