## A System of Interaction and Structure II: The Need for Deep Inference (2005)

Venue: | Logical Methods in Computer Science |

Citations: | 12 - 1 self |

### BibTeX

@INPROCEEDINGS{Tiu05asystem,

author = {Alwen Tiu},

title = {A System of Interaction and Structure II: The Need for Deep Inference},

booktitle = {Logical Methods in Computer Science},

year = {2005},

pages = {2006}

}

### OpenURL

### Abstract

This paper studies properties of the logic , which is an extension of multiplicative linear logic (MLL) with a self-dual non-commutative operator.

### Citations

261 |
Investigations into logical deduction
- Gentzen
- 1969
(Show Context)
Citation Context ...alculus”. As a working definition, we regard sequent systems as a subset of “shallow systems” (its precise definition will be given later). For example, the one-sided sequent systems for Gentzen’s LK =-=[2]-=- and Girard’s linear logic [3] fit into this definition. We hope also to shed lights to the known open problem of the sequentialization of Retore’s Pomset logic [9]. Pomset logic is an extension of mu... |

191 | The logic of bunched implications
- O’Hearn, Pym
- 1999
(Show Context)
Citation Context ...the precise definition of which is given in Section 6. There are calculi other than the calculus of structures which also employ some sort of deep inference, notably, the logic of bunched implication =-=[OP99]-=- and the display calculus [Bel82], where the context of the sequent is a tree-like structure and inference rules can access any part of the tree. But there has been so far no technical analyses in tho... |

162 | Basic Proof Theory
- Troelstra, Schwichtenberg
- 2000
(Show Context)
Citation Context ...tructures, started in [6]. The calculus of structures is a new formalism for presenting logics. It is a generalization of the one-sided sequent calculus (also known as Gentzen-Shütte sequent calculus =-=[13]-=-). One of the main features of calculus of structures is that it allows deep-inference, i.e., inference rules in the calculus of structures can be applied arbitrarily deep inside logical expressions. ... |

106 |
Display logic
- Belnap
- 1982
(Show Context)
Citation Context ...is given in Section 6. There are calculi other than the calculus of structures which also employ some sort of deep inference, notably, the logic of bunched implication [OP99] and the display calculus =-=[Bel82]-=-, where the context of the sequent is a tree-like structure and inference rules can access any part of the tree. But there has been so far no technical analyses in those calculi concerning the necessi... |

85 | A system of interaction and structure
- Guglielmi
(Show Context)
Citation Context ...e inference rules ofs£¡ would result in a strictly less expressive logical system. 1 Introduction This paper is the second part of a planned series of papers on the calculus of structures, started in =-=[6]-=-. The calculus of structures is a new formalism for presenting logics. It is a generalization of the one-sided sequent calculus (also known as Gentzen-Shütte sequent calculus [13]). One of the main fe... |

55 | Non-commutativity and MELL in the calculus of structures
- Guglielmi, Straßburger
- 2001
(Show Context)
Citation Context ...hose premise and conclusion are T and R, and whose rules are in S T is denoted by ρ is called non-trivial if R �= T , otherwise it is trivial. R Definition 8 System ∆ S . An instance of a rule R Ts¢¡ =-=[7]-=- is shown in Figure 6. The rules in R is called redex and the structure T is called contractum.s£¡ are of the form S{T } ρ S{R} . The structure The ruless↓ ands↑ are called the interaction rules. They... |

55 |
Computationally tractable classes of ordered sets
- Möhring
- 1989
(Show Context)
Citation Context ...his section we consider a different representation for structures, a special kind of graph called relation webs [6] (originally called traces [4]), which is a generalisation of series-parallel orders =-=[8]-=-. In this representation, structures can be characterized by the absence of certain certain forbidden configurations in its relation web. This characterisation is useful when we want to study general ... |

49 | Deep Inference and Symmetry for Classical Proofs - Brünnler - 2003 |

47 | Brünnler and Alwen Fernanto Tiu. A local system for classical logic - Kai - 2001 |

37 | Linear Logic and Noncommutativity in the Calculus of Structures
- Straßburger
- 2003
(Show Context)
Citation Context ... there has to date been no known sequentialization theorem yet. The logics¢¡ and Pomset logic are closely related; in fact, there is a sound translation froms£¡ to Pomset logic preserving provability =-=[11]-=-. We conjecture that they are actually the same logic. The result on the necessity of deep-inference ofs¢¡ therefore explains to some extent the difficulty in the sequentialization of Pomset logic. ∗ ... |

36 | Pomset logic: A non-commutative extension of classical linear logic
- Retoré
- 1997
(Show Context)
Citation Context ...ed sequent systems for Gentzen’s LK [2] and Girard’s linear logic [3] fit into this definition. We hope also to shed lights to the known open problem of the sequentialization of Retore’s Pomset logic =-=[9]-=-. Pomset logic is an extension of multiplicative linear logic with a self-dual non-commutative connective. It is presented in the calculus of proofnets, and there has to date been no known sequentiali... |

30 | A local system for classical logic
- Brünnler, Tiu
- 2001
(Show Context)
Citation Context ...ctures is that it allows deep-inference, i.e., inference rules in the calculus of structures can be applied arbitrarily deep inside logical expressions. In [6], and several other related works (e.g., =-=[1, 7]-=-), it has been shown that deep-inference gives rise to many interesting proof theoretical properties for the logical systems formalized in the calculus of structures that is not observable in sequent ... |

12 | Pomset logic as a calculus of directed cographs
- Retoré
- 1999
(Show Context)
Citation Context ...ture from sequent calculus as a formalism for presenting logics. System BV is not the only logical system that incorporates both commutativity and non-commutativity connectives. Retoré’s Pomset logic =-=[10]-=- achieves the same thing. However, Pomset logic is based on proofnets, and its sequentialisation is still an open question. Retoré uses a presentation of proof-nets called the R&B-directed cographs, w... |

8 | A calculus of order and interaction
- Guglielmi
- 1999
(Show Context)
Citation Context ...s the depth of S{ }. 4 Characterisation of Structures In this section we consider a different representation for structures, a special kind of graph called relation webs [6] (originally called traces =-=[4]-=-), which is a generalisation of series-parallel orders [8]. In this representation, structures can be characterized by the absence of certain certain forbidden configurations in its relation web. This... |

4 |
A system of interaction and order
- Guglielmi
- 2001
(Show Context)
Citation Context ...s of a down fragment {s¢ ¡ ¡s¡ ¢ ↓, , ↓} and an up fragment { ↑, , ↑}. The rules ↓ and ↑ in the up fragment are admissible (for proofs) as a consequence of the cut elimination theorem for this system =-=[4, 5]-=-.s¢¡ Therefore, it is enough to study the down fragment for its proof search properties. We need an extra rule, a logical axiom rule, to define proofs in . Definition 9 The down fragment ofs£¡ , toget... |

3 | Properties of a logical system in the calculus of structures - Tiu - 2001 |