Domains and Lambda-Calculi by Roberto M. Amadio

By Roberto M. Amadio

This publication describes the mathematical points of the semantics of programming languages. the most targets are to supply formal instruments to evaluate the which means of programming constructs in either a language-independent and a machine-independent method and to turn out houses approximately courses, similar to whether or not they terminate, or no matter if their result's an answer of the matter they're imagined to resolve. for you to accomplish that the authors first current, in an straight forward and unified means, the speculation of sure topological areas that experience proved of use within the modeling of assorted households of typed lambda calculi regarded as middle programming languages and as meta-languages for denotational semantics. This conception is referred to now as area thought, and was once based as an issue through Scott and Plotkin. one of many major matters is to set up hyperlinks among mathematical constructions and extra syntactic techniques to semantics, also known as operational semantics, that's additionally defined. This twin process has the double benefit of motivating computing device scientists to perform a little arithmetic and of fascinating mathematicians in unusual program components from laptop technology.

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G : (D ! E ) ! (D0 ! E 0) is de ned by (f ! g)(h) = g h f: Notice the \reversal" of the direction: f goes from D0 to D, not from D to D0. This is called contravariance (cf. appendix B). 57 58 CHAPTER 3. D1 MODELS AND INTERSECTION TYPES The association D 7! D ! D is not functorial in Cpo, because it is both contravariant and covariant in D. But it becomes functorial in the category of cpo's and injection-projection pairs. 2 (injection-projection pair) An injection-projection pair be- tween two cpo's D and D0 is a pair (i : D !

12 is sequential. ) 56 CHAPTER 2. SYNTACTIC THEORY OF THE -CALCULUS Chapter 3 D1 Models and Intersection Types In this chapter we address the fundamental domain equation D = D ! D which serves to de ne models of the -calculus. By \equation", we actually mean that we seek a D together with an order-isomorphism D = D ! D. g ! g. But we are interested in a non-trivial solution, that is a D of cardinality at least 2, so that not all -terms will be identi ed! Domain equations will be treated in generality in chapter 7.

2. 3 (op/den equivalence) The following equivalence holds, for any c, , 0: hc i! 0 , c] = 0 . Proof hint. ()): This is easily proved by induction on derivations. ((): This is proved by structural induction, and in the while case, by mathematical induction. )) for some n. )( ) # ) hwhile b do c i ! )( )) CHAPTER 1. CONTINUITY AND COMPUTABILITY 32 skip] c0 c1] if b then c0 else c1] while b do c] = = = = id c1] c0] cond h b] h c0] c1] ii x ( :cond h b] h c] id ii) is the set-theoretical pairing of f and g (cf.

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