R. O. Gandy. An early proof of normalization by A. M. Turing. In Seldin and Hindley [SH80], pp. 453--455.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Tait [70] The technique is very AEexible and has been generalized to a variety of calculi. 4 For some notions of reduction in some typed calculi there is a technique to prove weak normalization that is simpler than the Tait Girard technique to prove strong normalization. For instance, Turing [27] proves weak normalization for fi reduction in simply typed calculus by giving an explicit measure which decreases in every step of a certain fi reduction sequence. Prawitz [53] independently uses the same technique to prove weak normalization for reduction of natural deduction derivations in ....

R.O. Gandy. An early proof of normalization by A.M. Turing. In Seldin and Hindley [63], pages 453455.

....ones, implicitly use properties of perpetual (and maximal) redexes. That proof makes use of a syntactically de ned class of strongly normalizable terms. Melli es [40] gives a strong normalization proof for simply typed calculus using a perpetual strategy. The syntactic proofs mentioned above [42, 35, 21, 61, 25, 32] all use a decreasing metric de ned on types. The proofs that we present in this paper are syntactic, but they use information from types to verify that typable terms have a certain shape. This kind of shape is analyzed and used to prove normalization, without referring to types again. In other ....

R.O. Gandy. An early proof of normalization by A.M.Turing. In: [53], p. 453-455.

....for N) Type assignment for terms of a certain syntactic form is caused in the obvious way. 1) N x : A ) x:A) 2 : 2) N PQ : B ) N P : A B) and N Q : A; for some type A. 3) N x:P : C ) x:A N P : B and C A B; for some types A; B. Proof. i) See e.g. Turing s proof in Gandy [1980]. The idea is that reduction of the rightmost redex of highest rank decreases the number of such redexes, where rank is de ned by rank(p) 0; rank(A B) maxfrank(A) 1; rank(B)g: ii) and (iii) See Barendregt [1992] Actually, even strong normalisation holds for terms typeable in N (see ....

Gandy, R. [1980]. An early proof of normalization by A. M. Turing, in: J. P. Seldin and J. R. Hindley (eds.), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, New York, pp. 453-457.

....sequences from t, it goes straightforward to establish a bound on (t) for I terms t if we can find any normalisation sequences for them. In order to get a tighter bound, the key is to find shorter normalisation sequences. We start with a weak normalisation proof due to Turing according to [Gan80], which can also be found in many other literatures such as [And71, GLT89] Definition 5.5 The rank ae(T) of a simple type T is defined as follows. ae(T) 0 if T is atomic; 1 maxfae(T 0 ) ae(T 1 )g if T = T 0 T 1 . The rank ae(r) of a fi redex r = x U :v V )u U is ae(U V ) ....

R.O. Gandy (1980), An early proof of normalization by A.M. Turing. To: H.B. Curry: Essays on combinatory logic, lambda calculus and formalism, edited by J.P. Seldin and J.R. Hindley, Academic press, pp. 453-456.

....sequences from t, it goes straightforward to establish a bound on (t) for I terms t if we can find any normalization sequences for them. In order to get a tighter bound, the key is to find shorter normalization sequences. We start with a weak normalization proof due to Turing according to [Gan80], which can also be found in many other literatures such as [And71] and [GLT89] Definition 23 The rank ae(T ) of a simple type T is defined as follows. ae(T ) 0 if T is atomic; 1 maxfae(T 0 ) ae(T 1 )g if T = T 0 T 1 . The rank ae(r) of a fi redex r = x U :v V )u U is ae(U V ....

R.O. Gandy (1980), An early proof of normalization by A.M. Turing, To: H.B. Curry: Essays on combinatory logic, lambda calculus and formalism, edited by J.P. Seldin and J.R. Hindley, Academic press, pp. 453-456.

....assignment for terms of a certain syntactic form is caused in the obvious way. 1) Gamma N x : A ) x:A) 2 Gamma: 2) Gamma N PQ : B ) Gamma N P : A B) Gamma N Q : A; for some type A. 3) Gamma N x:P : C ) Gamma; x:A N P : B C j A B; for some types A; B. Proof. See e.g. Gandy [1980] for (i) and Barendregt [1992] for (ii) and (iii) Actually, even strong normalization holds for terms typeable in N (see e.g. de Vrijer [1987] or Barendregt [1992] 4. Relating N , L and L cf Now the proof of the equivalence between systems N and L will be lifted to that of N and L. 4.1. ....

Gandy, R. [1980] An early proof of normalization by A. M. Turing, in: J. P. Seldin and J. R.

....as the size of the type of the abstraction of the redex. The paper claims that there is a fi I fi S reduction strategy such that any newly created redexes are of lower order. The proof for the case of reducing a fi I redex is given (taken directly from the classic normalization proof by Turing [Gan80a] although it only deals with the possible new fi I redexes. The proof for the case of reducing a fi S redex is not given. Consider this fi S redex M : M j ( x: y:P )Q) N)O) where x 62 ( y:P )Q) Suppose y 62 P . Then ( y:P )Q) is a fi K redex and is not counted by de Groote s metric. Note ....

