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
Proof normalization with nonstandard objects. / Goto, Shigeki.
In: Theoretical Computer Science, Vol. 85, No. 2, 12.08.1991, p. 333-351.Research output: Contribution to journal › Article
}
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
UR - http://www.scopus.com/inward/citedby.url?scp=0026206523&partnerID=8YFLogxK
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 -