Abstract
This paper is a comparative study of a number of (intensional-semantically distinct) least and greatest fixed point operators that natural-deduction proof systems for intuitionistic logics can be extended with in a proof-theoretically defendable way. Eight pairs of such operators are analysed. The exposition is centred around a cube-shaped classification where each node stands for an axiomatization of one pair of operators as logical constants by intended proof and reduction rules and each arc for a proof- and reduction-preserving encoding of one pair in terms of another. The three dimensions of the cube reflect three orthogonal binary options: conventional-style vs. Mendler-style, basic ("[co]iterative") vs. enhanced ("primitive-[co]recursive"), simple vs. course-of-value [co]induction. Some of the axiomatizations and encodings are well known; others, however, are novel; the classification into a cube is also new. The differences between the least fixed point operators considered are illustrated on the example of the corresponding natural number types.
| Original language | English |
|---|---|
| Pages (from-to) | 315-339 |
| Journal | Theoretical Computer Science |
| Volume | 272 |
| Issue number | 1-2 |
| DOIs | |
| Publication status | Published - 6 Feb 2002 |
| Event | Theories of Types and Proofs 1997 (TTP'97) - Tokyo, Japan Duration: 8 Sept 1997 → 18 Sept 1997 |
Bibliographical note
Funding Information: T.U., V.V.: Partially supported by the Estonian Science Foundation under grant No. 2976. Publisher Copyright: © 2002 Elsevier Science B.V.Other keywords
- (Co)Inductive types
- Coding styles
- Least and greatest fixed points
- Natural deduction
- Schemes of (total) (co)recursion
- Typed lambda calculi