Population protocols are a distributed computation model in which a collection of anonymous, finite-state agents interact in randomly chosen pairs and update their states according to a fixed transition function. The computation is defined by the eventual stabilization of the population to a consensus that represents the output. In practice, it is natural to allow each agent to carry a unique identifier and compare it with that of another agent before interacting. We model this extension by having agents be totally ordered and interactions between two agents to be fireable only if their pair of identifiers falls in some condition set. For instance, PP[<] allows for two agents to interact only if the first one appears before the second one. We study population protocols over ordered agents PP[𝒩] where 𝒩 is a set of predicates available to restrict transition firing. We also study IO-PP[𝒩], the immediate observation fragment of PP[𝒩] where only one agent changes state per interaction. Our main result is that IO-PP[<] recognizes exactly the unambiguous star-free languages, which admits many other characterizations, such as two-variable first-order logic or two-way deterministic partially-ordered automata. We also provide a logic and an automaton model that fits in PP[<]. We further show that if the successor predicate appears in a set 𝒩 of NSPACE(n)-computable predicates, then IO-PP[𝒩] = PP[𝒩] = NSPACE(n). Finally, we investigate the problem of deciding whether a given population protocol always stabilizes to a consensus. While this problem is decidable for unordered population protocols, we show that this is undecidable already for PP[<] and IO-PP[+1], but conditionally decidable for IO-PP[<].

Abstract Ordered binary decision diagrams (OBDDs) are a fundamental data structure for the manipulation of Boolean functions, with strong applications to finite-state symbolic model checking. OBDDs allow for efficient algorithms using top-down dynamic programming. From an automata-theoretic perspective, OBDDs essentially are minimal deterministic finite automata recognizing languages whose words have a fixed length (the arity of the Boolean function). We introduce weakly acyclic diagrams (WADs), a generalization of OBDDs that maintains their algorithmic advantages, but can also represent infinite languages. We develop the theory of WADs and show that they can be used for symbolic model checking of various models of infinite-state systems.

Workflow nets are a well-established variant of Petri nets for the modeling of process activities such as business processes. The standard correctness notion of workflow nets is soundness, which comes in several variants. Their decidability was shown decades ago, but their complexity was only identified recently. In this work, we are primarily interested in two popular variants: 1-soundness and generalised soundness.

Leroux has proved that unreachability in Petri nets can be witnessed by a Presburger separator, i.e. if a marking $\vec{m}_\text{src}$ cannot reach a marking $\vec{m}_\text{tgt}$, then there is a formula $\varphi$ of Presburger arithmetic such that: $\varphi(\vec{m}_\text{src})$ holds; $\varphi$ is forward invariant, i.e., $\varphi(\vec{m})$ and $\vec{m} \rightarrow \vec{m}'$ imply $\varphi(\vec{m}'$); and $\neg \varphi(\vec{m}_\text{tgt})$ holds. While these separators could be used as explanations and as formal certificates of unreachability, this has not yet been the case due to their worst-case size, which is at least Ackermannian, and the complexity of checking that a formula is a separator, which is at least exponential (in the formula size). We show that, in continuous Petri nets, these two problems can be overcome. We introduce locally closed separators, and prove that: (a) unreachability can be witnessed by a locally closed separator computable in polynomial time; (b) checking whether a formula is a locally closed separator is in NC (so, simpler than unreachability, which is P-complete). We further consider the more general problem of (existential) set-to-set reachability, where two sets of markings are given as convex polytopes. We show that, while our approach does not extend directly, we can efficiently certify unreachability via an altered Petri net.

Population protocols form a well-established model of computation of passively mobile anonymous agents with constant-size memory. It is well known that population protocols compute Presburger-definable predicates, such as absolute majority and counting predicates. In this work, we initiate the study of population protocols operating over arbitrarily large data domains. More precisely, we introduce population protocols with unordered data as a formalism to reason about anonymous crowd computing over unordered sequences of data. We first show that it is possible to determine whether an unordered sequence from an infinite data domain has a datum with absolute majority. We then establish the expressive power of the "immediate observation" restriction of our model, namely where, in each interaction, an agent observes another agent who is unaware of the interaction.

