A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic

Ian Shillito; Dominik Kirst

doi:10.1145/3636501.3636957

A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic

Ian Shillito
, Dominik Kirst

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

7 Scopus citations

Abstract

Using the Coq proof assistant, we investigate the minimal non-constructive principles needed to show soundness and completeness of propositional bi-intuitionistic logic. Before being revisited and corrected by Goré and Shillito, the completeness of bi-intuitionistic logic, an extension of intuitionistic logic with a dual operation to implication, had a rather erratic history, making it an ideal case for computer mechanisation. Moreover, contributing a constructive perspective, we observe that the completeness of bi-intuitionistic logic explicates the same characteristics already observed in an ongoing effort to analyse completeness theorems in general.

Original language	English
Title of host publication	CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with
Subtitle of host publication	POPL 2024
Editors	Amin Timany, Dmitriy Traytel, Brigitte Pientka, Sandrine Blazy
Publisher	Association for Computing Machinery, Inc
Pages	218-229
Number of pages	12
ISBN (Electronic)	9798400704888
DOIs	https://doi.org/10.1145/3636501.3636957
State	Published - 9 Jan 2024
Event	13th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2024, in affiliation with the annual Symposium on Principles of Programming, Languages, ,POPL 2024 - London, United Kingdom Duration: 15 Jan 2024 → 16 Jan 2024

Publication series

Name	CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024

Conference

Conference	13th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2024, in affiliation with the annual Symposium on Principles of Programming, Languages, ,POPL 2024
Country/Territory	United Kingdom
City	London
Period	15/01/24 → 16/01/24

Keywords

Coq
bi-intuitionistic logic
completeness
constructive reverse mathematics

ASJC Scopus subject areas

Computer Science Applications
Software

Access to Document

10.1145/3636501.3636957

Cite this

Shillito, I., & Kirst, D. (2024). A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic. In A. Timany, D. Traytel, B. Pientka, & S. Blazy (Eds.), CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024 (pp. 218-229). (CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024). Association for Computing Machinery, Inc. https://doi.org/10.1145/3636501.3636957

Shillito, Ian ; Kirst, Dominik. / A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic. CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024. editor / Amin Timany ; Dmitriy Traytel ; Brigitte Pientka ; Sandrine Blazy. Association for Computing Machinery, Inc, 2024. pp. 218-229 (CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024).

@inproceedings{337c587d73f040839fe617ad47fad4c1,

title = "A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic",

abstract = "Using the Coq proof assistant, we investigate the minimal non-constructive principles needed to show soundness and completeness of propositional bi-intuitionistic logic. Before being revisited and corrected by Gor{\'e} and Shillito, the completeness of bi-intuitionistic logic, an extension of intuitionistic logic with a dual operation to implication, had a rather erratic history, making it an ideal case for computer mechanisation. Moreover, contributing a constructive perspective, we observe that the completeness of bi-intuitionistic logic explicates the same characteristics already observed in an ongoing effort to analyse completeness theorems in general.",

keywords = "Coq, bi-intuitionistic logic, completeness, constructive reverse mathematics",

author = "Ian Shillito and Dominik Kirst",

note = "Publisher Copyright: {\textcopyright} 2024 Owner/Author.; 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2024, in affiliation with the annual Symposium on Principles of Programming, Languages, ,POPL 2024 ; Conference date: 15-01-2024 Through 16-01-2024",

year = "2024",

month = jan,

day = "9",

doi = "10.1145/3636501.3636957",

language = "English",

series = "CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024",

publisher = "Association for Computing Machinery, Inc",

pages = "218--229",

editor = "Amin Timany and Dmitriy Traytel and Brigitte Pientka and Sandrine Blazy",

booktitle = "CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with",

}

Shillito, I & Kirst, D 2024, A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic. in A Timany, D Traytel, B Pientka & S Blazy (eds), CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024. CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024, Association for Computing Machinery, Inc, pp. 218-229, 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2024, in affiliation with the annual Symposium on Principles of Programming, Languages, ,POPL 2024, London, United Kingdom, 15/01/24. https://doi.org/10.1145/3636501.3636957

A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic. / Shillito, Ian; Kirst, Dominik.
CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024. ed. / Amin Timany; Dmitriy Traytel; Brigitte Pientka; Sandrine Blazy. Association for Computing Machinery, Inc, 2024. p. 218-229 (CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic

AU - Shillito, Ian

AU - Kirst, Dominik

PY - 2024/1/9

Y1 - 2024/1/9

N2 - Using the Coq proof assistant, we investigate the minimal non-constructive principles needed to show soundness and completeness of propositional bi-intuitionistic logic. Before being revisited and corrected by Goré and Shillito, the completeness of bi-intuitionistic logic, an extension of intuitionistic logic with a dual operation to implication, had a rather erratic history, making it an ideal case for computer mechanisation. Moreover, contributing a constructive perspective, we observe that the completeness of bi-intuitionistic logic explicates the same characteristics already observed in an ongoing effort to analyse completeness theorems in general.

AB - Using the Coq proof assistant, we investigate the minimal non-constructive principles needed to show soundness and completeness of propositional bi-intuitionistic logic. Before being revisited and corrected by Goré and Shillito, the completeness of bi-intuitionistic logic, an extension of intuitionistic logic with a dual operation to implication, had a rather erratic history, making it an ideal case for computer mechanisation. Moreover, contributing a constructive perspective, we observe that the completeness of bi-intuitionistic logic explicates the same characteristics already observed in an ongoing effort to analyse completeness theorems in general.

KW - Coq

KW - bi-intuitionistic logic

KW - completeness

KW - constructive reverse mathematics

UR - https://www.scopus.com/pages/publications/85182948274

U2 - 10.1145/3636501.3636957

DO - 10.1145/3636501.3636957

M3 - Conference contribution

AN - SCOPUS:85182948274

T3 - CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024

SP - 218

EP - 229

BT - CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with

A2 - Timany, Amin

A2 - Traytel, Dmitriy

A2 - Pientka, Brigitte

A2 - Blazy, Sandrine

PB - Association for Computing Machinery, Inc

T2 - 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2024, in affiliation with the annual Symposium on Principles of Programming, Languages, ,POPL 2024

Y2 - 15 January 2024 through 16 January 2024

ER -

Shillito I, Kirst D. A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic. In Timany A, Traytel D, Pientka B, Blazy S, editors, CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024. Association for Computing Machinery, Inc. 2024. p. 218-229. (CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024). doi: 10.1145/3636501.3636957

A Mechanised and Constructive Reverse Analysis of Soundness and Completeness of Bi-intuitionistic Logic

Abstract

Publication series

Conference

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this