D.S. Scott. Relating theories of the -calculus. In J. Hinhley and J. Seldin, editors, To H.B. Curry: Essays on combinatory logic, lambda calculus and formalism, pages 403-450. New York and London, Academic Press, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The attempts to build a transparent semantic models [BLR97] brought an advance in provision the applied elds as Information Systems. A general and 2 Viacheslav Wolfengagen still important computational approach to understand the concepts in calculi and relative systems was incorporated in [Sco80]. The bene ts of concepts for event driven computations were outlined in [Wol99] An attempt to rearrange the useful ideas will be done here. The main attention is paid to establishing the parallelism between a theory of computations and the concept notions. The main aim of this paper is to ....

D.S. Scott. Relating theories of the -calculus. In J. Hinhley and J. Seldin, editors, To H.B. Curry: Essays on combinatory logic, lambda calculus and formalism, pages 403-450. New York and London, Academic Press, 1980.

....F(M) where M is the monoid of endomorphisms on V . 6.4 COROLLARY If F(C) is a topos, then it is equivalent to the topos constructed using the monoid of endomorphisms of the universal object in C. Putting these observations together, we can see that if F(C) ex is a topos, then, much as in Scott [21], V is a partial combinatory algebra, and if C is concrete, then the topos obtained is the conventional realizability topos from this algebra. ....

D.S. Scott. Relating theories of the -calculus. In R. Hindley and J. Seldin, editors, To H.B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalisms, pages 403-450. Academic Press, 1980.

....be extended to interpret calculus. At the end we get a formal system, the computational lambda calculus ( c calculus for short) for proving equivalence of programs, which is sound and complete w.r.t. the categorical semantics of computations. 1 The methodology outlined above is inspired by [13] 2 , and it is followed in [11, 8] to obtain the p calculus. The view that category theory comes, logically, before the calculus led us to consider a categorical semantics of computations rst, rather than to modify directly the rules of conversion to get a correct calculus. A type ....

D.S. Scott. Relating theories of the -calculus. In R. Hindley and J. Seldin, editors, To H.B. Curry: essays in Combinarory Logic, lambda calculus and Formalisms. Academic Press, 1980.

....the end we get a formal system, the computational lambda calculus ( c calculus for short) similar to PP (see [GMW79] for proving equivalence and existence of programs, which is sound and complete w.r.t. the categorical semantics of computations. The methodology outlined above is inspired by [Sco80] 2 , in particular the view that category theory comes, logically, before the calculus led us to consider a categorical semantics of computations rst, rather than trying to hack directly on the rules of conversion to get a correct calculus. 2 I am trying to nd out where calculus ....

.... small category C can be lifted to a computational model ( T ; over the topos C of presheaves (i.e. the functor category Set C op ) and that such a lifting commutes with the Yoneda embedding Y of C into C, i.e. T(Y ) Y(T ) Y = Y( Y = Y( As pointed out in [Sco80] such an embedding enable us to switch from the equational (and rather inexpressive) calculus of an arbitrary computational model to the intuitionistic higher order logic of (a computational model over) a topos. The monad ( T ; is de ned by using the Yoneda embedding Y: C C and ....

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D.S. Scott. Relating theories of the -calculus. In R. Hindley and J. Seldin, editors, To H.B. Curry: essays in Combinarory Logic, lambda calculus and Formalisms. Academic Press, 1980.

....M( C( where M is the functor from C op to SET s.t. M(f : a b) m 2 M(b) 7 f 1 (m) 2 M(a) A dominance is de nable by more familiar data types: Proposition 17 [ id 1 : 1 , 1 ] 2 M( 1 ) is a dominance Therefore a pCCC always has a dominance. In [Sco80] the category Sh(C) of presheaves over C is proposed as conservative cartesian closed extension of C, because Sh(C) has all (small) limits and function spaces and the Yoneda embedding of C in Sh(C) preserves limits and function spaces that already exist in C. Therefore, a construction ....

.... L (E n 1 ; E n 1 ) E n 2 (n 2:5) similar to case (1:5) but using (3) of Lemma 48 instead of Prop 38 4 Numbered Sets and Sheaves Instead of introducing GEN L we could have used a more canonical cartesian closed extension of EN L , namely the topos of presheaves over EN L (see [Sco80] Moreover, the Yoneda embedding preserves limits and (partial) function spaces, so Theor 39 is for free. A more elaborate construction, that sometimes makes it possible to preserve even more structure (e.g. colimits) is the topos of sheaves for a subcanonical Grothendieck topology (for ....

D. S. Scott. Relating theories of the -calculus. In R. Hindley and J. Seldin, editors, To H.B. Curry: essays in Combinarory Logic, lambda calculus and Formalisms. Academic Press, 1980.

....categorical semantics and formal system in order to interpret richer languages, in particular the calculus. 4. We show that w.l.o.g. one may consider only (monads over) toposes, and we exploit this fact to establish conservative extension results. The methodology outlined above is inspired by [Sco80] 1 , and it is followed in [Ros86, Mog86] to obtain the p calculus. The view that category theory comes, logically, before the calculus led us to consider a categorical semantics of computations rst, rather than to modify directly the rules of conversion to get a correct calculus. ....

....so that g can be de ned as hid c ; g 1 i; t c;c1 ; g 2 . 4 If the metalanguage does not have nite products, we conjecture that its theories would no longer correspond to categories with nite products and a strong monad (even by taking as objects contexts and or the Karoubi envelope, used in [Sco80] to associate a cartesian closed category to an untyped theory) but instead to multicategories with a Kleisli triple. We felt the greater generality (of not having products in the metalanguage) was not worth the mathematical complications. 15 De nition 3.2 A strong monad over a category C ....

D.S. Scott. Relating theories of the -calculus. In R. Hindley and J. Seldin, editors, To H.B. Curry: essays in Combinarory Logic, lambda calculus and Formalisms. Academic Press, 1980.

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