## Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λ-Calculi (1992)

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@MISC{Gallier92constructivelogics.,

author = {Jean Gallier},

title = {Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λ-Calculi},

year = {1992}

}

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451 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ...en and Prawitz, we are also led quite naturally to the encoding of proofs as certain typed -terms, and to the correspondence between proof normalization ands-conversion (the Curry/Howard isomorphism [=-=16]). Se-=-quent calculi can be motivated by the desire to obtain more \symmetric" systems, but also systems in which proof search is easier to perform (due to the subformula property). Atsrst, the cut rule... |

285 | Computational interpretations of linear logic
- Abramsky
- 1993
(Show Context)
Citation Context ...y, the contraction rule plays a crucial role in linear logic, and in the understanding of the correspondence between proofs and computations, in particular strict versus lazy evaluation (see Abramsky =-=[1-=-]). In order to obtain a system for intuitionistic logic, we restrict the righthand side of a sequent to consist of at most one formula. We also modify the (: left) rule and the (_: right) rule which ... |

265 |
Investigations into logical deductions
- Gentzen
- 1935
(Show Context)
Citation Context ...owing that they are indispensable to achieve certain natural goals. For pedagogical reasons, it seems that it is best to begin with proof systems in natural deduction style (originally due to Gentzen =-=[8]-=- and thoroughly investigated by Prawitz [23] in the sixties). This way, it is fairly natural to introduce the distinction between intuitionistic and classical logic. By adopting a description of natur... |

242 |
Interpre'tation fonctionnelle et e'limination des coupures de l'arithme'tique d'ordre supe'rieur
- Girard
- 1972
(Show Context)
Citation Context ...property, Prawitz (1971)] Reduction in !;;+;? (as in Definition 3.3) is strongly normalizing. A proof can be given by adapting Tait's reducibility method [28], [30], as done in Girard [10] (1971), [11] (1972) (see also Gallier [7]). If one looks at the rules of the systems N ;^;_;? (or !;;+;? ), one notices a number of unpleasant features: (1) There is an asymmetry between the lefthand side and... |

228 |
Une extension de l’interpretation de Gödel a l’analyse, et son application a l’elimination des coupure dans l’analyse et la theorie des types
- Girard
- 1971
(Show Context)
Citation Context ...ormalization property, Prawitz (1971)] Reduction in !;;+;? (as in Definition 3.3) is strongly normalizing. A proof can be given by adapting Tait's reducibility method [28], [30], as done in Girard [10] (1971), [11] (1972) (see also Gallier [7]). If one looks at the rules of the systems N ;^;_;? (or !;;+;? ), one notices a number of unpleasant features: (1) There is an asymmetry between the left... |

226 | Natural Deduction, A Proof-Theoretical Study. Almquist and Wiksell - Prawitz - 1965 |

214 | Intensional interpretation of functionals of finite type I - Tait - 1967 |

174 | Introduction to Combinators and Calculus - Hindley, Seldin - 1986 |

163 |
A new constructive logic: Classical logic
- Girard
- 1991
(Show Context)
Citation Context ...kground material (with some exceptions, such as \contraction-free" systems for intuitionistic propositional logic and the Girard-translation of classical logic into intuitionistic logic, which is=-= new [14-=-]). The second part is devoted to more current topics such as linear logic, proof nets, the geometry of interaction, and unied systems of logic (LU ). In our presentation of background material, we ha... |

141 | Proofs and Types, volume 7 of Cambridge Tracts - Girard, Lafont, et al. - 1989 |

136 | Contraction-free sequent calculi for intui tionistic logic - Dyckhoff - 1992 |

127 |
Ideas and results in proof theory
- Prawitz
- 1971
(Show Context)
Citation Context ...P ); M;N) ! 5C (P ): A fundamental result about natural deduction is the fact that every proof (term) reduces to a normal form, which is unique up to -renaming. This result wassrst proved by Prawitz [24] for the system N ;^;_;? i . Theorem 3.4 [Church-Rosser property, Prawitz (1971)] Reduction in !;;+;? (specied in Definition 3.3) is con uent. Equivalently, conversion in !;;+;? is Church-Ross... |

