DS-1994-01:
Schellinx, Harold
(1994)
*The Noble Art of Linear Decorating.*
Doctoral thesis, University of Amsterdam.

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## Abstract

Linear logic (Girard, 1987) is a refinement of the formulation of classical

logic as a sequent calculus (Gentzen, 1935). The `intervention' is simple:

in the `classical' formulation a formula, present as a hypothesis in a

derivation, can be used an unlimited number of times, and any formula can

be introduced as a hypothesis, even it is not used. Furthermore, deriving

a theorem a number of times is equivalent to deriving it only once, and

whenever a certain conclusion has been reached, say $X$, it is possible to

add another one, say $Y$ (as in `$X$ or $Y$').

In the linear formulation this kind of practice is restricted. Contraction

and weakening are only allowed if the formula in question is marked with

a modality. This in turn leads to the introduction of two variants

of familiar logical connectives: an {\em additive} and a {\em multiplicative}

variant.

If linear typing is ignored, a derivation in classical logic is nothing but

a derivation in classical or intuitionistic logic.

Schellinx main question is the following: Take some derivation $\pi$ in

sequent calculus for classical or intuitionistic logic. Can we transform it into

a sequent derivation in linear logic in a way that essentially preserves

its structure, i.e. can we define a {\em linear decoration} of the original

proof? And if `yes', then is there an {\em optimal} way to do this?

Schellinx shows, for example, that for some fragments, for instance that

consisting of rules for implication, universal first order quantification,

and universal second order (propositional) quantification, this linear

decorating is completely {\em deterministic}.

Schellinx main concern is with {\em mappings}: from formulas to linear

formulas, from classical and intuitionistic proofs to linear proofs, from

classical (intuitionistic) proofs to classical (intuitionistic) proof, and

from linear proofs to linear proofs, mappings that in most cases will

preserve, at least, what he calls the {\em skeleton} of the original, in the

case `linear to linear' moreover its {\em dynamics} (i.e. behaviour under

cut elimination)

Schellinx introduces a number of technical devices for these purposes.

In chapter 5 the {\em exponential graph} of a derivation is

introduced. This graph permits the characterization of those derivations in

linear logic that are {\em dilatable}, that is, derivations in which

{\em all} modalized formulas can be replaced by non-model formulas, withou

changing the structure and the dynamics of the original. The most important

result is that a fully expanded linear derivation is dilatable iff her

exponential graph is acyclic.

In chapter 6, the notion {\em constrictive morphism} is introduced, which

can be used to optimalize modal translations. This leads to well--defined

restrictions on rules of the sequent calculus, that maintain completeness

with respect to provability. Schellinx formulates `alternative'

sequent calculi for intuitionistic and classical logic for which the

optimal modal translations are decorated: {\bf ILU}, {\bf LKT}

and {\bf LKQ}.

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-1994-01 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 1994 |

Subjects: | Logic |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/1964 |

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