## Higher Type Recursion, Ramification and Polynomial Time (1999)

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Venue: | Annals of Pure and Applied Logic |

Citations: | 22 - 3 self |

### BibTeX

@INPROCEEDINGS{Bellantoni99highertype,

author = {S. Bellantoni and K.-H. Niggl and H. Schwichtenberg},

title = {Higher Type Recursion, Ramification and Polynomial Time},

booktitle = {Annals of Pure and Applied Logic},

year = {1999},

pages = {17--30}

}

### Years of Citing Articles

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### Abstract

It is shown how to restrict recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types !oe, and by adding linear concepts to the lambda calculus. 1 Introduction Recursion in all finite types was introduced by Hilbert [9] and later became known as the essential part of Godel's system T [8]. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time. Simmons [16] showed that ramification can be used to characterize the primitive recursive functions by higher type recursion, and Leivant and Marion [14] showed that another form of ramification can be used to restrict higher type recursion to PSPACE. However, to characterize the much smaller class of polynomial-time computable functions by higher type recursion, it seems that an additional principle is required. By introducing linear...

### Citations

621 | Linear Logic
- Girard
- 1987
(Show Context)
Citation Context ... the present work there are no explicit polynomials and there is no attribution of work to occurrences of !. There are some connections between the present work and the "light linear logic" =-=of Girard [7]-=-; but due to differing frameworks an exact comparison has not been made. The modality ! used in the present work is different from the linear modality because the ! here includes concepts of ramificat... |

281 | Computational interpretations of linear logic
- Abramsky
- 1993
(Show Context)
Citation Context ...oof methods of the two papers are also different, as Hofmann uses a category-theoretic approach. This work has also connections to work in linear logic (see Girard, Scedrov and Scott [6] and Abramsky =-=[1] for backg-=-round on linear logic), where "!" refers to a knowledge of the input that allows it to be used nonlinearly. However, the logic behind the present work differs significantly from the polytime... |

179 | A new recursion-theoretic characterization of the polytime functions
- Bellantoni, Cook
- 1992
(Show Context)
Citation Context ... types have no free variables of incomplete types. Input positions of types !' and ' correspond to normal / tier 1 and safe / tier 0 input positions, common in earlier work on ramified recursion (cf. =-=[16, 13, 2, 4, 15]-=-). Affinability is central to the system and expresses the linearity constraints for bound variables of incomplete types. Affinability is designed such that the system RA is closed under reduction. Or... |

162 | Basic Proof Theory
- Troelstra, Schwichtenberg
- 2000
(Show Context)
Citation Context ...+1. Redexes are subterms shown on the left side of conversion rules above. A term is in normal form if it does not contain a redex. For every term t there is a unique normal-form term nf(t) (see e.g. =-=[17, 12] for proofs of norma-=-lisation in Godel's system T ). Two terms are equivalent if they have the same normal form. 1 Linear logicians may read "!" as "(". 3 One writes FV(t) for the set of free variables... |

142 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica 12
- Gödel
- 1958
(Show Context)
Citation Context ... !oe, and by adding linear concepts to the lambda calculus. 1 Introduction Recursion in all finite types was introduced by Hilbert [9] and later became known as the essential part of Godel's system T =-=[8]-=-. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time. Simmons [16] showed that ramification can be used to characterize th... |

62 |
Bounded linear logic: A modular approach to polynomial-time computability
- Girard, Scedrov, et al.
- 1992
(Show Context)
Citation Context ... coincide. The proof methods of the two papers are also different, as Hofmann uses a category-theoretic approach. This work has also connections to work in linear logic (see Girard, Scedrov and Scott =-=[6] and Abram-=-sky [1] for background on linear logic), where "!" refers to a knowledge of the input that allows it to be used nonlinearly. However, the logic behind the present work differs significantly ... |

32 |
Über das Unendliche
- Hilbert
- 1926
(Show Context)
Citation Context ... obtained by enriching the type structure with the formation of types !oe, and by adding linear concepts to the lambda calculus. 1 Introduction Recursion in all finite types was introduced by Hilbert =-=[9]-=- and later became known as the essential part of Godel's system T [8]. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time.... |