Constant-rate multi-mode systems (MMS) are hybrid systems with finitely many modes and real-valued variables that evolve over continuous time according to mode-specific constant rates. We introduce a variant of linear temporal logic (LTL) for MMS, and we investigate the complexity of the model-checking problem for syntactic fragments of LTL. We obtain a complexity landscape where each fragment is either P-complete, NP-complete or undecidable. These results generalize and unify several results on MMS and continuous counter systems.

We study the reachability problem for continuous one-counter automata, COCA for short. In such automata, transitions are guarded by upper- and lower-bound tests against the counter value. Additionally, the counter updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor between zero and one. Our three main results are as follows: we prove (1) that the reachability problem for COCA with global upper- and lower-bound tests is in NC 2 ; (2) that, in general, the problem is decidable in polynomial time; and (3) that it is NP-complete for COCA with parametric counter updates and bound tests.

Abstract Workflow nets are a well-established mathematical formalism for the analysis of business processes arising from either modeling tools or process mining. The central decision problems for workflow nets are k-soundness, generalised soundness and structural soundness. Most existing tools focus on k-soundness. In this work, we propose novel scalable semi-procedures for generalised and structural soundness. This is achieved via integral and continuous Petri net reachability relaxations. We show that our approach is competitive against state-of-the-art tools.

Abstract Leroux has proved that unreachability in Petri nets can be witnessed by a Presburger separator, i.e. if a marking $$\boldsymbol{m}_\text {src}$$ m src cannot reach a marking $$\boldsymbol{m}_\text {tgt}$$ m tgt , then there is a formula $$\varphi $$ φ of Presburger arithmetic such that: $$\varphi (\boldsymbol{m}_\text {src})$$ φ ( m src ) holds; $$\varphi $$ φ is forward invariant, i.e., $$\varphi (\boldsymbol{m})$$ φ ( m ) and $$\boldsymbol{m} \rightarrow \boldsymbol{m}'$$ m → m ′ imply $$\varphi (\boldsymbol{m}'$$ φ ( m ′ ); and $$\lnot \varphi (\boldsymbol{m}_\text {tgt})$$ ¬ φ ( m tgt ) holds. While these separators could be used as explanations and as formal certificates of unreachability, this has not yet been the case due to their (super-)Ackermannian worst-case size and the (super-)exponential complexity of checking that a formula is a separator. We show that, in continuous Petri nets, these two problems can be overcome. We introduce locally closed separators, and prove that: (a) unreachability can be witnessed by a locally closed separator computable in polynomial time; (b) checking whether a formula is a locally closed separator is in NC (so, simpler than unreachablity, which is P-complete).

Workflow nets are a popular variant of Petri nets that allow for the algorithmic formal analysis of business processes. The central decision problems concerning workflow nets deal with soundness, where the initial and final configurations are specified. Intuitively, soundness states that from every reachable configuration one can reach the final configuration. We settle the widely open complexity of the three main variants of soundness: classical, structural and generalised soundness. The first two are EXPSPACE-complete, and, surprisingly, the latter is PSPACE-complete, thus computationally simpler.

We study the reachability problem for continuous one-counter automata, COCA for short. In such automata, transitions are guarded by upper and lower bound tests against the counter value. Additionally, the counter updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor between zero and one. Our three main results are as follows: (1) We prove that the reachability problem for COCA with global upper and lower bound tests is in NC2; (2) that, in general, the problem is decidable in polynomial time; and (3) that it is decidable in the polynomial hierarchy for COCA with parametric counter updates and bound tests.

Abstract Numerous tasks in program analysis and synthesis reduce to deciding reachability in possibly infinite graphs such as those induced by Petri nets. However, the Petri net reachability problem has recently been shown to require non-elementary time, which raises questions about the practical applicability of Petri nets as target models. In this paper, we introduce a novel approach for efficiently semi-deciding the reachability problem for Petri nets in practice. Our key insight is that computationally lightweight over-approximations of Petri nets can be used as distance oracles in classical graph exploration algorithms such as $$\mathsf {A}^{*}$$ A ∗ and greedy best-first search. We provide and evaluate a prototype implementation of our approach that outperforms existing state-of-the-art tools, sometimes by orders of magnitude, and which is also competitive with domain-specific tools on benchmarks coming from program synthesis and concurrent program analysis.

