One-Dimensional Guarded Fragments

Author Emanuel Kieroński

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Emanuel Kieroński
  • University of Wrocław, Poland


The author would like to thank Sebastian Rudolph and the anonymous reviewers for their helpful comments.

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Emanuel Kieroński. One-Dimensional Guarded Fragments. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We call a first-order formula one-dimensional if every maximal block of existential (or universal) quantifiers in it leaves at most one variable free. We consider the one-dimensional restrictions of the guarded fragment, GF, and the tri-guarded fragment, TGF, the latter being a recent extension of GF in which quantification for subformulas with at most two free variables need not be guarded, and which thus may be seen as a unification of GF and the two-variable fragment, FO^2. We denote the resulting formalisms, resp., GF_1, and TGF_1. We show that GF_1 has an exponential model property and NExpTime-complete satisfiability problem (that is, it is easier than full GF). For TGF_1 we show that it is decidable, has the finite model property, and its satisfiability problem is 2-ExpTime-complete (NExpTime-complete in the absence of equality). All the above-mentioned results are obtained for signatures with no constants. We finally discuss the impact of their addition, observing that constants do not spoil the decidability but increase the complexity of the satisfiability problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • guarded fragment
  • two-variable logic
  • satisfiability
  • finite model property


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