## Light Functional Interpretation (2005)

Venue: | Lecture Notes in Computer Science, 3634:477 – 492, July 2005. Computer Science Logic: 19th International Workshop, CSL |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Hernest05lightfunctional,

author = {Mircea-dan Hernest},

title = {Light Functional Interpretation},

booktitle = {Lecture Notes in Computer Science, 3634:477 – 492, July 2005. Computer Science Logic: 19th International Workshop, CSL},

year = {2005},

pages = {477}

}

### OpenURL

### Abstract

an optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs

### Citations

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Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes
- Gödel
- 1958
(Show Context)
Citation Context ...ut-elimination has opened the path to the obtention of better practical results by means of such modular techniques (see [15] for a discussion on this). Gödel’s functional (Dialectica) interpretation =-=[13, 2]-=- directly applies to (far) more complex proofs than (its weaker form) Kreisel’s ⋆ Project LogiCal - Pôle Commun de Recherche en Informatique du Plateau de Saclay, CNRS, École Polytechnique, INRIA et U... |

101 |
Proofs as programs
- Bates, Constable
- 1985
(Show Context)
Citation Context ...ertenkolleg Logik in der Informatik (GKLI) - München, GERMANY. Partly financed by Deutsche Forschungsgemeinschaft. 1 Constructive proofs have a more or less explicit computational content, see, e.g., =-=[4]-=-.sModified Realizability [23]. Refinements [1, 6, 9] of the latter’s combination with Friedman’s A-translation [12] only partly repair this disparity (see [15] or [20] for discussions on this issue). ... |

78 |
Interpretation of analysis by means of constructive functionals of finite types
- Kreisel
- 1959
(Show Context)
Citation Context ...ormatik (GKLI) - München, GERMANY. Partly financed by Deutsche Forschungsgemeinschaft. 1 Constructive proofs have a more or less explicit computational content, see, e.g., [4].sModified Realizability =-=[23]-=-. Refinements [1, 6, 9] of the latter’s combination with Friedman’s A-translation [12] only partly repair this disparity (see [15] or [20] for discussions on this issue). However, the (rough) exact re... |

74 |
Classicaly and intuitionistically provable recursive functions
- Friedman
- 1978
(Show Context)
Citation Context ... 1 Constructive proofs have a more or less explicit computational content, see, e.g., [4].sModified Realizability [23]. Refinements [1, 6, 9] of the latter’s combination with Friedman’s A-translation =-=[12]-=- only partly repair this disparity (see [15] or [20] for discussions on this issue). However, the (rough) exact realizers yielded by the Dialectica interpretation are generally (far) more complex than... |

69 | Gödel’s functional (“Dialectica”) interpretation
- Avigad, Feferman
- 1998
(Show Context)
Citation Context ...ut-elimination has opened the path to the obtention of better practical results by means of such modular techniques (see [15] for a discussion on this). Gödel’s functional (Dialectica) interpretation =-=[13, 2]-=- directly applies to (far) more complex proofs than (its weaker form) Kreisel’s ⋆ Project LogiCal - Pôle Commun de Recherche en Informatique du Plateau de Saclay, CNRS, École Polytechnique, INRIA et U... |

54 | Program extraction from classical proofs
- Berger, Schwichtenberg
- 1994
(Show Context)
Citation Context ...d the realizers for the Dialectica interpretation of higher-order Symmetry and Transitivity are immediate (see Section 3 of [14]). On the other hand we thus stay as close as possible to the axiomatic =-=[34, 6]-=- of system Z rendering easier the task of implementing program-extraction by (light) Dialectica interpretation in Minlog [36]. We do, however, have to deviate from system Z when it comes to the Compat... |

53 | Analysing proofs in analysis
- Kohlenbach
- 1993
(Show Context)
Citation Context ...e functional interpretation We sketch below the combination of our refinement of Goedel’s Dialectica interpretation [19, 2, 13] with its optimization for the extraction of bounds 21 due to Kohlenbach =-=[21]-=-. We add to systems WE−Z − , WE−Z and WE−Z + a functional inequality constant for naturals (of type ιιo), denoted ≥. We define predicate inequality between terms s ι and t ι as an abbreviation for at(... |

