The Boundary Between Decidability and Undecidability for Transitive-Closure Logics

  • Neil Immerman
  • Alex Rabinovich
  • Tom Reps
  • Mooly Sagiv
  • Greta Yorsh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3210)

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 ∃ ∀ (DTC + [E])that goes beyond trees, includes a version of transitive closure, and is decidable. We show that satisfiability of ∃ ∀ (DTC + [E]) is NEXPTIME complete. We furthermore show that essentially any reasonable extension of ∃ ∀ (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 ∃ ∀ (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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Neil Immerman
    • 1
  • Alex Rabinovich
    • 2
  • Tom Reps
    • 3
  • Mooly Sagiv
    • 2
  • Greta Yorsh
    • 2
  1. 1.Dept. of Comp. Sci.Univ. of Massachusetts 
  2. 2.School of Comp. Sci.Tel Aviv Univ. 
  3. 3.Comp. Sci. Dept.Univ. of Wisconsin 

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