Towards a Unifying View on Monotone Constructive Definitions

Linde Vanbesien; Samuele Pollaci; Bart Bogaerts; Marc Denecker

doi:10.1007/978-3-031-63735-3_13

Towards a Unifying View on Monotone Constructive Definitions

Linde Vanbesien, Samuele Pollaci, Bart Bogaerts, Marc Denecker

Research output: Chapter in Book/Report/Conference proceeding › Conference paper

Abstract

Constructive definitions, including inductive and recursive definitions, are ubiquitous in mathematical texts and occur in a wide variety of computer science fields and Knowledge Representation applications. While in different areas there is a high level of familiarity with certain types of constructive definitions, fairly little interaction between different areas seems to exist, resulting in a lack of deep understanding of principles and their applications. This paper aims to fill this void by laying the foundations for a single unifying framework, bringing together a wide variety of definitions. First, we recall the principle of (monotone) inductive definition and its formalization in fixpoint theory. We discuss the constructive and the non-constructive interpretation of inductive definitions and the induction process. We then analyze examples, including but not limited to (co)inductive and (co)recursive definitions, found in a wide range of areas through the lens of our proposed framework.

Original language	English
Title of host publication	Artificial Intelligence and Image Analysis
Subtitle of host publication	18th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2024, and 22nd International Workshop on Combinatorial Image Analysis, IWCIA 2024, Fort Lauderdale, FL, USA, January 8–10, 2024, Revised Selected Papers
Editors	Reneta Barneva, Valentin Brimkov, Claudio Gentile, Aldo Pacchiano
Publisher	Springer Cham
Pages	218-235
Number of pages	18
Edition	1
ISBN (Electronic)	9783031637353
ISBN (Print)	9783031637346
DOIs	https://doi.org/10.1007/978-3-031-63735-3_13
Publication status	Published - 23 Jul 2024
Event	18th International Symposium on Artificial Intelligence and Mathematic - Fort Lauderdale, Fort Lauderdale, United States Duration: 8 Jan 2024 → 10 Jan 2024 Conference number: 18

Publication series

Name	Lecture Notes in Computer Science
Publisher	Springer Cham
Volume	14494
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	18th International Symposium on Artificial Intelligence and Mathematic
Abbreviated title	ISAIM
Country/Territory	United States
City	Fort Lauderdale
Period	8/01/24 → 10/01/24

Bibliographical note

Funding Information:
This work was supported by Fonds Wetenschappelijk Onderzoek \u2013 Vlaanderen (project G0B2221N), by the Flanders AI Impulse programme (project DTAI-EWI-IMPULS-AI) and the KU Leuven C1 grant C14/19/082.

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

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10.1007/978-3-031-63735-3_13

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Cite this

Vanbesien, L., Pollaci, S., Bogaerts, B., & Denecker, M. (2024). Towards a Unifying View on Monotone Constructive Definitions. In R. Barneva, V. Brimkov, C. Gentile, & A. Pacchiano (Eds.), Artificial Intelligence and Image Analysis: 18th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2024, and 22nd International Workshop on Combinatorial Image Analysis, IWCIA 2024, Fort Lauderdale, FL, USA, January 8–10, 2024, Revised Selected Papers (1 ed., pp. 218-235). (Lecture Notes in Computer Science; Vol. 14494). Springer Cham. https://doi.org/10.1007/978-3-031-63735-3_13

Vanbesien, Linde ; Pollaci, Samuele ; Bogaerts, Bart et al. / Towards a Unifying View on Monotone Constructive Definitions. Artificial Intelligence and Image Analysis: 18th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2024, and 22nd International Workshop on Combinatorial Image Analysis, IWCIA 2024, Fort Lauderdale, FL, USA, January 8–10, 2024, Revised Selected Papers. editor / Reneta Barneva ; Valentin Brimkov ; Claudio Gentile ; Aldo Pacchiano. 1. ed. Springer Cham, 2024. pp. 218-235 (Lecture Notes in Computer Science).

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abstract = "Constructive definitions, including inductive and recursive definitions, are ubiquitous in mathematical texts and occur in a wide variety of computer science fields and Knowledge Representation applications. While in different areas there is a high level of familiarity with certain types of constructive definitions, fairly little interaction between different areas seems to exist, resulting in a lack of deep understanding of principles and their applications. This paper aims to fill this void by laying the foundations for a single unifying framework, bringing together a wide variety of definitions. First, we recall the principle of (monotone) inductive definition and its formalization in fixpoint theory. We discuss the constructive and the non-constructive interpretation of inductive definitions and the induction process. We then analyze examples, including but not limited to (co)inductive and (co)recursive definitions, found in a wide range of areas through the lens of our proposed framework.",

author = "Linde Vanbesien and Samuele Pollaci and Bart Bogaerts and Marc Denecker",

note = "Funding Information: This work was supported by Fonds Wetenschappelijk Onderzoek \u2013 Vlaanderen (project G0B2221N), by the Flanders AI Impulse programme (project DTAI-EWI-IMPULS-AI) and the KU Leuven C1 grant C14/19/082. Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.; 18th International Symposium on Artificial Intelligence and Mathematic, ISAIM ; Conference date: 08-01-2024 Through 10-01-2024",

