## A Proof-Theoretic Characterization of the Basic Feasible Functionals (2002)

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Venue: | Theoretical Computer Science |

Citations: | 7 - 6 self |

### BibTeX

@TECHREPORT{Strahm02aproof-theoretic,

author = {Thomas Strahm},

title = {A Proof-Theoretic Characterization of the Basic Feasible Functionals},

institution = {Theoretical Computer Science},

year = {2002}

}

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### Abstract

We provide a natural characterization of the type two Mehlhorn-CookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27].

### Citations

186 | A new recursion-theoretic characterization of the poly-time functions
- Bellantoni, Cook
- 1992
(Show Context)
Citation Context ...Last but not least, let us mention the important activities in the program of so-called implicit computational complexity and tiered formalisms in the sense of Bellantoni, Cook, and Leivant (cf. e.g. =-=[4, 20, 22]-=-). There questions regarding higher types have recently been of interest, see for example Leivant [21], Bellantoni, Niggl, Schwichtenberg [5], and Hofmann [17]. For applicative theories based on safe ... |

122 |
Bounded Arithmetic. Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ... ranging over W, however. Assuming that the bounding operation f in a Σb W formula has polynomial growth, Σb W formulas can be seen as a very abstract applicative analogue of Buss’ Σb 1 formulas =-=(cf. [6]) o-=-r Ferreira’s NP formulas (cf. [14, 15]). Notice, however, whereas the latter classes of formulas define exactly the NP predicates, Σb W formulas of L in general define highly undecidable sets in th... |

119 |
Introduction to Combinators and λCalculus
- Hindley, Seldin
- 1986
(Show Context)
Citation Context ...ants of L so that all the axioms of PT are true in PRO. A further important model of PT is the open term model M(λη). This model is based on the usual λη reduction of the untyped lambda calculus (=-=cf. [1, 16]) an-=-d exploits the well-known equivalence between combinatory logic with extensionality and λη. In order to deal with the constants different from k and s, one extends λη reduction by the obvious redu... |

90 |
A language and axioms for explicit mathematics
- Feferman
- 1975
(Show Context)
Citation Context ...erpins their importance as a key candidate for the notion of type two feasibility. 1 Introduction In this paper we deal with applicative theories in the spirit of Feferman’s explicit mathematics (cf=-=. [12, 13]-=-). The paper is a successor to Strahm [27] (cf. also [28]), where so-called bounded applicative theories with a strong relationship to classes of computational complexity have been introduced and anal... |

87 | Functional interpretations of feasibly constructive arithmetic
- Cook, Urquhart
- 1993
(Show Context)
Citation Context ...nd, most importantly, oracle Turing machine characterizations. The basic feasible functionals of type 2, Bff2, were first studied in Melhorn [23]. More than ten years later in 1989, Cook and Urquhart =-=[11]-=- introduced the basic feasible functionals at all finite types in order to provide functional interpretations of feasibly constructive arithmetic; in particular, they defined a typed formal system PV ... |

72 |
Foundations of constructive mathematics, Metamathematical Studies
- Beeson
- 1985
(Show Context)
Citation Context ...fine highly undecidable sets in the standard models of PT described below. We now introduce the applicative theory PT. The underlying logic of PT is the classical logic of partial terms due to Beeson =-=[2, 3]; it-=- corresponds to E + logic with strictness and equality of Troelstra and Van Dalen [29]. According to this logic, quantifiers range over defined objects only, so that the usual axioms for ∃ and ∀ a... |

66 |
Constructive theories of functions and classes
- Feferman
- 1979
(Show Context)
Citation Context ...erpins their importance as a key candidate for the notion of type two feasibility. 1 Introduction In this paper we deal with applicative theories in the spirit of Feferman’s explicit mathematics (cf=-=. [12, 13]-=-). The paper is a successor to Strahm [27] (cf. also [28]), where so-called bounded applicative theories with a strong relationship to classes of computational complexity have been introduced and anal... |

63 |
Predicative recurrence and computational complexity I: word recurrence and poly-time
- Leivant
- 1994
(Show Context)
Citation Context ...Last but not least, let us mention the important activities in the program of so-called implicit computational complexity and tiered formalisms in the sense of Bellantoni, Cook, and Leivant (cf. e.g. =-=[4, 20, 22]-=-). There questions regarding higher types have recently been of interest, see for example Leivant [21], Bellantoni, Niggl, Schwichtenberg [5], and Hofmann [17]. For applicative theories based on safe ... |

39 | Computation models and function algebras
- Clote
- 1995
(Show Context)
Citation Context ...ined by PV ω terms. Subsequently, much work has been devoted to Bff, cf. e.g. Cook and Kapron [10, 19], Irwin, Kapron and Royer [18], Pezzoli [24], Royer [25], and Seth [26]. The survey article Clote=-= [9]-=- contains some of the key results concerning the basic feasible functionals. The main result obtained in this article states that the provably total type two functionals of PT coincide with the basic ... |

