By Paul Blain Levy (auth.)

Call-by-push-value is a programming language paradigm that, strangely, breaks down the call-by-value and call-by-name paradigms into uncomplicated primitives. This monograph, written for graduate scholars and researchers, exposes the call-by-push-value constitution underlying a impressive diversity of semantics, together with operational semantics, domain names, attainable worlds, continuations and video games.

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M and x M have different denotations: an environment for the latter must provide a binding for x, even though this binding is not used. • Writing x M implicitly includes the assumption that x otherwise I' , x: A would not be a context. rf. 1) We do not need to state explicitly that x rf. I', as is traditionally done. In fact, to reduce clutter when writing equations, we will generally omit the context, turnstile and type, and omit too all the assumptions required to make the equations well typed.

M'} 7rM 7r'M as as O'M I'M Because of these resemblances, we can understand the key issues in CBV and CBN without having to include products. That is why, in this chapter, we will not consider product types further. They are dealt with fully in Appendix A. There are more type constructors we could include while remaining simply typed, and in Appendix A we will include them so that our treatment of CBV and CBN there is as thorough as possible. The type system used in this chapter, therefore, provides only a fragment of the full type system that a (simply typed) CBN or CBV language can allow.

Ij. ij. m , T print c. ij. 3. Big-Step Semantics for CBN with print D By cont rast with Prop. 5 we have the following. Proposition 7 T he relation ~anytype is st rictly finer than ~ground. D Perh aps the simplest example of this proposition is print "hello ". M ~ground AX. (print "hello" . J, which is not ground, distinguishes the two sides . 4) is t hat, inside a ground CBN term, the only way to cause a subte rm of type A ----7 B to be evalua ted is to apply it. i,verge 20 CALL-BY-PUSH- VALUE This general CBN phenomenon can be described as "effects commute with A".

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