Abstract Population protocols are a well established model of computation by anonymous, identical finite-state agents. A protocol is well-specified if from every initial configuration, all fair executions of the protocol reach a common consensus. The central verification question for population protocols is the well-specification problem: deciding if a given protocol is well-specified. Esparza et al. have recently shown that this problem is decidable, but with very high complexity: it is at least as hard as the Petri net reachability problem, which is -hard, and for which only algorithms of non-primitive recursive complexity are currently known. In this paper we introduce the class $${ WS}^3$$ WS 3 of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that $${ WS}^3$$ WS 3 has the same computational power as general well-specified protocols, and captures standard protocols from the literature. Moreover, we show that the membership and correctness problems for $${ WS}^3$$ WS 3 reduce to solving boolean combinations of linear constraints over $${\mathbb {N}}$$ N . This allowed us to develop the first software able to automatically prove correctness for all of the infinitely many possible inputs.

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.

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.

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.

We present a sound and complete method for the verification of qualitative liveness properties of replicated systems under stochastic scheduling. These are systems consisting of a finite-state program, executed by an unknown number of indistinguishable agents, where the next agent to make a move is determined by the result of a random experiment. We show that if a property of such a system holds, then there is always a witness in the shape of a Presburger stage graph : a finite graph whose nodes are Presburger-definable sets of configurations. Due to the high complexity of the verification problem (non-elementary), we introduce an incomplete procedure for the construction of Presburger stage graphs, and implement it on top of an SMT solver. The procedure makes extensive use of the theory of well-quasi-orders, and of the structural theory of Petri nets and vector addition systems. We apply our results to a set of benchmarks, in particular to a large collection of population protocols, a model of distributed computation extensively studied by the distributed computing community.

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 w.r.t. arbitrary classes of affine operations.

Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula $φ$ of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with $2^{O(\text{poly}(|φ|))}$ states that computes $φ$. More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree. In this paper, we prove that every formula $φ$ of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with $O(\text{poly}(|φ|))$ states. Our proof is based on several new constructions, which may be of independent interest. Given a formula $φ$ of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with $O(|φ|^3)$ leaders) that computes $φ$; this completes the work initiated in [STACS'18], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes $φ$. Our last construction gets rid of this leader for small inputs.

Population protocols are a formal model of computation by identical, anonymous mobile agents interacting in pairs. Their computational power is rather limited: Angluin et al. have shown that they can only compute the predicates over $\mathbb{N}^k$ expressible in Presburger arithmetic. For this reason, several extensions of the model have been proposed, including the addition of devices called cover-time services, absence detectors, and clocks. All these extensions increase the expressive power to the class of predicates over $\mathbb{N}^k$ lying in the complexity class NL when the input is given in unary. However, these devices are difficult to implement, since they require that an agent atomically receives messages from all other agents in a population of unknown size; moreover, the agent must know that they have all been received. Inspired by the work of the verification community on Emerson and Namjoshi's broadcast protocols, we show that NL-power is also achieved by extending population protocols with reliable broadcasts, a simpler, standard communication primitive.

We introduce P eregrine , the first tool for the analysis and parameterized verification of population protocols. Population protocols are a model of computation very much studied by the distributed computing community, in which mobile anonymous agents interact stochastically to achieve a common task. P eregrine allows users to design protocols, to simulate them both manually and automatically, to gather statistics of properties such as convergence speed, and to verify correctness automatically. This paper describes the features of P eregrine and their implementation.

In this interactive paper, which you should preferably read connected to the Internet, the Black Ninjas introduce you to population protocols, a fundamental model of distributed computation, and to recent work by the authors and their colleagues on their automatic verification.

