First-Order and Temporal Logics for Nested Words

TL;DR

This paper introduces new temporal logics for nested words, capturing their call-return structure, and proves their expressive completeness and computational complexity, advancing formal verification methods for procedural programs and tree-structured data.

Contribution

It presents two novel temporal logics for nested words, proves their first-order expressiveness, and analyzes their satisfiability and model-checking complexities.

Findings

NWTL satisfiability is EXPTIME-complete

Model-checking is polynomial in model size and exponential in formula size

First-order logic over nested words has the three-variable property

Abstract

Nested words are a structured model of execution paths in procedural programs, reflecting their call and return nesting structure. Finite nested words also capture the structure of parse trees and other tree-structured data, such as XML. We provide new temporal logics for finite and infinite nested words, which are natural extensions of LTL, and prove that these logics are first-order expressively-complete. One of them is based on adding a "within" modality, evaluating a formula on a subword, to a logic CaRet previously studied in the context of verifying properties of recursive state machines (RSMs). The other logic, NWTL, is based on the notion of a summary path that uses both the linear and nesting structures. For NWTL we show that satisfiability is EXPTIME-complete, and that model-checking can be done in time polynomial in the size of the RSM model and exponential in the size of the NWTL formula (and is also EXPTIME-complete). Finally, we prove that first-order logic over nested words has the three-variable property, and we present a temporal logic for nested words which is complete for the two-variable fragment of first-order.