44 | Classical Logic, Storage Operators and Second-Order lambda-Calculus - Krivine - 1994 |

41 |
Hereditarily majorizable functionals of finite type
- Howard
- 1973
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Citation Context ... + , hence full extensionality would be derivable and Dialectica interpretation would fail to interpret all proofs of our system. 13 axiom Eιι,ι by Gödel primitive recursive functionals is exposed in =-=[17]-=-. See also [35] for a counterexample to the Dialectica realizability of Eιι,ι by Van de Pol - Schwichtenberg monotone majorizable functionals (a class of functionals intersecting but independent of Gö... |

32 | Extracting constructive content from classical logic via control-like reductions
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Citation Context ...of [14]. [A] . . . /B A → B →+ n+1 � �� � We are given, with n ≥ 1, z ≡ z, . . . , z and x ≡ xn+2, . . . , xm, that : {AD(z; Ti(z, x, y))} n+1 i=1 , {Ci D(xi; Ti(z, x, y))} m i=n+2 ⊢− BD(T (z, x); y) =-=(3)-=- It has been assumed that n + 1 ≤ m , where n + 1 is the number of copies of the assumption A which get discharged in this → + . Each of these n + 1 instances of A produces the same tuple z of existen... |

26 |
Extensional Gödel Functional Interpretation
- Luckhardt
- 1973
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Citation Context ... be called the light monotone 17 functional interpretation (abbreviated LMD-interpretation). The D-interpretation was first introduced in [13] for a Hilbert-style formulation of Arithmetic – see also =-=[38, 26, 10, 2, 20]-=- for other (more modern) formulations within Hilbert-style systems. Natural Deduction formulations of the Diller-Nahm [10] variant of D-interpretation were provided by Diller’s students Rath [33] and ... |

23 | The Warshall Algorithm and Dickson’s Lemma: Two Examples of Realistic Program Extraction
- Berger, Schwichtenberg, et al.
(Show Context)
Citation Context ...field of extractive Proof Theory (also dubbed proof mining [22]). The implemented algorithms coming from metamathematical research have yielded interesting and in many cases quite unexpected programs =-=[7, 15, 31, 32]-=-. Various approaches to program extraction from classical 1 proofs have been developed over years of research [1, 3, 6, 8, 9, 21, 24, 25, 27–30, 32]. The use of proof interpretations instead of the di... |

21 |
Interpreting classical theories in constructive ones
- Avigad
(Show Context)
Citation Context ...ünchen, GERMANY. Partly financed by Deutsche Forschungsgemeinschaft. 1 Constructive proofs have a more or less explicit computational content, see, e.g., [4].sModified Realizability [23]. Refinements =-=[1, 6, 9]-=- of the latter’s combination with Friedman’s A-translation [12] only partly repair this disparity (see [15] or [20] for discussions on this issue). However, the (rough) exact realizers yielded by the ... |

20 |
Minimalkalkül, ein reduzierter intuitionistischer Formalismus
- Der
- 1936
(Show Context)
Citation Context ...tional content. They also bring an important optimization with respect to the maximal type degree of programs extracted from those proofs for which the 6 The prominent place Johansson’s Minimal Logic =-=[18]-=- has in the presentation of system Z in [6] or [34] is no longer needed for our (light) Dialectica-extraction exposition. 7 Notice that ∀ is as computationally meaningful as ∃ in the context of progra... |

17 | Finding computational content in classical proofs - Constable, Murthy - 1991 |

16 | A new method of establishing conservativity of classical systems over their intuitionistic version
- Coquand, Hofmann
- 1999
(Show Context)
Citation Context ...ünchen, GERMANY. Partly financed by Deutsche Forschungsgemeinschaft. 1 Constructive proofs have a more or less explicit computational content, see, e.g., [4].sModified Realizability [23]. Refinements =-=[1, 6, 9]-=- of the latter’s combination with Friedman’s A-translation [12] only partly repair this disparity (see [15] or [20] for discussions on this issue). However, the (rough) exact realizers yielded by the ... |