69 |
Proof theory: Some applications of cut-elimination
- Schwichtenberg
- 1977
(Show Context)
Citation Context ...d by Gentzen [8] (1935). Gentzen's proof was later simplied by Tait [29] and Girard [13] (especially the induction measure). A simplied version of Tait's proof is nicely presented by Schwichtenberg [2=-=6]-=-. The proof given here combines ideas from Tait and Girard. The induction measure used is due to Tait [29] (the cut-rank), but the explicit transformations are adapted from Girard [13], [9]. We need t... |

65 | An intuitionistic theory of types
- Martin-Löf
- 1972
(Show Context)
Citation Context ...tz (1971)] Reduction in !;;+;? (specied in Definition 3.3) is con uent. Equivalently, conversion in !;;+;? is Church-Rosser. A proof can be given by adapting the method of Tait and Martin-Lof [21] using a form of parallel reduction (see also Barendregt [2], Hindley and Seldin [15], or Stenlund [27]). 13 Theorem 3.5 [Strong normalization property, Prawitz (1971)] Reduction in !;;+;? (as in D... |

61 |
Natural Deduction: a Proof-Theoretical Study
- Prawitz
- 1965
(Show Context)
Citation Context ...e certain natural goals. For pedagogical reasons, it seems that it is best to begin with proof systems in natural deduction style (originally due to Gentzen [8] and thoroughly investigated by Prawitz =-=[23]-=- in the sixties). This way, it is fairly natural to introduce the distinction between intuitionistic and classical logic. By adopting a description of natural deduction in terms of judgements, as oppo... |

58 | Linear logic. Theoretical computer science - Girard - 1987 |

57 |
Mathematical intuitionism: Introduction to proof theory, Transla- tions of mathematical monographs
- DRAGALIN
- 1988
(Show Context)
Citation Context ...limination aresner thans-conversion. This is the reason why it is quite dicult to prove a strong version of cut-elimination where all reduction sequences terminate. Such a proof was given by Dragalin [4]. If the alternate rules . N : A x: A; . M : C . M [N=x]: C (cut) 50 . M : A x: B; . N : C z: A B; . N [(zM)=x]: C (: left) x: A[=t]; . M : C z: 8tA; . M [(z)=x]: C (8: left) are used, then the r... |

49 |
A realizability interpretation of the theory of species
- Tait
- 1975
(Show Context)
Citation Context ...13 Theorem 3.5 [Strong normalization property, Prawitz (1971)] Reduction in !;;+;? (as in Definition 3.3) is strongly normalizing. A proof can be given by adapting Tait's reducibility method [28], [30], as done in Girard [10] (1971), [11] (1972) (see also Gallier [7]). If one looks at the rules of the systems N ;^;_;? (or !;;+;? ), one notices a number of unpleasant features: (1) There is an as... |

48 |
1974] The correspondence between cut-elimination and normalization
- Zucker
(Show Context)
Citation Context ...here is no free lunch! The above considerations lead naturally to the following question: what is the exact relationship between cut-elimination and (strong) normalization in intuitionistic logic? In =-=[37-=-], Je Zucker makes an in-depth study of this relationship. Although Zucker obtains some very nice results in 68 this formidable paper, he does not provide a complete answer to the problem. Intuitively... |

44 |
Normal derivability in classical logic
- Tait
- 1968
(Show Context)
Citation Context ... Lincoln, Scedrov and Shankar [20], and on the complexity of cut-elimination by Hudelmaier [17]. The cut elimination theorem is proved in full for the Gentzen system LK using Tait's induction measure =-=[29-=-] and some twists due to Girard [13]. We conclude with a fairly extensive discussion of the reduction of classical logic to intuitionistic logic. Besides the standard translations due to Godel, Gentze... |

