An O ((n·log n)3)-Time Transformation from Grz into Decidable Fragments of Classical First-Order Logic

@inproceedings{Demri1998AnO,
  title={An O ((n·log n)3)-Time Transformation from Grz into Decidable Fragments of Classical First-Order Logic},
  author={St{\'e}phane Demri and Rajeev Gor{\'e}},
  booktitle={FTP},
  year={1998}
}
The provability logic Grz is characterized by a class of modal frames that is not first-order definable. We present a simple embedding of Grz into decidable fragments of classical first-order logic such as FO2 and the guarded fragment. The embedding is an O((n.log n)3)-time transformation that neither involves first principles about Turing machines (and therefore is easy to implement), nor the semantical characterization of Grz (and therefore does not use any second-order machinery). Instead… 
View via Publisher
lsv.ens-cachan.fr
9 Citations
Background Citations
1

9 Citations

Citation Type

Citation Type

Tractable Transformations from Modal Provability Logics into First-Order Logic
TLDR
Using Belnap's proof-theoretical framework Display Logic, this work characterise the "pseudo-displayable" subclass of LF and shows how to define polynomial-time transformations from each such LF into the corresponding L, and hence into first-order classical logic.
Theoremhood-preserving Maps Characterizing Cut Elimination for Modal Provability Logics
TLDR
The results show that all pseudo-displayable logics are ‘essentially first- order’ even though their characteristic axiom may not be first-order definable.
Deciding Regular Grammar Logics with Converse Through First-Order Logic
TLDR
The results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed-point operators.
Display Calculi for Nominal Tense Logics
TLDR
It is shown that every -valid formula admits a cut-free derivation in Æ and that a restricted display calculus Æ, is not only complete for , but that it enjoys cut-elimination for arbitrary sequents.
Translations: generalizing relative expressiveness between logics
TLDR
The aim of this paper is to propose a sufficiently general and reasonable formal criterion for expressiveness, so as to apply not only to model-theoretic logics, but also to Tarskian and proof-the theoretical logics.
Resolution-based decision procedures for the universal theory of some classes of distributive lattices with operators
A survey on temporal logics for specifying and verifying real-time systems
  • Savas Konur
  • Philosophy, Computer Science
    Frontiers of Computer Science
  • 2013
TLDR
Decidability, axiomatizability, expressiveness, model checking results for each logic analysed are given and a comparison of features of the temporal logics discussed is provided.
Theoremhood Preserving Maps as a Characterisation of Cut Elimination for Provability Logics
We deene cut-free display calculi for provability (modal) logics that are not properly displayable according to Kracht's analysis. We also show that a weak form of the cut-elimination theorem (for…
Encoding Two-Valued Nonclassical Logics in Classical Logic

References

SHOWING 1-10 OF 63 REFERENCES
A Polynomial Translation of S4 into T and Contraction-Free Tableaux for S4
TLDR
Tableau proof-systems are used to show that Fitting's translation of the transitive modal logic S4 into T can be constructed in deterministic polynomial time, and Mints'stranslation of modal formulae into modal clauses is used to obtain a complete contraction free fragment of the sequent calculus for S4.
On Modal Systems Having Arithmetical Interpretations
TLDR
It is shown that cut-elimination fails for QGL (the extension of GL to a language with quantifiers), and it is further shown that, despite this failure, QGL still has some of GL' s interesting properties (e.g., the disjunction property).
Cut-free sequent and tableau systems for propositional Diodorean modal logics
TLDR
Each tableau system has a cut-free sequent analogue proving that Gentzen's cut-elimination theorem holds for these latter systems.
Complexity Results for Classes of Quantificational Formulas
Combining Hilbert Style and Semantic Reasoning in a Resolution Framework
TLDR
A combination of Hilbert style and semantic reasoning is proposed, particularly tailored for cases where either the semantics of some operators is not known, or it is second-order, orIt is just too complicated to handle, or flexibility in experimenting with different versions of a logic is required.
A Resolution Calculus for Modal Logics
TLDR
A syntax transformation is presented that eliminates themodal logic operators from modal logic formulae by shifting the modal context information to the term level and can be brought into conjunctive normal form such that a clause based resolution calculus without any additional inference rule, but with special modal unification algorithms, can be defined.
Resolution-Based Calculi for Modal and Temporal Logics
TLDR
A technique is presented which provides a means to develop resolution-based calculi for (first-order) modal and temporal logics and an extraction of either suitable “simpler” background theories or logic-specific inference rules is presented.
Provability interpretations of modal logic
We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let…
Tableau Methods for Modal and Temporal Logics
TLDR
This chapter gives a systematic and unified introduction to tableau methods for automating deduction in modal and temporal logics and focuses on the propositional fragments restricted to a two-valued (classical) basis.
Relations Between Propositional Normal Modal Logics: An Overview
TLDR
The modal logic literature is notorious for multiple axiomatizations of the same logic and for conflicting overloading of axiom names, so this work catalogs the most interesting axioms, their numerous variants, and explores the relationships between them in terms of interderivability.
...
...