15 | Proof mining: a systematic way of analysing proofs in mathematics
- Kohlenbach, Oliva
- 2003
(Show Context)
Citation Context ...ica) interpretation, Proof Mining, Proof-Carrying Code. 1 Introduction Important practical results have been obtained in recent years in the field of extractive Proof Theory (also dubbed proof mining =-=[22]-=-). The implemented algorithms coming from metamathematical research have yielded interesting and in many cases quite unexpected programs [7, 15, 31, 32]. Various approaches to program extraction from ... |

14 |
Eine Variante zur Dialectica Interpretation der Heyting-Arithmetik endlicher Typen,” Archive Mathematische Logik Grundlagenforschung 16
- Diller, Nahm
- 1974
(Show Context)
Citation Context ... be called the light monotone 17 functional interpretation (abbreviated LMD-interpretation). The D-interpretation was first introduced in [13] for a Hilbert-style formulation of Arithmetic – see also =-=[38, 26, 10, 2, 20]-=- for other (more modern) formulations within Hilbert-style systems. Natural Deduction formulations of the Diller-Nahm [10] variant of D-interpretation were provided by Diller’s students Rath [33] and ... |

13 | Syntactic translations and provably recursive functions - Leivant - 1985 |

9 | Proof interpretations and the computational content of proofs. Draft available online, http://www.mathematik.tu-darmstadt.de/∼kohlenbach
- Kohlenbach
(Show Context)
Citation Context ...computational content, see, e.g., [4].sModified Realizability [23]. Refinements [1, 6, 9] of the latter’s combination with Friedman’s A-translation [12] only partly repair this disparity (see [15] or =-=[20]-=- for discussions on this issue). However, the (rough) exact realizers yielded by the Dialectica interpretation are generally (far) more complex than those produced by, e.g., the technique of Berger, B... |

7 |
A complexity analysis of functional interpretations
- Hernest, Kohlenbach
- 2005
(Show Context)
Citation Context ...ues normalize to basically the same programs (see [15] for such an example), the normalization of programs extracted by proof interpretations is generally far more expensive than their synthesis (see =-=[16]-=- for a detailed complexity exposition). The simpler the rough extracted term is, the lower the overall cost of producing the final normalized realizer becomes. In order to handle this contraction prob... |

5 |
A comparison between two techniques of program extraction from classical proofs. Preprint 14pp., 2003. Title Suppressed Due to Excessive Length – supply \combirunning
- Hernest
(Show Context)
Citation Context ...field of extractive Proof Theory (also dubbed proof mining [22]). The implemented algorithms coming from metamathematical research have yielded interesting and in many cases quite unexpected programs =-=[7, 15, 31, 32]-=-. Various approaches to program extraction from classical 1 proofs have been developed over years of research [1, 3, 6, 8, 9, 21, 24, 25, 27–30, 32]. The use of proof interpretations instead of the di... |

4 |
Finite type arithmetic
- Jørgensen
- 2001
(Show Context)
Citation Context ... restriction set by Berger on his {∀} + rule - we must take into account the inclusion of the computationally relevant contractions into the Dialectica realizing terms. We build on top of Jørgensen’s =-=[19]-=- Natural Deduction formulation of pure Dialectica interpretation which we transform by eliminating his “Contraction Lemma” 3 and by allowing free variables in the extracted terms 4 . We call light 5 f... |

2 |
2005), Bounded functional interpretation, Annals of Pure and Applied Logic 135, 73-112. the strength of some semi-constructive theories 21
- Ferreira, Oliva
- 1973
(Show Context)
Citation Context ...the Dialectica interpretation, Kohlenbach [21] devised the monotone functional interpretation, an adaptation of Gödel’s technique to the extraction of uniform bounds for the exact realizers (see also =-=[11]-=- for the more recent bounded functional interpretation of Ferreira and Oliva). While so prolific (see [20] or [22]) in the context of mathematical Analysis, the monotone functional interpretation, use... |

1 |
Uniform Heyting Arithmetic. Annals of Pure and
- Berger
- 2005
(Show Context)
Citation Context ...imination already from the primary extracted terms of a number of contraction formulas which are identified as computationally redundant by means of an adaptation of Berger’s uniform quantifiers from =-=[5]-=- to the Dialectica-extraction context. These are called “quantifiers without computational content” in [34] and we will here call them quantifiers without computational meaning (abbreviated ncm, with ... |

1 | Technical Appendix to this paper. Available in the author’s webpage - Hernest |