# Atomic Cut Elimination for classical Logic

@inproceedings{Brnnler2003AtomicCE,
title={Atomic Cut Elimination for classical Logic},
author={Kai Br{\"u}nnler},
booktitle={CSL},
year={2003}
}
System SKS is a set of rules for classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction systems, it has an explicit cut rule, which is admissible. In contrast to sequent systems, the cut rule can easily be reduced to atomic form. This allows for a very simple cut elimination procedure based on plugging in parts of a proof, like normalisation in natural deduction and unlike cut elimination in the sequent calculus. It should thus… Expand
48 Citations

#### Figures and Topics from this paper

Locality for Classical Logic
• Kai Brünnler
• Mathematics, Computer Science
• Notre Dame J. Formal Log.
• 2006
D deductive systems for classical propositional and predicate logic in the calculus of structures are seen, which have a cut rule which is admissible and which leads to inference rules that are local : they do not require the inspection of expressions of arbitrary size. Expand
Cut Elimination inside a Deep Inference System for Classical Predicate Logic
This paper sees a cut elimination procedure for a deep inference system for classical predicate logic and derives Herbrand's Theorem, which is expressed as a factorisation of derivations. Expand
Deep Inference and Symmetry in Classical Proofs
• Mathematics
• 2003
In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry,Expand
Ingredients of a Deep Inference Theorem Prover
Deep inference deductive systems for classical logic provide exponentially shorter proofs than the sequent calculus systems, however with the cost of higher nondeterminism and larger search space inExpand
Symmetric normalisation for intuitionistic logic
• Mathematics, Computer Science
• CSL-LICS
• 2014
It is proved a generalisation of cut elimination, that is symmetric normalisation, where all rules dual to standard ones are permuted up in the derivation, and the result is a decomposition theorem having cut elimination and interpolation as corollaries. Expand
Hybrid Logic in the Calculus of Structures
Hybrid logic is an extension of modal logic which allows to access the states of a Kripke structure directly from within the logic. This is achieved with nominals which are an additional kind ofExpand
MELL in the calculus of structures
This paper studies a system for MELL, the multiplicative exponential fragment of linear logic, in the calculus of structures, which has the following features: a local promotion rule, no non-deterministic splitting of the context in the times rule and a modular proof for the cut elimination theorem. Expand
Cut-elimination and proof-search for bi-intuitionistic logic using nested sequents
• Mathematics, Computer Science
• 2008
A new sequent calculus for bi-intuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents and has an easily derivable variant calculus which is amenable to automated proof search. Expand
From Deep Inference to Proof Nets via Cut Elimination
This article shows how derivations in the deep inference system SKS for classical propositional logic can be translated into proof nets and observes a loss of information which can be understood as ‘eliminating bureaucracy’. Expand

#### References

SHOWING 1-10 OF 15 REFERENCES
A Local System for Classical Logic
• Computer Science
• LPAR
• 2001
The calculus of structures is a framework for specifying logical systems, which is similar to the one-sided sequent calculus but more general, and a system of inference rules for propositional classical logic is presented, with the main novelty that all the rules are local. Expand
A Local System for Linear Logic
A deductive system for linear logic, in which all rules are local, and the contraction rule is reduced to an atomic version, and there is no global promotion rule. Expand
MELL in the calculus of structures
This paper studies a system for MELL, the multiplicative exponential fragment of linear logic, in the calculus of structures, which has the following features: a local promotion rule, no non-deterministic splitting of the context in the times rule and a modular proof for the cut elimination theorem. Expand
A Purely Logical Account of Sequentiality in Proof Search
This work is argued about as being the first step in a two-step research for capturing most of CCS in a purely logical fashion, and should become a common basis for several possible extensions. Expand
Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction
This paper presents a way of extending the paradigm "proofs as programs" to classical proofs, which can be seen as a simple extension of intuitionistic natural deduction, whose algorithmic interpretation is very well known. Expand
Computational Adequacy in an Elementary Topos
It is proved that computational adequacy holds if and only if the topos is 1-consistent (i.e. its internal logic validates only true Σ$$^{\rm 0}_{\rm 1}$$-sentences). Expand
A Calculus of Order and Interaction
System MV is a simple, propositional linear calculus that deals with the commuta- tive as well as the non-commutative composition of structures. The multiplicative fragment of linear logic is aExpand
Uniform Proofs as a Foundation for Logic Programming
• Computer Science
• Ann. Pure Appl. Log.
• 1991
A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Expand
Non-commutativity and MELL in the Calculus of Structures
• Mathematics, Computer Science
• CSL
• 2001
The calculus of structures is introduced: it is more general than the sequent calculus and it allows for cut elimination and the subformula property, and it is shown that multiplicative exponential linear logic benefits from its presentation in the calculus of structure. Expand
Constructive Logics Part I: A Tutorial on Proof Systems and Typed gamma-Calculi
This paper has attempted to cover the basic material on natural deduction, sequent calculus, and typed λ-calculus, but also to provide an introduction to Girard's linear logic, one of the most exciting developments in logic these past six years. Expand