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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Springer, Heidelberg (1996)
Dong, G., Su, J.: Space-bounded foies. In: Principles of Database Systems, pp. 139–150. ACM Press, New York (1995)
Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. In: Proc. 29th IEEE Symposium on Foundations of Computer Science, pp. 328–337. IEEE Computer Society Press, Los Alamitos (1988)
Grädel, E.: On the decision problem for two-variable first-order logic. Bulletin of Symbolic Logic 3, 53–69 (1997)
Grädel, E., Otto, M., Rosen, E.: Undecidability results on two-variable logics. Archive of Math. Logic 38, 313–354 (1999)
Grädel, E., Walukiewicz, I.: Guarded fixed point logic. In: Proc. 14th IEEE Symposium on Logic in Computer Science, pp. 45–54. IEEE Computer Society Press, Los Alamitos (1999)
Henriksen, J.G., Jensen, J., Jørgensen, M., Klarlund, N., Paige, B., Rauhe, T., Sandholm, A.: Mona: Monadic second-order logic in practice. In: Brinksma, E., Steffen, B., Cleaveland, W.R., Larsen, K.G., Margaria, T. (eds.) TACAS 1995. LNCS, vol. 1019. Springer, Heidelberg (1995)
Immerman, N.: Descriptive Complexity. Springer, Heidelberg (1999)
Immerman, N., Rabinovich, A., Reps, T., Sagiv, M., Yorsh, G.: Verification via structure simulation. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 281–294. Springer, Heidelberg (2004)
Meyer, A.R.: Weak monadic second-order theory of successor is not elementary recursive. In: Logic Colloquium. Proc. Symposium on Logic, Boston, pp. 132–154 (1975)
Mortimer, M.: On languages with two variables. Zeitschr. f. math. Logik u. Grundlagen d. Math 21, 135–140 (1975)
Papadimitriou, C.: Computational Complexity. Addison Wesley, Reading (1994)
Rabin, M.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141, 1–35 (1969)
Sagiv, M., Reps, T., Wilhelm, R.: Parametric shape analysis via 3-valued logic. In: Trans. On Prog. Lang. and Syst., pp. 217–298 (2002)
Yorsh, G., Reps, T., Sagiv, M.: Symbolically computing most-precise abstract operations for shape analysis. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 530–545. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Immerman, N., Rabinovich, A., Reps, T., Sagiv, M., Yorsh, G. (2004). The Boundary Between Decidability and Undecidability for Transitive-Closure Logics. In: Marcinkowski, J., Tarlecki, A. (eds) Computer Science Logic. CSL 2004. Lecture Notes in Computer Science, vol 3210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30124-0_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-30124-0_15
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23024-3
Online ISBN: 978-3-540-30124-0
eBook Packages: Springer Book Archive