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Logic and CS / Girard. Proofs and Types.pdf
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Chapter 9

Denotational Semantics of T

The constructions of chapter 8 provide a nice denotational semantics of the systems we have already considered.

9.1Simple typed calculus

We propose here to interpret the simple typed calculus, based on ! and . The essential idea is that:

-abstraction turns a function (x 7!t[x]) into an object;

application associates to an object t of type U!V a function u 7!t u.

In other words, application and -abstraction are two mutually inverse operations which identify objects of type U!V and functions from U to V .

So we shall interpret them as follows:

-abstraction by the operation which maps a stable function from A to B to its trace, a point of A ! B;

application by the operation which maps a point of A ! B to the function of which it is the trace.

9.1.1Types

Suppose we have xed for each atomic type Si a coherence space [[Si]]; then we de ne [[T ]] for each type T by:

[[U V ]] = [[U]] N [[V ]]

[[U!V ]] = [[U]] ! [[V ]]

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9.1. SIMPLE TYPED CALCULUS

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9.1.2Terms

If t[x1; : : : ; xn] is a term of type T depending on free variables xi of type Si (some of the xi may not actually occur in t), we associate to it a stable function