Reasoning about Data Repetitions with Counter Systems

TL;DR

This paper explores the complexity of linear-time temporal logics over data words with multiple attributes, establishing deep connections with counter systems and Petri nets, and providing tight complexity bounds for satisfiability problems.

Contribution

It introduces a new class of counter systems with pointer-accessed counters, and characterizes the complexity of various logic fragments and extensions in relation to these systems.

Findings

2EXPSPACE upper bounds for satisfiability problems

Matching lower bounds via reductions from counter system reachability

Precise complexity characterizations based on attribute and counter counts

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.