Lambek, J., Logic without structural rules (Another look at cut elimination) , in: eds. K. Dosen and P. Schroeder-Heister, "Substructural Logics", pp. 179--206, Oxford University Press, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....one to one between provable formulae in the sequent system of logic and inhabited types in L since the type part of L is a sequent system as well. 4 . Term cut elimination We shall show for sequent lambda calculus L an analogue of the cut elimination property in the usual manner (see Lambek [5]) by showing that for each statement derivable by the application of the (term cut) rule there is a corresponding statement with the same predicate (type) derivable without it. The addition that we have here is that the subject (term) of the latter statement is the normal form of the subject ....

Lambek, J., Logic without structural rules (Another look at cut elimination) , in: eds. K. Dosen and P. Schroeder-Heister, "Substructural Logics", pp. 179--206, Oxford University Press, 1993.

....of the symbols , I , n or = Consequently this calculus is decidable. For any sequent x S there are only a nite number of possible proof trees, since there are only a nite number of rules and each rule introduces a symbol. Lambek proved that this calculus is equivalent to L. To do 3 [4]. 120 this it is convenient to have an additional rule that enables us to use existing results in new proofs, the cut: z A x; A; y B x; z; y; B (11) It is clear that this cut is a potential threat to the decidability of the calculus, because it does not introduce new symbols. ....

J. Lambek, Logic without structural rules. Another look at cut elimination, Ms. McGill University, Montreal, 1990.

....of derivability. The pure residuation logic is then obtained by imposing the additional rules of inference of Def 2. 1, which establish the residuation laws for #, # and , One can now study equality of proofs in terms of appropriate categorical equations for the labelling system, cf. Lambek 93] and [Tro e l stra 92]for discussion in the context of combinatorial linear logic. Definition 2.1 The pure logic of residuation: combinator proof terms ( Lambek 88] 3 1A : A # A f : A #Bg: B # C g # f : A # C f : #A # B A,B (f) A # #B g : A # #B 1 A,B (g) #A # B f : A ....

Joachim Lambek. Logic Without Structural Rules. In [Dosen & Schroder-Heister 93], 179--206, 1993. 17

.... Lambda Terms Although the Curry Howard procedure of associating typed lambda terms to natural deduction proofs (cf [12] is by now quite familiar, a similar process applied to Gentzen sequent calculus appears less so, despite the related work of Lambek connected to categorical coherence theorems [17, 18]. One motivation of this term assignment is to think of an intuitionist sequent 0 B (where 0 = fA 1 ; A k g) as an input output device, accepting k inputs of types A 1 ; A k and returning an output of type B. To this end, recall that our language of typed lambda calculus has ....

.... treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg Kelly [5] Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22] Of course, for the simple types of this paper, normalization or cut elimination poses no problems. But even for these types, we shall obtain more: cut elimination implies that the internal language supports a compositional dinatural interpretation (between definable functors) In ....

J. Lambek. Logic without structural rules, Manuscript. (Mcgill University), 1990.

.... Lambda Terms Although the Curry Howard procedure of associating typed lambda terms to natural deduction proofs (cf [12] is by now quite familiar, a similar process applied to Gentzen sequent calculus appears less so, despite the related work of Lambek connected to categorical coherence theorems [17, 18]. One motivation of this term assignment is to think of an intuitionist sequent Gamma B (where Gamma = fA 1 ; A k g) as an input output device, accepting k inputs of types A 1 ; A k and returning an output of type B. To this end, recall that our language of typed lambda ....

.... treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg Kelly [5] Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22] Of course, for the simple types of this paper, normalization or cut elimination poses no problems. But even for these types, we shall obtain more: cut elimination implies that the internal language supports a compositional dinatural interpretation (between definable functors) In ....

J. Lambek. Logic without structural rules, Manuscript. (Mcgill University), 1990.

....of derivability. The pure residuation logic is then obtained by imposing the additional rules of inference of Def 2.1, which establish the residuation laws for ; 2 and = n. One can now study equality of proofs in terms of appropriate categorical equations for the labelling system, cf. Lambek 93] and [Troelstra 92] for discussion in the context of combinatorial linear logic. De nition 2.1 The pure logic of residuation: combinator proof terms ( Lambek 88] 3 1A : A A f : A B g : B C g f : A C f : A B A;B (f) A 2B g : A 2B 1 A;B (g) A B f : A B ....

Joachim Lambek. Logic Without Structural Rules. In [Dosen & Schroder-Heister 93], 179-206, 1993. 17

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC