First-Order and Temporal Logics for Nested Words

Authors:
  • RARajeev Alur
    Rajeev Alur
  • MAMarcelo Arenas
    Marcelo Arenas
  • PBPablo Barcelo
    Pablo Barcelo
  • KEKousha Etessami
    Kousha Etessami
  • NINeil Immerman
    Neil Immerman
  • LLLeonid Libkin
    Leonid Libkin
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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.