# Refining Labelled Systems for Modal and Constructive Logics with Applications

This thesis introduces the "method of structural refinement", which serves as a means of transforming the relational semantics of a modal and/or constructive logic into an 'economical' proof system by connecting two proof-theoretic paradigms: labelled and nested sequent calculi. The formalism of labelled sequents has been successful in that cut-free calculi in possession of desirable proof-theoretic properties can be automatically generated for large classes of logics. Despite these qualities, labelled systems make use of a complicated syntax that explicitly incorporates the semantics of the associated logic, and such systems typically violate the subformula property to a high degree. By contrast, nested sequent calculi employ a simpler syntax and adhere to a strict reading of the subformula property, making such systems useful in the design of automated reasoning algorithms. However, the downside of the nested sequent paradigm is that a general theory concerning the automated construction of such calculi (as in the labelled setting) is essentially absent, meaning that the construction of nested systems and the confirmation of their properties is usually done on a case-by-case basis. The refinement method connects both paradigms in a fruitful way, by transforming labelled systems into nested (or, refined labelled) systems with the properties of the former preserved throughout the transformation process. To demonstrate the method of refinement and some of its applications, we consider grammar logics, first-order intuitionistic logics, and deontic STIT logics. The introduced refined labelled calculi will be used to provide the first proof-search algorithms for deontic STIT logics. Furthermore, we employ our refined labelled calculi for grammar logics to show that every logic in the class possesses the effective Lyndon interpolation property.

## Authors

• 10 publications
02/13/2018

### Proof systems: from nestings to sequents and back

In this work, we explore proof theoretical connections between sequent, ...
04/19/2021

### On the Correspondence between Nested Calculi and Semantic Systems for Intuitionistic Logics

This paper studies the relationship between labelled and nested calculi ...
08/29/2019

### Automating Agential Reasoning: Proof-Calculi and Syntactic Decidability for STIT Logics

This work provides proof-search algorithms and automated counter-model e...
10/15/2019

### On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

This paper shows how to derive nested calculi from labelled calculi for ...
07/05/2021

### Nested Sequents for Intuitionistic Modal Logics via Structural Refinement

We employ a recently developed methodology – called "structural refineme...
05/23/2021

### Uniform interpolation via nested sequents and hypersequents

A modular proof-theoretic framework was recently developed to prove Crai...
02/12/2020

### Labelled calculi for quantified modal logics with definite descriptions

We introduce labelled sequent calculi for quantified modal logics with d...
##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.