36 | A new characterization of type-2 feasibility
- Kapron, Cook
- 1996
(Show Context)
Citation Context ...n of Buss’ theory S 1 2. The basic feasible functionals Bff are exactly those functionals which can be defined by PV ω terms. Subsequently, much work has been devoted to Bff, cf. e.g. Cook and Kapr=-=on [10, 19]-=-, Irwin, Kapron and Royer [18], Pezzoli [24], Royer [25], and Seth [26]. The survey article Clote [9] contains some of the key results concerning the basic feasible functionals. The main result obtain... |

30 |
A foundational delineation of poly-time
- Leivant
- 1994
(Show Context)
Citation Context ...Last but not least, let us mention the important activities in the program of so-called implicit computational complexity and tiered formalisms in the sense of Bellantoni, Cook, and Leivant (cf. e.g. =-=[4, 20, 22]-=-). There questions regarding higher types have recently been of interest, see for example Leivant [21], Bellantoni, Niggl, Schwichtenberg [5], and Hofmann [17]. For applicative theories based on safe ... |

27 | Characterizations of the basic feasible functions of finite type
- Cook, Kapron
- 1990
(Show Context)
Citation Context ...n of Buss’ theory S 1 2. The basic feasible functionals Bff are exactly those functionals which can be defined by PV ω terms. Subsequently, much work has been devoted to Bff, cf. e.g. Cook and Kapr=-=on [10, 19]-=-, Irwin, Kapron and Royer [18], Pezzoli [24], Royer [25], and Seth [26]. The survey article Clote [9] contains some of the key results concerning the basic feasible functionals. The main result obtain... |

23 |
Proving programs and programming proofs
- Beeson
- 1986
(Show Context)
Citation Context ...fine highly undecidable sets in the standard models of PT described below. We now introduce the applicative theory PT. The underlying logic of PT is the classical logic of partial terms due to Beeson =-=[2, 3]; it-=- corresponds to E + logic with strictness and equality of Troelstra and Van Dalen [29]. According to this logic, quantifiers range over defined objects only, so that the usual axioms for ∃ and ∀ a... |

23 | Higher type recursion, ramification and polynomial time
- Bellantoni, Niggl, et al.
- 2000
(Show Context)
Citation Context ...e sense of Bellantoni, Cook, and Leivant (cf. e.g. [4, 20, 22]). There questions regarding higher types have recently been of interest, see for example Leivant [21], Bellantoni, Niggl, Schwichtenberg =-=[5]-=-, and Hofmann [17]. For applicative theories based on safe induction, see Cantini [8]. Recently and independently, Leivant has given a proof-theoretic characterization of Bff in terms of second order ... |

23 |
Polynomial Time Computable Arithmetic and Conservative Extensions
- Ferreira
- 1988
(Show Context)
Citation Context ...hat the bounding operation f in a Σb W formula has polynomial growth, Σb W formulas can be seen as a very abstract applicative analogue of Buss’ Σb 1 formulas (cf. [6]) or Ferreira’s NP formula=-=s (cf. [14, 15])-=-. Notice, however, whereas the latter classes of formulas define exactly the NP predicates, Σb W formulas of L in general define highly undecidable sets in the standard models of PT described below. ... |

17 |
Complexity theory of higher type functionals
- Seth
- 1994
(Show Context)
Citation Context ...se functionals which can be defined by PV ω terms. Subsequently, much work has been devoted to Bff, cf. e.g. Cook and Kapron [10, 19], Irwin, Kapron and Royer [18], Pezzoli [24], Royer [25], and Seth=-= [26]-=-. The survey article Clote [9] contains some of the key results concerning the basic feasible functionals. The main result obtained in this article states that the provably total type two functionals ... |

15 |
Type systems for polynomial-time computation, Habilitation thesis
- Hofmann
- 1999
(Show Context)
Citation Context ...toni, Cook, and Leivant (cf. e.g. [4, 20, 22]). There questions regarding higher types have recently been of interest, see for example Leivant [21], Bellantoni, Niggl, Schwichtenberg [5], and Hofmann =-=[17]-=-. For applicative theories based on safe induction, see Cantini [8]. Recently and independently, Leivant has given a proof-theoretic characterization of Bff in terms of second order logic with positiv... |

15 |
On characterizations of the basic feasible functionals
- Irwin, Kapron, et al.
(Show Context)
Citation Context ... feasible functionals Bff are exactly those functionals which can be defined by PV ω terms. Subsequently, much work has been devoted to Bff, cf. e.g. Cook and Kapron [10, 19], Irwin, Kapron and Royer=-= [18]-=-, Pezzoli [24], Royer [25], and Seth [26]. The survey article Clote [9] contains some of the key results concerning the basic feasible functionals. The main result obtained in this article states that... |

14 |
Predicative recurrence in finite type
- Leivant
- 1994
(Show Context)
Citation Context ...l complexity and tiered formalisms in the sense of Bellantoni, Cook, and Leivant (cf. e.g. [4, 20, 22]). There questions regarding higher types have recently been of interest, see for example Leivant =-=[21]-=-, Bellantoni, Niggl, Schwichtenberg [5], and Hofmann [17]. For applicative theories based on safe induction, see Cantini [8]. Recently and independently, Leivant has given a proof-theoretic characteri... |