Population protocols are a well established model of distributed computation by mobile finite-state agents with very limited storage. A classical result establishes that population protocols compute exactly predicates definable in Presburger arithmetic. We initiate the study of the minimal amount of memory required to compute a given predicate as a function of its size. We present results on the predicates $x \geq n$ for $n \in \mathbb{N}$, and more generally on the predicates corresponding to systems of linear inequalities. We show that they can be computed by protocols with $O(\log n)$ states (or, more generally, logarithmic in the coefficients of the predicate), and that, surprisingly, some families of predicates can be computed by protocols with $O(\log\log n)$ states. We give essentially matching lower bounds for the class of 1-aware protocols.

This paper studies sets of rational numbers definable by continuous Petri nets and extensions thereof. First, we identify a polynomial-time decidable fragment of existential FO(ℚ,+,<;) and show that the sets of rationals definable in this fragment coincide with reachability sets of continuous Petri nets. Next, we introduce and study continuous vector addition systems with states (CVASS), which are vector addition systems with states in which counters may hold non-negative rational values, and in which the effect of a transition can be scaled by a positive rational number smaller or equal to one. This class strictly generalizes continuous Petri nets by additionally allowing for discrete control-state information. We prove that reachability sets of CVASS are equivalent to the sets of rational numbers definable in existential FO(ℚ,+,<;) from which we can conclude that reachability in CVASS is NP-complete. Finally, our results explain and yield as corollaries a number of polynomial-time algorithms for decision problems that have recently been studied in the literature.

Population protocols are a well established model of computation by anonymous, identical finite state agents. A protocol is well-specified if from every initial configuration, all fair executions of the protocol reach a common consensus. The central verification question for population protocols is the well-specification problem: deciding if a given protocol is well-specified. Esparza et al. have recently shown that this problem is decidable, but with very high complexity: it is at least as hard as the Petri net reachability problem, which is EXPSPACE-hard, and for which only algorithms of non-primitive recursive complexity are currently known.

The well-quasi-ordering (i.e., a well-founded quasi-ordering such that all antichains are finite) that defines well-structured transition systems (WSTS) is shown not to be the weakest hypothesis that implies decidability of the coverability problem. We show coverability decidable for monotone transition systems that only require the absence of infinite antichains and call well behaved transitions systems (WBTS) the new strict superclass of the class of WSTS that arises. By contrast, we confirm that boundedness and termination are undecidable for WBTS under the usual hypotheses, and show that stronger monotonicity conditions can enforce decidability. Proofs are similar or even identical to existing proofs but the surprising message is that a hypothesis implicitely assumed minimal for twenty years in the theory of WSTS can meaningfully be relaxed, allowing more orderings to be handled in an abstract way. Comment: 19 pages, 3 figures

Continuous Petri nets are a relaxation of classical discrete Petri nets in which transitions can be fired a fractional number of times, and consequently places may contain a fractional number of tokens. Such continuous Petri nets are an appealing object to study, since they over-approximate the set of reachable configurations of their discrete counterparts, and their reachability problem is known to be decidable in polynomial time. The starting point of this article is to show that the reachability relation for continuous Petri nets is definable by a sentence of linear size in the existential theory of the rationals with addition and order. Using this characterization, we obtain decidability and complexity results for a number of classical decision problems for continuous Petri nets. In particular, we settle the open problem about the precise complexity of reachability set inclusion. Finally, we show how continuous Petri nets can be incorporated inside the classical backward coverability algorithm for discrete Petri nets as a pruning heuristic to tackle the symbolic state explosion problem. The cornerstone of the approach we present is that our logical characterization enables us to leverage the power of modern SMT-solvers to yield a highly performant and robust decision procedure for coverability in Petri nets. We demonstrate the applicability of our approach on a set of standard benchmarks from the literature.

Known to be decidable since 1981, there still remains a huge gap between the best known lower and upper bounds for the reach ability problem for vector addition systems with states (VASS). Here the problem is shown PSPACE-complete in the two-dimensional case, vastly improving on the doubly exponential time bound established in 1986 by Howell, Rosier, Huynh and Yen. Cover ability and bounded ness for two-dimensional VASS are also shown PSPACE-complete, and reach ability in two-dimensional VASS and in integer VASS under unary encoding are considered.