Computational Logic

ion is modelled by currying: [r IAX : A.M : A,-t B] = A([r, x: A IM : B]) : [r) -t [A] ::} [B]. The interpretation of application makes use of the evaluation map ev : (A ::} B) x A -t B obtained by uncurrying the identity on A ::} B. If r IM : A -t Band r IN: A then [r IM N) = ([r IM), [r IN]} ; ev. We shall now show how PCF can be interpreted in any of the cartesian closed categories C, Ci , Cb and Cib. The interpretation of exp is the flat game N of natural numbers, defined as follows. MN = {q}U{nJnEw} AN(q) OQ AN(n) = PA * I-N q q I-N n for each n ~ = {c,q}U{qnJnEw} Thus N has a single initial question q to which P can respond by playing a natural number. The strategies for N are..L = {c} and [n] = {c, qn} for each number n. These strategies are all innocent and well-bracketed, so we have an interpretation of the numeric constants in each of our cartesian closed categories. The arithmetic operation succ is interpreted using the strategy depicted below.