39 |
Introduction to Metamathematics. North-Holland, seventh edition
- Kleene
- 1952
(Show Context)
Citation Context ...:P ) (contrac: left) :(P _ :P ) ::(P _ :P ) 17 Nevertheless, it is possible to formulate a cut-free system GK ;^;_;? i which is equivalent to G ;^;_;? i (see section 8). Such a system due to Kleene [1=-=8]-=- has no contraction rule, and the premise of every sequent can be interpreted as a set as opposed to a multiset (furthermore, in the case of intuitionistic propositional logic, it is possible to desig... |

35 |
The Lambda Calculus, volume 103
- Barendregt
- 1984
(Show Context)
Citation Context ...c and the lambda calculus. If the reader does not feel suciently comfortable with these topics, we suggest consulting Girard, Lafont, Taylor [9] or Gallier [6] for background on logic, and Barendregt =-=[2]-=-, Hindley and Seldin [15], or Krivine [19] for background on the lambda calculus. For an in-depth study of constructivism in mathematics, we highly recommend Troelstra and van Dalen [32]. 2 Natural De... |

33 | On Girard’s “Candidats de Réductibilite
- Gallier
- 1990
(Show Context)
Citation Context ...ction in !;;+;? (as in Definition 3.3) is strongly normalizing. A proof can be given by adapting Tait's reducibility method [28], [30], as done in Girard [10] (1971), [11] (1972) (see also Gallier [7]). If one looks at the rules of the systems N ;^;_;? (or !;;+;? ), one notices a number of unpleasant features: (1) There is an asymmetry between the lefthand side and the righthand side of a sequ... |

30 | Proofs and Types, volume 7 of Cambridge Tracts in Theoret Computer Science - Girard, Lafont, et al. - 1989 |

28 | Interprétation functionelle et élimination des coupures dans l’arithmétique d’ordre supérieure - Girard - 1972 |

26 | On Girard's "candidats de reductibilit'es - Gallier - 1990 |

22 |
Bounds for cut elimination in intuitionistic propositional logic
- Hudelmaier
(Show Context)
Citation Context ... in automated theorem proving by Dyckho [5], on the embedding of intuitionistic logic 3 into linear logic by Lincoln, Scedrov and Shankar [20], and on the complexity of cut-elimination by Hudelmaier [=-=17]-=-. The cut elimination theorem is proved in full for the Gentzen system LK using Tait's induction measure [29] and some twists due to Girard [13]. We conclude with a fairly extensive discussion of the ... |

16 |
Contraction-free sequent calculi for intuitionistic logic
- Dyckho
- 1992
(Show Context)
Citation Context ...ch always terminates. Such systems were discovered in the earlysfties by Vorob'ev [35, 36]. Interest in such systems has been revived recently due to some work in automated theorem proving by Dyckho [=-=5]-=-, on the embedding of intuitionistic logic 3 into linear logic by Lincoln, Scedrov and Shankar [20], and on the complexity of cut-elimination by Hudelmaier [17]. The cut elimination theorem is proved ... |

16 | Linearizing intuitionistic implication
- Lincoln, Scedrov, et al.
- 1993
(Show Context)
Citation Context ...st in such systems has been revived recently due to some work in automated theorem proving by Dyckho [5], on the embedding of intuitionistic logic 3 into linear logic by Lincoln, Scedrov and Shankar [=-=20]-=-, and on the complexity of cut-elimination by Hudelmaier [17]. The cut elimination theorem is proved in full for the Gentzen system LK using Tait's induction measure [29] and some twists due to Girard... |

15 | Une extension de l'interpr'etation de G"odel `a l'analyse et son application `a l'elimination de coupures dans l'analyse et la th'eorie des types - Girard - 1971 |

14 | On an interpretation of second-order quantification in first-order intuitionistic propositional logic - Pitts - 1992 |

12 |
Intensional interpretation of functionals of type I
- Tait
- 1967
(Show Context)
Citation Context ...27]). 13 Theorem 3.5 [Strong normalization property, Prawitz (1971)] Reduction in !;;+;? (as in Definition 3.3) is strongly normalizing. A proof can be given by adapting Tait's reducibility method [28], [30], as done in Girard [10] (1971), [11] (1972) (see also Gallier [7]). If one looks at the rules of the systems N ;^;_;? (or !;;+;? ), one notices a number of unpleasant features: (1) There is... |