R. O. Gandy. An early proof of normalization by A. M. Turing. In Seldin and Hindley [SH80], pp. 453--455.

.... 2 Gamma (variable) Gamma; x : ff Gamma M : fi Gamma Gamma (x : ff: M ) ff fi) abstraction) Gamma Gamma M : ff fi) Gamma Gamma N : ff Gamma Gamma (M N ) fi (application) We now establish the normalization of Church s simply typed calculus by giving a proof due to Turing [9]. Theorem 26. Normalization) Let Gamma be a context, and let M 2 and ff 2 S be such that Gamma Gamma M : ff then M has a fi normal form. Proof. One de nes the order of a fi redex ( x : ff: M ) N ) as the length of the type assigned to (x : ff: M ) Now, consider some fi contraction P ....

R.O. Gandy. An early proof of normalization by A.M. Turing. In J. P. Seldin and J. R. Hindley, editors, to H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 453455. Academic Press, 1980.

....by Tait [70] The technique is very flexible and has been generalized to a variety of calculi. For some notions of reduction in some typed calculi there is a technique to prove weak normalization that is simpler than the Tait Girard technique to prove strong normalization. For instance, Turing [27] proves weak normalization for fi reduction in simply typed calculus by giving an explicit measure which decreases in every step of a certain fi reduction sequence. Prawitz [53] independently uses the same technique to prove weak normalization for reduction of natural deduction derivations in ....

R.O. Gandy. An early proof of normalization by A.M. Turing. In Seldin and Hindley [63], pages 453--455.

....by Tait [68] It has since been generalized to a variety of calculi, see [3, 16, 21, 24, 43, 69] For notions of reduction in some typed calculi there is a technique to prove weak normalization that is simpler than the Tait Girard technique to prove strong normalization. For instance, Turing [17] proves weak normalization for fi reduction in simply typed calculus by giving an explicit measure which decreases in every step of a certain fi reduction sequence. Prawitz [55] independently uses the same technique to prove weak normalization for reduction of natural deduction derivations in ....

R.O. Gandy. An early proof of normalization by A.M. Turing. In Seldin and Hindley [63], pages 453--455.

....strategy must terminate (normalize) thus implying fi SN by the first part. This is done in the style of decreasing metric termination proofs often found in the term rewriting literature and is very similar to the original weak normalization proof for the simply typed calculus by Turing [Gan80a]. The required reduction strategy is very simple: just reduce innermost I redexes. The new proof method for proving fi SN which we present is important for more than one reason. Most importantly, this is the first fi SN proof for a polymorphic extension of the typed calculus which does not ....

....M such that jM j 2 fl . Lemma 4. 2 If M;N 2 ( fl and M Delta Gamma N where Delta is an innermost I redex, then order ffl (M ) order ffl (N ) Proof: Note that this proof follows ideas first presented in Turing s weak normalization proof for the simply typed calculus [Gan80a]. We give here only the barest sketch of the proof. See the full technical report for details [KW94] The residuals of any pre existing fi redexes (other than the one being reduced) retain the same order value, so for them we need only show they are not duplicated. The initial fi reduction step ....

R. O. Gandy. An early proof of normalization by A. M. Turing. In Seldin and Hindley [SH80], pp. 453--455.

....by Tait [58] The technique is very flexible and has been generalized to a variety of calculi. For some notions of reduction in some typed calculi there is a technique to prove weak normalization that is simpler than the Tait Girard technique to prove strong normalization. For instance, Turing [23] proves weak normalization for fi reduction in simply typed calculus by giving an explicit measure which decreases in every step of a certain fi reduction sequence. Prawitz [47] independently uses the same technique to prove weak normalization for reduction of natural deduction derivations in ....

R.O. Gandy. An early proof of normalization by A.M. Turing. In Seldin and Hindley [52], pages 453--455.

....by Tait [87] It has since been generalized to a variety of calculi, see [8, 38, 43, 50, 56, 88] For notions of reduction in some typed calculi there is a technique to prove weak normalization that is simpler than the Tait Girard technique to prove strong normalization. For instance, Turing [39] proves weak normalization for fi reduction in simply typed calculus by giving an explicit measure which decreases in every step of a certain fi reduction sequence. Prawitz [71] independently uses the same technique to prove weak normalization for reduction of natural deduction derivations in ....

R.O. Gandy. An early proof of normalization by A.M. Turing. In Seldin and Hindley [80], pages 453--455.

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R. O. Gandy. An early proof of normalization by A. M. Turing. In Seldin and Hindley [SH80], pp. 453--455.

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R. O. Gandy, An early proof of normalization by A. M. Turing, in: Hindley and Seldin [41], pp. 453455.

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R. O. Gandy. An early proof of normalization by A.M. Turing. In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, volume 267, pages 453-455. Academic Press, 1980.

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R. O. Gandy. An early proof of normalization by A. M. Turing. In Seldin and Hindley [SH80], pages 453--455. Cited on page 2.

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R. Gandy, An early proof of normalization by A.M. Turing, In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism (J.R. Hindley and J.P. Seldin, Eds.), Academic Press (1980) 453-455.

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R.O. Gandy, An early proof of normalization by A. M. Turing, in: [48, pp. 453--455], Academic Press, New York, 1980,

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R. O. Gandy. An early proof of normalization by A. M. Turing. In

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