The Boundary Between Decidability and Undecidability for Transitive-Closure Logics
- Neil ImmermanAffiliated withDept. of Comp. Sci., Univ. of Massachusetts
- , Alex RabinovichAffiliated withSchool of Comp. Sci., Tel Aviv Univ.
- , Tom RepsAffiliated withComp. Sci. Dept., Univ. of Wisconsin
- , Mooly SagivAffiliated withSchool of Comp. Sci., Tel Aviv Univ.
- , Greta YorshAffiliated withSchool of Comp. Sci., Tel Aviv Univ.
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.
- The Boundary Between Decidability and Undecidability for Transitive-Closure Logics
- Book Title
- Computer Science Logic
- Book Subtitle
- 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 20-24, 2004. Proceedings
- pp 160-174
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
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- Editor Affiliations
- 16. Wroclaw University
- 17. Polish Academy of Sciences, Institute of Computer Science
- Author Affiliations
- 18. Dept. of Comp. Sci., Univ. of Massachusetts,
- 19. School of Comp. Sci., Tel Aviv Univ.,
- 20. Comp. Sci. Dept., Univ. of Wisconsin,
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