## Non-Strictly Positive Fixed-Points for Classical Natural Deduction, accepted for publication in APAL (2004)

Citations: | 4 - 0 self |

### BibTeX

@TECHREPORT{Matthes04non-strictlypositive,

author = {Ralph Matthes},

title = {Non-Strictly Positive Fixed-Points for Classical Natural Deduction, accepted for publication in APAL},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

Termination for classical natural deduction is difficult in the presence of commuting/permutative conversions for disjunction. An approach based on reducibility candidates is presented that uses non-strictly positive inductive definitions. It covers second-order universal quantification and also the extension of the logic by fixed-points of non-strictly positive operators, which appears to be a new result. Finally, the relation to Parigot’s strictly-positive inductive definition of his set of reducibility candidates and to his notion of generalized reducibility candidates is explained. Key words: PACS:

### Citations

321 |
calculus: an algorithmic interpretation of classical natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...th the same operation recursively applied to s. 11 This approach is taken by Andou [11], and the admissible terms are called λ ⊥ -regular. A change to the syntax itself leads to Parigot’s λµ-calculus =-=[21]-=-: Variables that may be bound by µ are taken from another set of variables, the µ-variables, and free occurrences of those µ-variables a only come from an application a s to a term s (written as [a]s ... |

238 |
Interprétation fonctionnelle et elimination des coupures de l’arithmétique d’ordre supérieur. Thèse d’état, Université de Paris 7
- Girard
- 1972
(Show Context)
Citation Context ...rm (inj 0 x)y. (In [14], a more general definition of SN is given which captures this kind of “junk terms”.) ✷ 11sThe proof will now follow the lines of usual proofs by the candidate method by Girard =-=[15]-=-. The candidates will be subsets of SN with additional closure properties, modelled after the definition of SN. However, for our notion of saturated sets, in order to treat µx.r, we use a closure prop... |

124 |
Ideas and results in proof theory
- Prawitz
- 1971
(Show Context)
Citation Context ...variable. SN and SAT could be adapted to that formulation, and the problem with the 9 Definitions of validity such as the one used by Prawitz in the proof of strong normalization also for disjunction =-=[6]-=- are I-based. 10 Interestingly, Prawitz expressed in [6, p. 290] that disjunctions “seem impossible to handle in this way”. Certainly, the second proof only works due to the quantification over all P-... |

95 |
Inductive types and type constraints in the second-order lambda calculus
- Mendler
(Show Context)
Citation Context ...with non-interleaving positive fixed-points essentially has been studied by Geuvers [17] under the name Fret, and strong normalization has been shown by him through an embedding into Mendler’s system =-=[18]-=-. A direct 18sproof of strong normalization by saturated sets has been given by the author [19] under the name NPF. No embedding into system F exists [20]. One expects that this negative result also h... |

75 | Natural Deduction - Prawitz - 1971 |

51 | Inductive and Coinductive types with Iteration and Recursion
- Geuvers
- 1992
(Show Context)
Citation Context ...y [8]. As usual, we now assume that −→ and the related notions refer to the extended set of axiom schemes. System F with non-interleaving positive fixed-points essentially has been studied by Geuvers =-=[17]-=- under the name Fret, and strong normalization has been shown by him through an embedding into Mendler’s system [18]. A direct 18sproof of strong normalization by saturated sets has been given by the ... |

48 |
A realizability interpretation of the theory of species
- Tait
- 1975
(Show Context)
Citation Context ...rmutative conversions simplify case analyses also in call-by-name evaluation. Two proofs of strong normalization are given and carefully compared. Both use logical predicates, one is inspired by Tait =-=[3]-=-, the other by Parigot [1]. Strong normalization is captured by an inductive definition of a set SN in the style of van Raamsdonk and Severi [4]. The essential new ingredient is a notion of saturated ... |

37 |
Parallel Reductions in λ-calculus
- Takahashi
- 1995
(Show Context)
Citation Context ...trongly normalizing, see below, hence Newman’s lemma gives confluence for typable terms. We conjecture that every term is confluent, to be provable by the method of complete developments by Takahashi =-=[10]-=-. For a variant of λµ-calculus, this has been done by Andou [11]. Remark 5 In classical natural deduction in Curry-style, the simplification rules of Parigot [1] would be x(µy.r) −→µ r[y := x] µx.xr −... |

34 |
Proofs of Strong Normalisation for Second Order Classical Natural Deduction
- Parigot
- 1997
(Show Context)
Citation Context ...semantical method of candidates of reducibility or the proof-theoretic method of a Kolmogorov translation (corresponding to a translation in continuationpassing style) exist (for the second kind, see =-=[1,2]-=-), the author claims that there is no proof that combines second-order classical logic (with proof by contradiction) in the most natural formulation and disjunction with permutative/commuting conversi... |

33 | Short proofs of normalization for the simplytyped lambda-calculus, permutative conversions and Gödel’s
- Joachimski, Matthes
- 2003
(Show Context)
Citation Context ...tion of typable strongly normalizing terms in the spirit of van Raamsdonk and Severi [4] that modularizes the normalization proof. The proposed solution extends the work on permutative conversions in =-=[12]-=- to classical natural deduction. Multiple Eliminations. Inductively define multiple eliminations as a set of expressions as follows: ⋆ is a multiple elim. e is an elimination E is a multiple elim. e[⋆... |

23 | A short proof of the strong normalization of classical natural deduction with disjunction
- David, Nour
(Show Context)
Citation Context ...], x0. t0, x1. t1)] � PROOF. Existence of the decomposition is proven by induction on terms, uniqueness is obvious. ✷ Remark 7 The lemma is certainly not the only way to analyze terms. David and Nour =-=[13]-=- put all the terms r with a permutative redex visible in a decomposition E[t] of r into one category. With our notation, this can be described as follows: Let F + always denote a multiple elimination ... |

