# Uniform Interpolation and Propositional Quantifiers in Modal Logics

```@article{Blkov2007UniformIA,
title={Uniform Interpolation and Propositional Quantifiers in Modal Logics},
author={Marta B{\'i}lkov{\'a}},
journal={Studia Logica},
year={2007},
volume={85},
pages={1-31}
}```
• M. Bílková
• Published 1 February 2007
• Philosophy
• Studia Logica
We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view.Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.We shall present such a proof of the uniform interpolation theorem for normal modal logics K and T. It…
A Note on Uniform Interpolation Proofs in Modal Deep Inference Calculi
This paper shows how to perform a proof of uniform interpolation property in deep inference calculus for the basic modal logic K via forgetting a variable in a certain normal form constructed by backward proof search.
The Logic of Exact Covers: Completeness and Uniform Interpolation
• D. Pattinson
• Mathematics, Philosophy
2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
• 2013
We show that all (not necessarily normal or monotone) modal logics that can be axiomatised in rank-1 have the interpolation property, and that in fact interpolation is uniform if the logics just have
Intuitionistic modal logics and Dyckhoff's calculus
In 1992 Dyckhoff developed a sequent calculus for intuitionistic propositional logic in which proof search is terminating. In this paper this result is extended to several intuitionistic modal
Uniform Interpolation in Coalgebraic Modal Logic
• Philosophy, Mathematics
CALCO
• 2017
A notion of one-step (uniform) interpolation is introduced, which refers only to a restricted logic without nesting of modalities, and it is shown that a coalgebraic modal logic has uniform interpolation if it has one- step interpolation.
Uniform Interpolation for Monotone Modal Logic
• Computer Science, Philosophy
• 2010
It is proved that the standard modal language and the r -based one are effectively equi-expressive, meaning that there are effective translations in both directions.
Terminating sequent calculi for two intuitionistic modal logics
It is shown by proof–theoretic means that these terminating calculi are equivalent to the cutfree extensions of G3ip that form some of the standard calculi for intuitionistic modal logics.
Sequent Calculi and Interpolation for Non-Normal Modal and Deontic Logics
G3-style sequent calculi for the logics in the cube of non-normal modal logics and for their deontic extensions are studied. For each calculus we prove that weakening and contraction are
Uniform Interpolation by Resolution in Modal Logic
• Philosophy
JELIA
• 2008
This paper proposes a method, based on an extension of resolution to modal logics, to perform variable forgetting for formulae in conjunctive normal form, in the modal logic K.
Model Completeness, Uniform Interpolants and Superposition Calculus
• Computer Science
J. Autom. Reason.
• 2021
It is shown how covers are strictly related to model completions, a well-known topic in model theory, and it is shown that computing covers is computationally tractable for the fragment of the language used when tackling the verification of data-aware processes.

## References

SHOWING 1-10 OF 23 REFERENCES
On the Complexity of Propositional Quantification in Intuitionistic Logic
It is shown that H π + is recursively isomorphic to full second order classical logic and is the intuitionistic analogue of the modal systems studied by Fine.
Undefinability of propositional quantifiers in the modal system S4
• Philosophy
Stud Logica
• 1995
We show that (contrary to the parallel case of intuitionistic logic, see [7], [4]) there does not exist a translation fromS42 (the propositional modal systemS4 enriched with propositional
On an Interpretation of Second Order Quantification in First Order Intuitionistic Propositional Logic
It is shown that a model of IpC 2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra, and the result has a number of interesting consequences for the algebraic counterpart of I pC, the theory of Heyting algebras.
The Computational Complexity of Provability in Systems of Modal Propositional Logic
• Computer Science, Mathematics
SIAM J. Comput.
• 1977
The computational complexity of the provability problem in systems of modal propositional logic is investigated. Every problem computable in polynomial space is \$\log \$ space reducible to the
Structural proof theory
• Mathematics
• 2001
From natural deduction to sequent calculus, diversity and unity in structural proof theory are explored and the quantifiers are explained.
Sequent Systems for Modal Logics
This chapter surveys the application of various kinds of sequent systems to modal and temporal logic, also called tense logic, using ordinary Gentzen sequents as a starting point.
Bisimulations, model descriptions and propositional quantifiers
In this paper we give perspicuous proofs of the existence of model descriptions for finite Kripke models and of Uniform Interpolation for the theories IPC, K, GL and S4Grz, using bounded
On Modal Logic with Propositional Quantifiers
I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A .
Efficient Loop-Check for Backward Proof Search in Some Non-classical Propositional Logics
• Philosophy
TABLEAUX
• 1996
We consider the modal logics KT and S4, the tense logic Kt, and the fragment IPC(∧,→) of intuitionistic logic.
Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀p and ∃p
A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take a model to be one of these structures, α, together with some function or relation