The reachability problem in multi-pushdown automata (MPDA), or equivalently, interleaved Dyck reachability, has many applications in static analysis of recursive programs. An example is safety verification of multithreaded recursive programs with shared memory. Since these problems are undecidable, the literature contains many decidable (and efficient) underapproximations of MPDA. A uniform framework that captures many of these underapproximations is that of bounded treewidth: To each execution of the MPDA, we associate a graph; then we consider the subset of all graphs that have a treewidth at most k , for some constant k . In fact, bounding treewidth is a generic approach to obtain classes of systems with decidable reachability, even beyond MPDA underapproximations. The resulting systems are also called MSO-definable bounded-treewidth systems. While bounded treewidth is a powerful tool for reachability and similar types of analysis, the word languages (i.e. action sequences corresponding to executions) of these systems remain far from understood. For the slight restriction of bounded special treewidth, or “bounded-stw” (which is equivalent to bounded treewidth on MPDA, and even includes all bounded-treewidth systems studied in the literature), this work reveals a connection with multiple context-free languages (MCFL), a concept from computational linguistics. We show that the word languages of MSO-definable bounded-stw systems are exactly the MCFL. We exploit this connection to provide an optimal algorithm for computing downward closures for MSO-definable bounded-stw systems. Computing downward closures is a notoriously difficult task that has many applications in the verification of complex systems: As an example application, we show that in programs with dynamic spawning of MSO-definable bounded-stw processes, safety verification has the same complexity as in the case of processes with sequential recursive processes.

We lift metrics over words to metrics over word-to-word transductions, by defining the distance between two transductions as the supremum of the distances of their respective outputs over all inputs. This allows to compare transducers beyond equivalence. Two transducers are close (resp. $k$-close) with respect to a metric if their distance is finite (resp. at most $k$). Over integer-valued metrics computing the distance between transducers is equivalent to deciding the closeness and $k$-closeness problems. For common integer-valued edit distances such as, Hamming, transposition, conjugacy and Levenshtein family of distances, we show that the closeness and the $k$-closeness problems are decidable for functional transducers. Hence, the distance with respect to these metrics is also computable. Finally, we relate the notion of distance between functions to the notions of diameter of a relation and index of a relation in another. We show that computing edit distance between functional transducers is equivalent to computing diameter of a rational relation and both are a specific instance of the index problem of rational relations.

We study the satisfiability of string constraints where context-free membership constraints may be imposed on variables. Additionally a variable may be constrained to be a subword of a word obtained by shuffling variables and their transductions. The satisfiability problem is known to be undecidable even without rational transductions. It is known to be NExptime-complete without transductions, if the subword relations between variables do not have a cyclic dependency between them. We show that the satisfiability problem stays decidable in this fragment even when rational transductions are added. It is 2NExptime-complete with context-free membership, and NExptime-complete with only regular membership. For the lower bound we prove a technical lemma that is of independent interest: The length of the shortest word in the intersection of a pushdown automaton (of size $O(n)$) and $n$ finite-state automata (each of size $O(n)$) can be double exponential in $n$.

We study verification of reachability properties over Communicating Datalog Programs (CDPs), which are networks of relational nodes connected through unordered channels and running Datalog-like computations. Each node manipulates a local state database (DB), depending on incoming messages and additional input DBs from external services. Decidability of verification for CDPs has so far been established only under boundedness assumptions on the state and channel sizes, showing at the same time undecidability of reachability for unbounded states with only two unary relations or unbounded channels with a single binary relation. The goal of this paper is to study the open case of CDPs with bounded states and unbounded channels, under the assumption that channels carry unary relations only. We discuss the significance of the resulting model and prove the decidability of verification of variants of reachability, captured in fragments of first-order CTL. We do so through a novel reduction to coverability problems in a class of high-level Petri Nets that manipulate unordered data identifiers. We study the tightness of our results, showing that minor generalizations of the considered reachability properties yield undecidability of verification, both for CDPs and the corresponding Petri Net model.

We consider a variant of string constraints given by membership constraints in context-free languages and subword relation between variables. The satisfiability problem for this variant turns out to be undecidable. We consider a fragment in which the subword-order constraints do not impose any cyclic dependency between variables. We show that this fragment is NexpTime-complete. As an application of our result, we settle the complexity of control state reachability in acyclic lossy channel pushdown systems, an important distributed system model. The problem was shown to be decidable in [8]. However, no elementary upper bound was known. We show that this problem is NexpTime-complete.