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Vanbesien, L, Pollaci, S , Bogaerts, B & Denecker, M 2024, Towards a Unifying View on Monotone Constructive Definitions. in R Barneva, V Brimkov, C Gentile & A Pacchiano (eds), Artificial Intelligence and Image Analysis: 18th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2024, and 22nd International Workshop on Combinatorial Image Analysis, IWCIA 2024, Fort Lauderdale, FL, USA, January 8–10, 2024, Revised Selected Papers. 1 edn, Lecture Notes in Computer Science, vol. 14494, Springer Cham, pp. 218-235, 18th International Symposium on Artificial Intelligence and Mathematic, Fort Lauderdale, Florida, United States, 8/01/24. https://doi.org/10.1007/978-3-031-63735-3_13

Towards a Unifying View on Monotone Constructive Definitions. / Vanbesien, Linde; Pollaci, Samuele ; Bogaerts, Bart et al.
Artificial Intelligence and Image Analysis: 18th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2024, and 22nd International Workshop on Combinatorial Image Analysis, IWCIA 2024, Fort Lauderdale, FL, USA, January 8–10, 2024, Revised Selected Papers. ed. / Reneta Barneva; Valentin Brimkov; Claudio Gentile; Aldo Pacchiano. 1. ed. Springer Cham, 2024. p. 218-235 (Lecture Notes in Computer Science; Vol. 14494).

Research output: Chapter in Book/Report/Conference proceeding › Conference paper

TY - GEN

T1 - Towards a Unifying View on Monotone Constructive Definitions

AU - Vanbesien, Linde

AU - Pollaci, Samuele

AU - Bogaerts, Bart

AU - Denecker, Marc

N1 - Conference code: 18

PY - 2024/7/23

Y1 - 2024/7/23

N2 - Constructive definitions, including inductive and recursive definitions, are ubiquitous in mathematical texts and occur in a wide variety of computer science fields and Knowledge Representation applications. While in different areas there is a high level of familiarity with certain types of constructive definitions, fairly little interaction between different areas seems to exist, resulting in a lack of deep understanding of principles and their applications. This paper aims to fill this void by laying the foundations for a single unifying framework, bringing together a wide variety of definitions. First, we recall the principle of (monotone) inductive definition and its formalization in fixpoint theory. We discuss the constructive and the non-constructive interpretation of inductive definitions and the induction process. We then analyze examples, including but not limited to (co)inductive and (co)recursive definitions, found in a wide range of areas through the lens of our proposed framework.

AB - Constructive definitions, including inductive and recursive definitions, are ubiquitous in mathematical texts and occur in a wide variety of computer science fields and Knowledge Representation applications. While in different areas there is a high level of familiarity with certain types of constructive definitions, fairly little interaction between different areas seems to exist, resulting in a lack of deep understanding of principles and their applications. This paper aims to fill this void by laying the foundations for a single unifying framework, bringing together a wide variety of definitions. First, we recall the principle of (monotone) inductive definition and its formalization in fixpoint theory. We discuss the constructive and the non-constructive interpretation of inductive definitions and the induction process. We then analyze examples, including but not limited to (co)inductive and (co)recursive definitions, found in a wide range of areas through the lens of our proposed framework.

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SN - 9783031637346

T3 - Lecture Notes in Computer Science

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BT - Artificial Intelligence and Image Analysis

A2 - Barneva, Reneta

A2 - Brimkov, Valentin

A2 - Gentile, Claudio

A2 - Pacchiano, Aldo

PB - Springer Cham

T2 - 18th International Symposium on Artificial Intelligence and Mathematic

Y2 - 8 January 2024 through 10 January 2024

ER -

Vanbesien L, Pollaci S , Bogaerts B, Denecker M. Towards a Unifying View on Monotone Constructive Definitions. In Barneva R, Brimkov V, Gentile C, Pacchiano A, editors, Artificial Intelligence and Image Analysis: 18th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2024, and 22nd International Workshop on Combinatorial Image Analysis, IWCIA 2024, Fort Lauderdale, FL, USA, January 8–10, 2024, Revised Selected Papers. 1 ed. Springer Cham. 2024. p. 218-235. (Lecture Notes in Computer Science). doi: 10.1007/978-3-031-63735-3_13

Towards a Unifying View on Monotone Constructive Definitions

Abstract

Publication series

Conference

Bibliographical note

Access to Document

Handle.net

Other files and links

FWOAL990: Approximation Fixpoint Theory as a General Algebraic Theory of Constructive Knowledge

Cite this

Towards a Unifying View on Monotone Constructive Definitions

Abstract

Publication series

Conference

Bibliographical note

Access to Document

Handle.net

Other files and links

Projects

FWOAL990: Approximation Fixpoint Theory as a General Algebraic Theory of Constructive Knowledge

Cite this