Skip to main content
SpringerLink
Log in

A Proof of the Focusing Theorem via MALL Proof Nets

  • Conference paper
  • First Online:
Logic, Language, Information, and Computation (WoLLIC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13468))

  • 431 Accesses

Abstract

We present a demonstration of Andreoli’s focusing theorem for proofs of linear logic (MALL) that avoids directly reasoning on sequent calculus proofs. Following Andreoli-Maieli’s strategy, exploited in the MLL case, we prove the focusing theorem as a particular sequentialization strategy for MALL proof nets that are in canonical form. Canonical proof nets satisfy the property that asynchronous links are always ready to sequentialization while synchronous focusing links represent clusters of links that are hereditarily ready to sequentialization.

Supported by Istituto Nazionale di Alta Matematica “Francesco Severi” (INdAM).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Compared with the syntax of [6], monomial proof structures [8] are technically simpler; they allow us to easily extend to the MALL case arguments originally used for the MLL case such as Laurent’s Splitting Lemma [7] and Andreoli-Maieli’s Focusing Theorem [2]. Monomial proof structures have a natural presentation in terms of Coherent Spaces [4] and their correctness criterion can be also formulated in terms of “graph retraction steps” à la Danos [9].

  2. 2.

    The dependence condition corresponds to the resolution condition of [6].

  3. 3.

    Observe that in general substitutions may not preserve the property of being a proof structure.

  4. 4.

    The logical degree of a formula F, denoted \(\partial (F)\), is defined by induction on the height of F: if F is atomic then \(\partial (F)=0\), else F has the form \(F_1\circ F_2\), with , and \(\partial (F)=\partial (F_1)+\partial (F_2)+1\).

  5. 5.

    Note that a \(J(\pi )\) differs from a switching \(S(\pi )\) for the following facts: (i) we do not consider slices, (ii) we do not mutilate premises and (iii) there can be multiple (possibly, none) jumps exiting from a -node and going to different nodes depending on p or \(C_p\) nodes.

References

  1. Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. JLC 2(3), 297–347 (1992). https://doi.org/10.1093/logcom/2.3.297

    Article  Google Scholar 

  2. Andreoli, J.-M., Maieli, R.: Focusing and proof-nets in linear and non-commutative logic. In: Ganzinger, H., McAllester, D., Voronkov, A. (eds.) LPAR 1999. LNCS (LNAI), vol. 1705, pp. 320–336. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48242-3_20

    Chapter  Google Scholar 

  3. Danos, V., Regnier, L.: The structure of multiplicatives. AML 28, 181–203 (1989). https://doi.org/10.1007/BF01622878

    Article  Google Scholar 

  4. Girard, J.-Y.: Linear logic. TCS 50, 1–102 (1987). https://doi.org/10.1016/0304-3975(87)90045-4

    Article  Google Scholar 

  5. Girard, J.-Y.: Proof-nets: the parallel syntax for proof theory. In: Logic and Algebra (1996). https://doi.org/10.1201/9780203748671

  6. Hughes, D., Van Glabbeek, R.: Proof nets for unit-free multiplicative-additive linear logic. In: Proceedings of IEEE Logic in Computer Science (2003). https://doi.org/10.1109/LICS.2003.1210039

  7. Laurent, O.: Sequentialization of multiplicative proof nets. Manuscript, April 2013. https://perso.ens-lyon.fr/olivier.laurent/seqmll.pdf

  8. Laurent, O., Maieli, R.: Cut elimination for monomial MALL proof nets. In: LICS 2008 (2008). https://doi.org/10.1109/LICS.2008.31

  9. Maieli, R.: Retractile proof nets of the purely multiplicative and additive fragment of linear logic. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 363–377. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-75560-9_27

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, “Roma TRE” University, Rome, Italy

    Roberto Maieli

Authors
  1. Roberto Maieli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto Maieli .

Editor information

Editors and Affiliations

  1. Technische Universität Wien, Vienna, Austria

    Agata Ciabattoni

  2. University College London, London, UK

    Elaine Pimentel

  3. Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil

    Ruy J. G. B. de Queiroz

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Maieli, R. (2022). A Proof of the Focusing Theorem via MALL Proof Nets. In: Ciabattoni, A., Pimentel, E., de Queiroz, R.J.G.B. (eds) Logic, Language, Information, and Computation. WoLLIC 2022. Lecture Notes in Computer Science, vol 13468. Springer, Cham. https://doi.org/10.1007/978-3-031-15298-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-15298-6_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-15297-9

  • Online ISBN: 978-3-031-15298-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation