—The normalization theorem

1966 ◽  
pp. 110-122
Author(s):  
Raghavan Narasimhan
Keyword(s):  
Normalization Theorem

scholarly journals The normalization theorem for extended natural deduction

2017 ◽  
Vol 101 (115) ◽  
pp. 75-98
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Natural Deduction ◽  
Standard System ◽  
Cut Elimination ◽  
Normalization Theorem

The normalization theorem for the system of extended natural deduction will be proved as a consequence of the cut-elimination theorem, by using the connections between the system of extended natural deduction and a standard system of sequents.

Normalization theorems for full first order classical natural deduction

1991 ◽  
Vol 56 (1) ◽  
pp. 129-149 ◽  
Author(s):  
Gunnar Stålmarck
Keyword(s):  
Normal Form ◽  
Natural Deduction ◽  
Full System ◽  
Strong Normalization ◽  
Logical Constants ◽  
Restricted Version ◽  
First Order ◽  
Form Theorem ◽  
Normalization Theorem ◽  
Normal Form Theorem

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.

Church-Rosser theorem for typed functional systems

1985 ◽  
Vol 50 (3) ◽  
pp. 782-790 ◽  
Author(s):  
George Koletsos
Keyword(s):  
Upper Bound ◽  
Type Structure ◽  
Functional Systems ◽  
Normalization Theorem ◽  
One Step ◽  
The One ◽  
Reduction Relation ◽  
The Church ◽  
The Way

Introduction. This paper contains a new proof of the Church-Rosser theorem for the typed λ-calculus, which also applies to systems with infinitely long terms.The ordinary proof of the Church-Rosser theorem for the general untyped calculus goes as follows (see [1]). If is the binary reduction relation between the terms we define the one-step reduction 1 in such a way that the following lemma is valid.Lemma. For all terms a and b we have: ab if and only if there is a sequence a = a0, …, an = b, n ≥ 0, such that aiiai + 1for 0 ≤ i < n.We then prove the Church-Rosser property for the relation 1 by induction on the length of the reductions. And by combining this result with the above lemma we obtain the Church-Rosser theorem for the relation .Unfortunately when we come to infinite terms the above lemma is not valid anymore. The difficulty is that, assuming the hypothesis for the infinitely many premises of the infinite rule, there may not exist an upper bound for the lengths n of the sequences ai = a0, …, an = bi (i < α); cf. the infinite rule (iv) in §6.A completely new idea in the case of the typed λ-calculus would be to exploit the type structure in the way Tait did in order to prove the normalization theorem. In this we succeed by defining a suitable predicate, the monovaluedness predicate, defined over the type structure and having some nice properties. The key notion permitting to define this predicate is the notion of I-form term (see below). This Tait-type proof has a merit, namely that it can be extended immediately to the case of infinite terms.

Gentzen's Proof of Normalization for Natural Deduction

2008 ◽  
Vol 14 (2) ◽  
pp. 240-257 ◽  
Author(s):  
Jan von Plato
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Direct Proof ◽  
Cut Elimination ◽  
Doctoral Thesis ◽  
Detailed Proof ◽  
Normalization Theorem ◽  
The One ◽  
The Right ◽  
Special Case

AbstractGentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen (Investigations into logical reasoning) that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933.

scholarly journals Normalization as a consequence of cut elimination

2009 ◽  
Vol 86 (100) ◽  
pp. 27-34
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Cut Elimination ◽  
Deduction System ◽  
Normalization Theorem ◽  
Natural Deduction System

Pairs of systems, which consist of a system of sequents and a natural deduction system for some part of intuitionistic logic, are considered. For each of these pairs of systems the property that the normalization theorem is a consequence of the cut-elimination theorem is presented.

A basis result in combinatory logic

1988 ◽  
Vol 53 (4) ◽  
pp. 1224-1226
Author(s):  
Remi Legrand
Keyword(s):  
Combinatory Logic ◽  
Reduction Rule ◽  
Reduction System ◽  
Normalization Theorem ◽  
Finite Set

The aim of this article is to show that a basis for combinatory logic [2] must contain at least one combinator with rank strictly greater than two. We use notation of [1].Let Q be a primitive combinator given by its reduction rule Qx1 … xn → C, where C is a pure combination of the variables x1,…, xn. n is called the rank of the combinator.A set {Q1,…,Qn} of combinators is a basis for combinatory logic if for every finite set {x1,…,xm} of variables and every pure combination C of these variables, there exists a pure combinator Q of Q1,…,Qn such that Qx1…xm↠C.Property. The Church-Rosser theorem and the (quasi-)normalization theorem are valid for the combinatory reduction system under consideration.Proof. See [4], [5], and [6].Any basis must contain at least one combinator with rank strictly greater than two.Let us assume a basis B with combinators of rank strictly less than 3; then there exists a pure combination X of the combinators in B such that: XABC ↠ CAB.(*) First of all, we remark that if XABC ↠ M ↠ CAB, M must contain at least one occurrence of each of the variables A, B and C.Notation. We denote by E[X1,…,Xn] expressions that contain at least one occurrence of every term X1,…,Xn.

A normalization theorem for set theory

1988 ◽  
Vol 53 (3) ◽  
pp. 673-695 ◽  
Author(s):  
Sidney C. Bailin
Keyword(s):  
Set Theory ◽  
Natural Deduction ◽  
Inference Rule ◽  
Major Premise ◽  
Elimination Rule ◽  
Introduction Rule ◽  
Normalization Theorem ◽  
Image Position ◽  
Well Foundedness

In this paper we present a normalization theorem for a natural deduction formulation of Zermelo set theory. Our result gets around M. Crabbe's counterexample to normalizability (Hallnäs [3]) by adding an inference rule of the formand requiring that this rule be used wherever it is applicable. Alternatively, we can regard the result as pertaining to a modified notion of normalization, in which an inferenceis never considered reducible if A is T Є T, even if R is an elimination rule and the major premise of R is the conclusion of an introduction rule. A third alternative is to regard (1) as a derived rule: using the general well-foundedness rulewe can derive (1). If we regard (2) as neutral with respect to the normality of derivations (i.e., (2) counts as neither an introduction nor an elimination rule), then the resulting proofs are normal.

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