On Logics with Two Variables

@article{Grdel1999OnLW,
  title={On Logics with Two Variables},
  author={Erich Gr{\"a}del and Martin Otto},
  journal={Theor. Comput. Sci.},
  year={1999},
  volume={224},
  pages={73-113}
}
  • E. GrädelM. Otto
  • Published 6 August 1999
  • Computer Science, Philosophy
  • Theor. Comput. Sci.

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...

References

SHOWING 1-10 OF 41 REFERENCES

Undecidability results on two-variable logics

It is shown that going beyond L2 by adding any one of the following leads to an undecidable logic: very weak forms of recursion, such as transitive closure or monadic fixed-point operations.

Why is Modal Logic So Robustly Decidable?

  • Moshe Y. Vardi
  • Philosophy, Computer Science
    Descriptive Complexity and Finite Models
  • 1996
It is argued that the robust decidability of modal logic can be explained by the so-called tree- model property, and it is shown how the tree-model property leads to automata-based decision procedures.

On the Restraining Power of Guards

  • E. Grädel
  • Mathematics, Computer Science
    Journal of Symbolic Logic
  • 1999
It is proved that the satisfiability problems for the guarded fragment and the loosely guarded fragment of first-order logic are complete for deterministic double exponential time and that some natural, modest extensions of the guarded fragments are undecidable.

On the boundedness problem for two-variable first-order logic

  • Phokion G. KolaitisM. Otto
  • Mathematics
    Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226)
  • 1998
This paper asserts that the boundedness problem for FO/sup 2/ is undecidable, even when restricted to negation-free and equality-free formulas /spl phi/(X, x) in which x is the only free variable and X is a monadic relation symbol that occurs within the scopes of universal quantifiers only.

Bounded variable logics: two, three, and more

  • M. Otto
  • Mathematics
    Arch. Math. Log.
  • 1999
This paper deals with a translation that associates equivalence of structures in the k-variable fragments with bisimulation equivalence between derived structures and relates some interesting issues for the cases of an arbitrary number of variables to the case of just three variables.

Model Checking and Transitive-Closure Logic

We give a linear-time algorithm to translate any formula from computation tree logic (CTL or CTL*) into an equivalent expression in a variable-confined fragment of transitive-closure logic FO(TC).

On Model-Checking for Fragments of µ-Calculus

It is shown that the logic L2 is as expressive as ECTL* given in [13], and the model checking problem for the μ-calculus is equivalent to the non-emptiness problem of parity tree automata.

Automatic Veriication of Nite-state Concurrent Systems Using Temporal-logic Speciications. Acm

An algorithm for checking whether a GSMP satisses its TCTL-speciication is presented, which combines model- checking algorithm of this paper with model-checking for discrete-time Markov chains and ways to cope with the PSPACE complexity of the problem are devised.

An automata-theoretic approach to branching-time model checking

It is shown that alternating tree automata are the key to a comprehensive automata-theoretic framework for branching temporal logics, and can be used to obtain optimal decision procedures and make it possible to derive optimal model-checking algorithms.

Complexity of two-variable logic with counting

It is proved that the problem of satisfiability of sentences of C/sub 1//sup 2/ is NEXPTIME-complete, which easily implies that the satisfiability problem for C/sup2/ is in non-deterministic, doubly exponential time.