27 | Characterizations of the basic feasible functions of finite type
- Cook, Kapron
- 1990
(Show Context)
Citation Context ...ation from the study of ordinary ramified recursion. The approach to higher-type functions taken in this work contrasts with Cook and Kapron's wellknown Basic Feasible Functions defined by PV ! terms =-=[5]-=-. There, explicit size bounds are used and the critical value computed during the recursion is of ground type. A further difference can be seen by the fact that the system RA admits the iteration func... |

16 | A mixed modal/linear lambda calculus with applications to BellantoniCook safe recursion
- Hofmann
- 1997
(Show Context)
Citation Context ... nf(t~n) in time p t (j~nj). Thus, t denotes a polynomial time computable function. The converse also holds, as each polynomial time computable function is computed by some RA-term. Recently, Hofmann =-=[10, 11]-=- used modalities of ramification and of linearity in a lambda calculus, and defined them for all higher types. This interesting work also characterizes polynomial time computability. There, the two mo... |

15 |
Subrecursion and lambda representation over free algebras (preliminary summary
- Leivant
- 1990
(Show Context)
Citation Context ... types have no free variables of incomplete types. Input positions of types !' and ' correspond to normal / tier 1 and safe / tier 0 input positions, common in earlier work on ramified recursion (cf. =-=[16, 13, 2, 4, 15]-=-). Affinability is central to the system and expresses the linearity constraints for bound variables of incomplete types. Affinability is designed such that the system RA is closed under reduction. Or... |

15 |
Predicative recurrence and computational complexity IV: Predicative functionals and poly-space. should be published
- Leivant, Marion
- 1994
(Show Context)
Citation Context ... small complexity classes such as polynomial time. Simmons [16] showed that ramification can be used to characterize the primitive recursive functions by higher type recursion, and Leivant and Marion =-=[14]-=- showed that another form of ramification can be used to restrict higher type recursion to PSPACE. However, to characterize the much smaller class of polynomial-time computable functions by higher typ... |

10 | and K-H Niggl. Ranking primitive recursions: The low Grzegorczyk classes revisited
- Bellantoni
- 1999
(Show Context)
Citation Context ... types have no free variables of incomplete types. Input positions of types !' and ' correspond to normal / tier 1 and safe / tier 0 input positions, common in earlier work on ramified recursion (cf. =-=[16, 13, 2, 4, 15]-=-). Affinability is central to the system and expresses the linearity constraints for bound variables of incomplete types. Affinability is designed such that the system RA is closed under reduction. Or... |

10 |
The realm of primitive recursion
- Simmons
- 1988
(Show Context)
Citation Context ...r became known as the essential part of Godel's system T [8]. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time. Simmons =-=[16]-=- showed that ramification can be used to characterize the primitive recursive functions by higher type recursion, and Leivant and Marion [14] showed that another form of ramification can be used to re... |

7 |
The µ-measure as a tool for classifying computational complexity
- Niggl
(Show Context)
Citation Context |

1 | Characterizing parallel polylog time using type 2 recursions with polynomial output length - Bellantoni - 1995 |

1 |
New results on linear/modal lambda calculus: Higher Result types and recursion on trees
- Hofmann
- 1998
(Show Context)
Citation Context ... nf(t~n) in time p t (j~nj). Thus, t denotes a polynomial time computable function. The converse also holds, as each polynomial time computable function is computed by some RA-term. Recently, Hofmann =-=[10, 11]-=- used modalities of ramification and of linearity in a lambda calculus, and defined them for all higher types. This interesting work also characterizes polynomial time computability. There, the two mo... |

1 |
Short proofs of normalisation for the simply-typed -calculus, permutative conversions and Godel's
- Joachimski, Matthes
- 1998
(Show Context)
Citation Context ...+1. Redexes are subterms shown on the left side of conversion rules above. A term is in normal form if it does not contain a redex. For every term t there is a unique normal-form term nf(t) (see e.g. =-=[17, 12] for proofs of norma-=-lisation in Godel's system T ). Two terms are equivalent if they have the same normal form. 1 Linear logicians may read "!" as "(". 3 One writes FV(t) for the set of free variables... |