TY - GEN
T1 - Formally Computing with the Non-computable
AU - Cohen, Liron
N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021/1/1
Y1 - 2021/1/1
N2 - Church–Turing computability, which is the standard notion of computation, is based on functions for which there is an effective method for constructing their values. However, intuitionistic mathematics, as conceived by Brouwer, extends the notion of effective algorithmic constructions by also admitting constructions corresponding to human experiences of mathematical truths, which are based on temporal intuitions. In particular, the key notion of infinitely proceeding sequences of freely chosen objects, known as free choice sequences, regards functions as being constructed over time. This paper describes how free choice sequences can be embedded in an implemented formal framework, namely the constructive type theory of the Nuprl proof assistant. Some broader implications of supporting such an extended notion of computability in a formal system are then discussed, focusing on formal verification and constructive mathematics.
AB - Church–Turing computability, which is the standard notion of computation, is based on functions for which there is an effective method for constructing their values. However, intuitionistic mathematics, as conceived by Brouwer, extends the notion of effective algorithmic constructions by also admitting constructions corresponding to human experiences of mathematical truths, which are based on temporal intuitions. In particular, the key notion of infinitely proceeding sequences of freely chosen objects, known as free choice sequences, regards functions as being constructed over time. This paper describes how free choice sequences can be embedded in an implemented formal framework, namely the constructive type theory of the Nuprl proof assistant. Some broader implications of supporting such an extended notion of computability in a formal system are then discussed, focusing on formal verification and constructive mathematics.
UR - https://www.scopus.com/pages/publications/85112138944
U2 - 10.1007/978-3-030-80049-9_12
DO - 10.1007/978-3-030-80049-9_12
M3 - Conference contribution
AN - SCOPUS:85112138944
SN - 9783030800482
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 135
EP - 145
BT - Connecting with Computability - 17th Conference on Computability in Europe, CiE 2021, Proceedings
A2 - De Mol, Liesbeth
A2 - Weiermann, Andreas
A2 - Manea, Florin
A2 - Fernández-Duque, David
PB - Springer Science and Business Media Deutschland GmbH
T2 - 17th Conference on Computability in Europe, CiE 2021
Y2 - 5 July 2021 through 9 July 2021
ER -