We consider weighted tiling systems to represent functions from graphs to a commutative semiring such as the Natural semiring or the Tropical semiring. The system labels the nodes of a graph by its states, and checks if the neighbourhood of every node belongs to a set of permissible tiles, and assigns a weight accordingly. The weight of a labeling is the semiring-product of the weights assigned to the nodes, and the weight of the graph is the semiring-sum of the weights of labelings. We show that we can model interesting algorithmic questions using this formalism - like computing the clique number of a graph or computing the permanent of a matrix. The evaluation problem is, given a weighted tiling system and a graph, to compute the weight of the graph. We study the complexity of the evaluation problem and give tight upper and lower bounds for several commutative semirings. Further we provide an efficient evaluation algorithm if the input graph is of bounded tree-width.

A prominent research direction of the database theory community is to develop techniques for verification of database-driven systems operating over relational and numerical data. Along this line, we lift the framework of database manipulating systems \citeAbdullaAAMR-pods-16 which handle relational data to also accommodate numerical data and the natural order on them. We study an under-approximation called recency bounding under which the most basic verification problem --reachability, is decidable. Even under this under-approximation the reachability space is infinite in multiple dimensions -- owing to the unbounded sizes of the active domain, the unbounded numerical domain it has access to, and the unbounded length of the executions. We show that, nevertheless, reachability is ExpTime complete. Going beyond reachability to LTL model checking renders verification undecidable.

We extend the classical model of multi-pushdown systems by considering systems that operate on a finite set of variables ranging over natural numbers. The conditions on variables are defined via gap-order constraints that allow to compare variables for equality, or to check that the gap between the values of two variables exceeds a given natural number. Furthermore, each message inside a stack is equipped with a data item representing its value. When a message is pushed to the stack, its value may be defined by a variable. When a message is popped, its value may be copied to a variable. Thus, we obtain a system that is infinite in multiple dimensions, namely we have a number of stacks that may contain an unbounded number of messages each of which is equipped with a natural number. It is well-known that the verification of any non-trivial property of multi-pushdown systems is undecidable, even for two stacks and for a finite data-domain. In this paper, we show the decidability of the reachability problem for the classes of data multi-pushdown system that admit a bounded split-width (or equivalently a bounded tree-width). As an immediate consequence, we obtain decidability for several subclasses of data multi-pushdown systems. These include systems with single stacks, restricted ordering policies on stack operations, bounded scope, bounded phase, and bounded context switches.

We extend the classical model of lossy channel systems by considering systems that operate on a finite set of variables ranging over an infinite data domain. Furthermore, each message inside a channel is equipped with a data item representing its value. Although we restrict the model by allowing the variables to be only tested for (dis-)equality, we show that the state reachability problem is undecidable. In light of this negative result, we consider bounded-phase reachability, where the processes are restricted to performing either send or receive operations during each phase. We show decidability of state reachability in this case by computing a symbolic encoding of the set of system configurations that are reachable from a given configuration.

We propose a formalism to model database-driven systems, called database manipulating systems (DMS). The actions of a (DMS) modify the current instance of a relational database by adding new elements into the database, deleting tuples from the relations and adding tuples to the relations. The elements which are modified by an action are chosen by (full) first-order queries. (DMS) is a highly expressive model and can be thought of as a succinct representation of an infinite state relational transition system, in line with similar models proposed in the literature. We propose monadic second order logic (MSO-FO) to reason about sequences of database instances appearing along a run. Unsurprisingly, the linear-time model checking problem of (DMS) against (MSO-FO) is undecidable. Towards decidability, we propose under-approximate model checking of (DMS), where the under-approximation parameter is the "bound on recency". In a k-recency-bounded run, only the most recent k elements in the current active domain may be modified by an action. More runs can be verified by increasing the bound on recency. Our main result shows that recency-bounded model checking of (DMS) against (MSO-FO) is decidable, by a reduction to the satisfiability problem of MSO over nested words.

We introduce an automata-theoretic method for the verification of distributed algorithms running on ring networks. In a distributed algorithm, an arbitrary number of processes cooperate to achieve a common goal (e.g., elect a leader). Processes have unique identifiers (pids) from an infinite, totally ordered domain. An algorithm proceeds in synchronous rounds, each round allowing a process to perform a bounded sequence of actions such as send or receive a pid, store it in some register, and compare register contents wrt. the associated total order. An algorithm is supposed to be correct independently of the number of processes. To specify correctness properties, we introduce a logic that can reason about processes and pids. Referring to leader election, it may say that, at the end of an execution, each process stores the maximum pid in some dedicated register. Since the verification of distributed algorithms is undecidable, we propose an underapproximation technique, which bounds the number of rounds. This is an appealing approach, as the number of rounds needed by a distributed algorithm to conclude is often exponentially smaller than the number of processes. We provide an automata-theoretic solution, reducing model checking to emptiness for alternating two-way automata on words. Overall, we show that round-bounded verification of distributed algorithms over rings is PSPACE-complete.

There are two schools of thought on reasoning about distributed systems: one following interleaving based semantics, and one following partial-order/graph based semantics. This paper compares these two approaches and argues in favour of the latter. An introductory treatment of the split-width technique is also provided.