21 |
Strong normalization of classical natural deduction with disjunction. TLCA’01
- Groote
- 2001
(Show Context)
Citation Context ...a s to a term s (written as [a]s and called a named term by Parigot). Parigot goes even further and avoids ⊥ altogether in his formulation of λµ-calculus. In the more liberal formulation by de Groote =-=[2]-=- 12 , the µ-reduction rule would become e[µa.r] −→µ µb.r[a ⋆ := be] with b a “new” µ-variable. SN and SAT could be adapted to that formulation, and the problem with the 9 Definitions of validity such ... |

20 |
A semantical proof of strong normalization theorem for full propositional classical natural deduction
- Nour, Saber
- 2005
(Show Context)
Citation Context ...[case (r, x0. t0, x1. t1)]] ∈ SN The definition only decrees it for r = F[x]. In the presence of RAA, it seems very hard to prove this rule. In the proofs by David and Nour [13] and by Nour and Saber =-=[16]-=- for λµ-calculus, a slightly simpler closure rule of the set sn of strongly normalizing terms plays an essential role (for F + , see remark 7): case (r, x0. F + [t0], x1. F + [t1]) ∈ sn F + [case (r, ... |

7 | Normalization theorems for full first order classical natural deduction - Stalmarck - 1991 |

7 |
Church-Rosser property of simple reduction for full first-order classical natural deduction. Annals of Pure and Applied logic 119
- Andou
- 2003
(Show Context)
Citation Context ...uence for typable terms. We conjecture that every term is confluent, to be provable by the method of complete developments by Takahashi [10]. For a variant of λµ-calculus, this has been done by Andou =-=[11]-=-. Remark 5 In classical natural deduction in Curry-style, the simplification rules of Parigot [1] would be x(µy.r) −→µ r[y := x] µx.xr −→µ r for x not free in r While the first rule does not make sens... |

6 |
Reduction Properties of ΠIE-Systems
- Joachimski
- 2001
(Show Context)
Citation Context ...le and strongly normalizing term is in SN. PROOF. This can be proven as in [12, p.74]. ✷ Remark 13 SN �= sn. PROOF. There are untypable terms in sn which are not in SN, e.g., the term (inj 0 x)y. (In =-=[14]-=-, a more general definition of SN is given which captures this kind of “junk terms”.) ✷ 11sThe proof will now follow the lines of usual proofs by the candidate method by Girard [15]. The candidates wi... |

4 |
Parigot’s second order λµ-calculus and inductive types
- Matthes
- 2001
(Show Context)
Citation Context ...rbitrary target types of the recursively defined functions. This is not done by extending the normalization proof for PF +¬ , but by a reductionpreserving embedding of that extension into PF +¬ , see =-=[22]-=- for a presentation with fixed types and λµ-calculus. µ-reduction and permutation rules are preserved since that embedding translates the recursor (and the iterator)— which gives rise to an eliminatio... |

3 |
English translation of Lambda-calcul, types et modeles
- Lambda-calculus, Masson, et al.
- 1993
(Show Context)
Citation Context ... s[x := r] −→ ∗ s[x := r ′ ]. Lemma 1 (Subject Reduction) If Γ ⊢ t : A and t −→ t ′ then Γ ⊢ t ′ : A. PROOF. This is not trivial since F + is formulated in Curry-style. Nevertheless, Krivine’s method =-=[5]-=- is very well-suited to cover also the permutative conversions. We omit the lengthy but boring adaptation of Krivine’s proof. Note that the proof would be obvious in the absence of the rules for the u... |

2 |
On normalisation, Tech. Rep. CS-R9545, Centrum voor Wiskunde en Informatica (CWI
- Raamsdonk, Severi
- 1995
(Show Context)
Citation Context ...d. Both use logical predicates, one is inspired by Tait [3], the other by Parigot [1]. Strong normalization is captured by an inductive definition of a set SN in the style of van Raamsdonk and Severi =-=[4]-=-. The essential new ingredient is a notion of saturated sets that relies on a clause that is positive but not strictly positive and thus far from being a Horn clause. It is shown that the proof method... |

2 |
Un plongement de la logique classique du 2nd ordre dans AF2 . Unpublished manuscript
- Joly
- 1996
(Show Context)
Citation Context ...no harm to strong normalization. This rule “reduces” stability of A + B to stability of arbitrary types C which is certainly no well-founded process. Therefore, there is no hope that Joly’s embedding =-=[8]-=- of classical natural deduction into system F by can be extended to sums. But one might argue that one could get sums just by an impredicative encoding. However, one would not get permutative conversi... |

2 | Monotone fixed-point types and strong normalization
- Matthes
- 1994
(Show Context)
Citation Context ...the name Fret, and strong normalization has been shown by him through an embedding into Mendler’s system [18]. A direct 18sproof of strong normalization by saturated sets has been given by the author =-=[19]-=- under the name NPF. No embedding into system F exists [20]. One expects that this negative result also holds for any reasonable extension of the two systems by some η-rules. Here, we show that strong... |

1 |
Type Fixpoints: Iteration vs
- Sp̷lawski, Urzyczyn
- 1999
(Show Context)
Citation Context ...im through an embedding into Mendler’s system [18]. A direct 18sproof of strong normalization by saturated sets has been given by the author [19] under the name NPF. No embedding into system F exists =-=[20]-=-. One expects that this negative result also holds for any reasonable extension of the two systems by some η-rules. Here, we show that strong normalization also holds with permutative conversions, the... |