Reasoning about Data Repetitions with Counter Systems

Abstract

Abstract. 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.

Logics of Repeating Values on Data Trees and Branching Counter Systems

“…More concretely, the data tests are formulas of the form u ≈ EFv, stating that the data value stored in attribute (also called variable here) u of the current node is equal to the data value stored in attribute v of some descendant. This logic of repeating values, or LRV, has been the center of a line of investigation studied in [6,7] on data words, evidencing tight correspondences between reachability problems for Vector Addition Systems and the satisfiability problem. The current work pursues this question further, exhibiting connections between the satisfiability problem of LRV over data trees and the bottom-up coverability problem for branching counter systems.…”

“…3 Other works have considered different simplifications of these structures, either having only one data value per node (e.g., [2]) or ignoring the label (e.g., [7]). …”

“…A wealth of specification formalisms on these structures (either for data trees or its 'word' version, data words) have been introduced, stemming from automata [25,28], first-order logic [2,19,16,4], XPath [20,18,15,14,13], or temporal logics [9,24,22,11,7,21]. In full generality, most formalisms lead to undecidable reasoning problems and a well-known research trend consists of finding a good trade-off between expressiveness and decidability.…”

“…However, the techniques needed to encode the satisfiability of the logic into a counter system are not simple extensions from the ones provided on data words. The reason for this difficulty is manyfold: a) the fact that now the future is non-linear in addition to the possibility of having a data value repeating at several descendants in different variables, makes the techniques of [7] for propagating values of configurations impractical; b) further, this seems to be impossible for the case of data trees, and we could only show a reduction for the fragment LRV D ; c) in order to reduce the satisfiability problem for the full logic we will need to augment the power of branching VASS with the possibility to 'merge' counters in a more powerful way, somewhat akin to what has been done for encoding the satisfiability for FO 2 [19].…”

On Functions Weakly Computable by Petri Nets and Vector Addition Systems

“…There are several extensions of Petri nets for which reachability (or coverability or termination) remains decidable: pushdown VASSes [23], nets with nested zerotests [27], recursive VASSes [4] and Branching VASSes [6], VASSes with pointers to counters [5], etc. In many cases, it is not known how these extensions compare in expressive power and in complexity.…”