A Systematic Approach to Canonicity in the Classical Sequent Calculus

@inproceedings{Chaudhuri2012ASA,
  title={A Systematic Approach to Canonicity in the Classical Sequent Calculus},
  author={Kaustuv Chaudhuri and Stefan Hetzl and Dale Miller},
  booktitle={CSL},
  year={2012}
}
The sequent calculus is often criticized for requiring proofs to contain large amounts of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps---such as instantiating a block of quantifiers---by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written… 

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References

SHOWING 1-10 OF 49 REFERENCES
Canonical Sequent Proofs via Multi-Focusing
TLDR
This paper recovers permutative canonicity directly in the cut-free sequent calculus by generalizing focused sequent proofs to admit multiple foci, and then considering the restricted class of maximally multi-focused proofs.
A Proof Calculus Which Reduces Syntactic Bureaucracy
TLDR
A logic-independent proof calculus, where proofs can be freely composed by connectives, and prove its basic properties, which allows to avoid certain types of syntactic bureaucracy inherent to all usual proof systems.
Logic Programming with Focusing Proofs in Linear Logic
TLDR
It is shown that the syntactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.
A compact representation of proofs
TLDR
This paper is able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe.
A New Deconstructive Logic: Linear Logic
TLDR
The method presented is powerful: it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD, delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of ‘programming-with-proofs’ to classical logic.
Classical proof forestry
Naming Proofs in Classical Propositional Logic
TLDR
With the Boolean semiring, the theory of classical proof nets is presented, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cut-elimination procedure, which gives a “Boolean” category, which is not a poset.
A framework for defining logics
TLDR
The Edinburgh Logical Framework provides a means to define (or present) logics through a general treatment of syntax, rules, and proofs by means of a typed λ-calculus with dependent types, whereby each judgment is identified with the type of its proofs.
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