### Abstract

It is well known that formal proof systems can serve as programming languages. A proof that describes an algorithm can be executed by Prawitz's normalization procedure. This paper extends the computational use of proofs to realize a lazy computation formally. It enables computation of a proof over stream objects (infinitely long lists) as in concurrent Prolog. In this paper we follow the natural deduction formalism. Our presentation of natural deduction differs from Gentzen's system in the existential elimination rule. We apply the Borkowski-Słupecki's device. There is no difference between Gentzen's rule and Borkowski-Słupecki's rule as far as formula provability is concerned. However, the new rule is essential to proof normalization. A new concept, the pseudonormal proof, is introduced to formalize our normalization method. To deal with infinitely long objects we extend the number theory to incorporate infinite numbers. This is an application of nonstandard analysis to computer programs. We show that the rule of mathematical induction can be extended to cover infinite numbers with appropriate computational meaning.

Original language | English |
---|---|

Pages (from-to) | 333-351 |

Number of pages | 19 |

Journal | Theoretical Computer Science |

Volume | 85 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1991 Aug 12 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Theoretical Computer Science*,

*85*(2), 333-351. https://doi.org/10.1016/0304-3975(91)90186-6

**Proof normalization with nonstandard objects.** / Goto, Shigeki.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 85, no. 2, pp. 333-351. https://doi.org/10.1016/0304-3975(91)90186-6

}

TY - JOUR

T1 - Proof normalization with nonstandard objects

AU - Goto, Shigeki

PY - 1991/8/12

Y1 - 1991/8/12

N2 - It is well known that formal proof systems can serve as programming languages. A proof that describes an algorithm can be executed by Prawitz's normalization procedure. This paper extends the computational use of proofs to realize a lazy computation formally. It enables computation of a proof over stream objects (infinitely long lists) as in concurrent Prolog. In this paper we follow the natural deduction formalism. Our presentation of natural deduction differs from Gentzen's system in the existential elimination rule. We apply the Borkowski-Słupecki's device. There is no difference between Gentzen's rule and Borkowski-Słupecki's rule as far as formula provability is concerned. However, the new rule is essential to proof normalization. A new concept, the pseudonormal proof, is introduced to formalize our normalization method. To deal with infinitely long objects we extend the number theory to incorporate infinite numbers. This is an application of nonstandard analysis to computer programs. We show that the rule of mathematical induction can be extended to cover infinite numbers with appropriate computational meaning.

AB - It is well known that formal proof systems can serve as programming languages. A proof that describes an algorithm can be executed by Prawitz's normalization procedure. This paper extends the computational use of proofs to realize a lazy computation formally. It enables computation of a proof over stream objects (infinitely long lists) as in concurrent Prolog. In this paper we follow the natural deduction formalism. Our presentation of natural deduction differs from Gentzen's system in the existential elimination rule. We apply the Borkowski-Słupecki's device. There is no difference between Gentzen's rule and Borkowski-Słupecki's rule as far as formula provability is concerned. However, the new rule is essential to proof normalization. A new concept, the pseudonormal proof, is introduced to formalize our normalization method. To deal with infinitely long objects we extend the number theory to incorporate infinite numbers. This is an application of nonstandard analysis to computer programs. We show that the rule of mathematical induction can be extended to cover infinite numbers with appropriate computational meaning.

UR - http://www.scopus.com/inward/record.url?scp=0026206523&partnerID=8YFLogxK

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U2 - 10.1016/0304-3975(91)90186-6

DO - 10.1016/0304-3975(91)90186-6

M3 - Article

AN - SCOPUS:0026206523

VL - 85

SP - 333

EP - 351

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 2

ER -