Добавил:
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:
Logic and CS / Girard. Proofs and Types.pdf
Скачиваний:
38
Добавлен:
10.08.2013
Размер:
947.15 Кб
Скачать
(a; b)

60

CHAPTER 8. COHERENCE SPACES

There remains another solution:

F3(ftg; ftg) = F3(ffg; ftg) = F3(ftg; ffg) = ftg

F3(ffg; ffg) = ffg

? otherwise.

The stability condition was used to eliminate the case of:

F0(ftg; ?) = F0(?; ftg) = ftg

What have we got against this example? It violates a principle of least data: we have F0(ftg; ftg) = ftg; we seek to nd a least approximant to the pair of arguments ftg; ftg which already gives ftg; now we have at our disposal ?; ftg and ftg; ? which are minimal (?; ? does not work) and distinct.

Of course, knowing that we always have a distinguished (least) solution (rather than many minimal solutions) for a problem of this kind radically simpli es a lot of calculations.

8.4Direct product of two coherence spaces

A function F of two arguments, mapping A; B to C is stable when:

i) a0 a 2 A ^ b0 b 2 B ) F (a0; b0) F (a; b)

"

bj) =

"

F (ai; bj) (directed union)

ii) F (Si"2I ai; Sj2J

S(i;j)2I J

iii) a1 [ a2 2 A ^ b1 [ b2 2 B ) F (a1 \ a2; b \ b2) = F (a1; b1) \ F (a2; b2)

Likewise we de ne stability in any number of arguments. Observe that, whereas separate continuity su ces for joint continuity, stability in two arguments is equivalent to stability in each separately, together with the additional condition that the pullback

I@

@

@

(a; b0)

(a0; b)

@I

(a0; b0) (where a0 a 2 A and b0 b 2 B) be preserved.

We would like to avoid studying stable functions of two (or more) variables and so reduce them to the unary case. For this we shall introduce the (direct) product A N B of two coherence spaces. The notation comes from linear logic.

Соседние файлы в папке Logic and CS