12 |
combinatory logic and positive safe induction
- Cantini
(Show Context)
Citation Context ...rding higher types have recently been of interest, see for example Leivant [21], Bellantoni, Niggl, Schwichtenberg [5], and Hofmann [17]. For applicative theories based on safe induction, see Cantini =-=[8]. -=-Recently and independently, Leivant has given a proof-theoretic characterization of Bff in terms of second order logic with positive comprehension (work to be presented at CSL ’02). We will compare ... |

12 | Theories with self-application and computational complexity
- Strahm
(Show Context)
Citation Context ...e provide a natural characterization of the type two Mehlhorn-CookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in =-=[27]-=-, thus providing a proof of a result claimed in the conclusion of [27]. This further characterization of the basic feasible functionals underpins their importance as a key candidate for the notion of ... |

11 | and uniformity in weak applicative theories
- Cantini
(Show Context)
Citation Context ...sses of computational complexity have been introduced and analyzed. For a more detailed background on applicative theories, we refer the reader to [27] and the articles cited there. Recently, Cantini =-=[7]-=- has studied substantial extensions of the theories introduced in [27] by choice and uniformity principles as well as a form of self-referential truth. The main emphasis in [27] is on four applicative... |

10 | Semantics vs. syntax vs. computations: Machine models for type-2 polynomial-time bounded functionals
- Royer
- 1997
(Show Context)
Citation Context ...are exactly those functionals which can be defined by PV ω terms. Subsequently, much work has been devoted to Bff, cf. e.g. Cook and Kapron [10, 19], Irwin, Kapron and Royer [18], Pezzoli [24], Royer=-= [25]-=-, and Seth [26]. The survey article Clote [9] contains some of the key results concerning the basic feasible functionals. The main result obtained in this article states that the provably total type t... |

10 |
Proof Theory and Logical Complexitiy
- Girard
- 1987
(Show Context)
Citation Context ...ules of PT + are positive, we now obtain the desired partial cut elimination theorem for PT + . Its proof is immediate from the well-known proof of the cut elimination theorem for LK (cf. e.g. Girard =-=[16]-=-) and is therefore omitted. Theorem 2 (Partial cut elimination for PT + ) We have for all sequents Γ ⇒ ∆. Γ ⇒ ∆ that PT + Γ ⇒ ∆ entails PT + The second crucial step in our upper bound argument consist... |

7 |
The Lambda Calculus, revised ed
- Barendregt
- 1985
(Show Context)
Citation Context ...ants of L so that all the axioms of PT are true in PRO. A further important model of PT is the open term model M(λη). This model is based on the usual λη reduction of the untyped lambda calculus (=-=cf. [1, 16]) an-=-d exploits the well-known equivalence between combinatory logic with extensionality and λη. In order to deal with the constants different from k and s, one extends λη reduction by the obvious redu... |

7 |
Polynomial time computable arithmetic. In Logic and Computation, Proceedings of a Workshop held at
- Ferreira
- 1987
(Show Context)
Citation Context ...hat the bounding operation f in a Σb W formula has polynomial growth, Σb W formulas can be seen as a very abstract applicative analogue of Buss’ Σb 1 formulas (cf. [6]) or Ferreira’s NP formula=-=s (cf. [14, 15])-=-. Notice, however, whereas the latter classes of formulas define exactly the NP predicates, Σb W formulas of L in general define highly undecidable sets in the standard models of PT described below. ... |

7 |
Polynomial and Abstract Subrecursive Classes
- Melhorn
- 1976
(Show Context)
Citation Context ... lambda calculus, function algebra, programming language, and, most importantly, oracle Turing machine characterizations. The basic feasible functionals of type 2, Bff2, were first studied in Melhorn =-=[23]-=-. More than ten years later in 1989, Cook and Urquhart [11] introduced the basic feasible functionals at all finite types in order to provide functional interpretations of feasibly constructive arithm... |

5 | Proof-theoretic Contributions to Explicit Mathematics - Strahm - 2001 |

3 |
On the computational complexity of type 2 functionals
- Pezzoli
- 1998
(Show Context)
Citation Context ...tionals Bff are exactly those functionals which can be defined by PV ω terms. Subsequently, much work has been devoted to Bff, cf. e.g. Cook and Kapron [10, 19], Irwin, Kapron and Royer [18], Pezzoli=-= [24]-=-, Royer [25], and Seth [26]. The survey article Clote [9] contains some of the key results concerning the basic feasible functionals. The main result obtained in this article states that the provably ... |

3 |
Implicit Computational Complexity for Higher Type Functionals
- Leivant
- 2002
(Show Context)
Citation Context ...interest, see for example Leivant [22], Bellantoni, Niggl, Schwichtenberg [5], and Hofmann [18]. For applicative theories based on safe induction, see Cantini [8]. Recently and independently, Leivant =-=[24]-=- has given a proof-theoretic characterization of Bff in terms of second order logic with positive comprehension We will compare our approach with Leivant’s elsewhere. Acknowledgments. We would like to... |