TY - GEN

T1 - Inductive Continuity via Brouwer Trees

AU - Cohen, Liron

AU - da Rocha Paiva, Bruno

AU - Rahli, Vincent

AU - Tosun, Ayberk

N1 - Funding Information:
Funding Liron Cohen: This research was partially supported by Grant No. 2020145 from the United States-Israel Binational Science Foundation (BSF).
Publisher Copyright:
© Liron Cohen, Bruno da Rocha Paiva, Vincent Rahli, and Ayberk Tosun;

PY - 2023/8/1

Y1 - 2023/8/1

N2 - Continuity is a key principle of intuitionistic logic that is generally accepted by constructivists but is inconsistent with classical logic. Most commonly, continuity states that a function from the Baire space to numbers, only needs approximations of the points in the Baire space to compute. More recently, another formulation of the continuity principle was put forward. It states that for any function F from the Baire space to numbers, there exists a (dialogue) tree that contains the values of F at its leaves and such that the modulus of F at each point of the Baire space is given by the length of the corresponding branch in the tree. In this paper we provide the first internalization of this “inductive” continuity principle within a computational setting. Concretely, we present a class of intuitionistic theories that validate this formulation of continuity thanks to computations that construct such dialogue trees internally to the theories using effectful computations. We further demonstrate that this inductive continuity principle implies other forms of continuity principles.

AB - Continuity is a key principle of intuitionistic logic that is generally accepted by constructivists but is inconsistent with classical logic. Most commonly, continuity states that a function from the Baire space to numbers, only needs approximations of the points in the Baire space to compute. More recently, another formulation of the continuity principle was put forward. It states that for any function F from the Baire space to numbers, there exists a (dialogue) tree that contains the values of F at its leaves and such that the modulus of F at each point of the Baire space is given by the length of the corresponding branch in the tree. In this paper we provide the first internalization of this “inductive” continuity principle within a computational setting. Concretely, we present a class of intuitionistic theories that validate this formulation of continuity thanks to computations that construct such dialogue trees internally to the theories using effectful computations. We further demonstrate that this inductive continuity principle implies other forms of continuity principles.

KW - Agda

KW - Constructive Type Theory

KW - Continuity

KW - Dialogue trees

KW - Extensional Type Theory

KW - Intuitionistic Logic

KW - Realizability

KW - Stateful computations

KW - Theorem proving

UR - http://www.scopus.com/inward/record.url?scp=85166081750&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2023.37

DO - 10.4230/LIPIcs.MFCS.2023.37

M3 - Conference contribution

AN - SCOPUS:85166081750

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023

A2 - Leroux, Jerome

A2 - Lombardy, Sylvain

A2 - Peleg, David

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023

Y2 - 28 August 2023 through 1 September 2023

ER -