Download PDFOpen PDF in browser

The Triguarded Fragment of First-Order Logic

16 pagesPublished: October 23, 2018


Past research into decidable fragments of first-order logic (FO) has produced two very prominent fragments: the guarded fragment GF, and the two-variable fragment FO2. These fragments are of crucial importance because they provide significant insights into decidabil- ity and expressiveness of other (computational) logics like Modal Logics (MLs) and various Description Logics (DLs), which play a central role in Verification, Knowledge Represen- tation, and other areas. In this paper, we take a closer look at GF and FO2, and present a new fragment that subsumes them both. This fragment, called the triguarded fragment (denoted TGF), is obtained by relaxing the standard definition of GF: quantification is required to be guarded only for subformulae with three or more free variables. We show that, in the absence of equality, satisfiability in TGF is N2ExpTime-complete, but becomes NExpTime-complete if we bound the arity of predicates by a constant (a natural assumption in the context of MLs and DLs). Finally, we observe that many natural extensions of TGF, including the addition of equality, lead to undecidability.

Keyphrases: complexity, decidability, Description Logics, fragments of first-order logic, modal logic, The guarded fragment, The two-variable fragment

In: Gilles Barthe, Geoff Sutcliffe and Margus Veanes (editors). LPAR-22. 22nd International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 57, pages 604--619

BibTeX entry
  author    = {Sebastian Rudolph and Mantas Simkus},
  title     = {The Triguarded Fragment of First-Order Logic},
  booktitle = {LPAR-22. 22nd International Conference on Logic for Programming, Artificial Intelligence and Reasoning},
  editor    = {Gilles Barthe and Geoff Sutcliffe and Margus Veanes},
  series    = {EPiC Series in Computing},
  volume    = {57},
  pages     = {604--619},
  year      = {2018},
  publisher = {EasyChair},
  bibsource = {EasyChair,},
  issn      = {2398-7340},
  url       = {},
  doi       = {10.29007/m8ts}}
Download PDFOpen PDF in browser