11 |
Proof Theory. volume 81
- Takeuti
- 1975
(Show Context)
Citation Context ...able. In fact, proof search should be conducted in the system GK ;^;_;8;9;? i , since we know that searching for an irredundant proof terminates in the propositional case (see theorem 8.10). Takeuti [=-=31]-=- has made an interesting observation about invertibility of rules in intuitionistic cut-free sequent calculi, but before discussing this observation, we shall discuss other contractionfree systems for... |

10 | Research Report 7: The Siphon: Managing Distant Replicated Repositories - Prusker, Wobber - 1991 |

9 | types et modèles. Etudes et recherches en informatique - Lambda-Calcul - 1990 |

8 |
A new algorithm for derivability in the constructive propositional calculus
- Vorob’ev
- 1970
(Show Context)
Citation Context ... In particular, we discuss some \contraction-free" systems for intuitionistic propositional logic for which proof search always terminates. Such systems were discovered in the earlysfties by Voro=-=b'ev [35, 36-=-]. Interest in such systems has been revived recently due to some work in automated theorem proving by Dyckho [5], on the embedding of intuitionistic logic 3 into linear logic by Lincoln, Scedrov and ... |

4 |
Regles Admissibles en calcul propositionnel intuitionniste
- Rozière
- 1992
(Show Context)
Citation Context ...s, with interesting applications to the theory of Heyting algebras. An indepth study of invertibility and admissibility of rules in intuitionistic logic can be found in Paul Roziere's elegant thesis [=-=25]-=-. We now come back to Takeuti's observation [31] (see Chapter 1, paragraph 8). The crucial fact about intuitionistic systems is not so much the fact that sequents are restricted so that righthand 46 s... |

4 | 6: Binary Periodic Synchronizing Sequences. Marcin Skubiszewski. May Research Report 7: The Siphon: Managing Distant Replicated Repositories - Prusker, Wobber |

3 |
On an interpretation of second-order quanti in intuitionistic propositional logic
- Pitts
- 1992
(Show Context)
Citation Context ... B, both of (strictly) smaller complexity. Thus, this system requires no test of circularity (test for the repetition of sequents tosnd irredundant proofs), unlike in the system GK ;^;_;? i . Pitts [2=-=2-=-] reports on applications of the system LJ T to intuitionistic logic with quantication over propositional letters, with interesting applications to the theory of Heyting algebras. An indepth study of ... |

3 |
The derivability problem in the constructive propositional calculus with strong negation
- Vorob'ev
- 1952
(Show Context)
Citation Context ... In particular, we discuss some \contraction-free" systems for intuitionistic propositional logic for which proof search always terminates. Such systems were discovered in the earlysfties by Voro=-=b'ev [35, 36-=-]. Interest in such systems has been revived recently due to some work in automated theorem proving by Dyckho [5], on the embedding of intuitionistic logic 3 into linear logic by Lincoln, Scedrov and ... |

2 |
Normalization, Cut-eliminations and the Theory of Proofs
- Ungar
- 1992
(Show Context)
Citation Context ...er thans-reduction (as alluded to in section 10). Thus, the exact relationship between cut-elimination and (strong) normalization remains a challenging open problem. For the latest results, see Ungar =-=[33-=-]. A few more remarks about the role of contraction and weakening will be useful as a motivation for linear logic. We already noticed with the cut rule that contexts (the , occurring in the premise(s... |

2 | Logic for Computer Science. Harper and Row - Gallier - 1986 |

1 | Linearizing ituitionistic implication - Lincoln, Scedrov, et al. - 1991 |

1 | Proof Theory, volume 81 of Studies in Logic. North-Holland - Takeuti - 1975 |

1 | 35] N.N. Vorob'ev. The derivability problem in the constructive propositional calculus with strong negation. Doklady Akademii Nauk SSSR - Logic, Universitext - 1980 |