The boundary between decidability and undecidability for transitive-closure logics

Neil Immerman, Alex Rabinovich, Tom Reps, Mooly Sagiv, Greta Yorsh

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

To reason effectively about programs, it is important to have some version of a transitive-closure operator so that we can describe such notions as the set of nodes reachable from a program's variables. On the other hand, with a few notable exceptions, adding transitive closure to even very tame logics makes them undecidable. In this paper, we explore the boundary between decidability and undecidability for transitive-closure logics. Rabin proved that the monadic second-order theory of trees is decidable, although the complexity of the decision procedure is not elementary. If we go beyond trees, however, undecidability comes immediately. We have identified a rather weak language called V(DTC+ [E]) that goes beyond trees, includes a version of transitive closure, and is decidable. We show that satisfiability of 3V(DTC +[E]) is NEXPTIME complete. We furthermore show that essentially any reasonable extension of 3V(DTC+[E]) is undecidable. Our main contribution is to demonstrate these sharp divisions between decidable and undecidable. We also compare the complexity and expressibility of Er(DTC +[E]) with related decidable languages including MSO(trees) and guarded fixed point logics. We mention possible applications to systems some of us are building that use decidable logics to reason about programs.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
EditorsJerzy Marcinkowski, Andrzej Tarlecki
PublisherSpringer Verlag
Pages160-174
Number of pages15
ISBN (Print)3540230246, 9783540230243
DOIs
StatePublished - 2004

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume3210
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

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