Playful, streamlike computation

We offer a short tour into the interactive interpretation of sequential programs. We emphasize streamlike computation – that is, computation of successive bits of information upon request. The core of the approach surveyed here dates back to the work of Berry and the author on sequential algorithms on concrete data structures in the late seventies, culminating in the design of the programming language CDS, in which the semantics of programs of any type can be explored interactively. Around one decade later, two major insights of Cartwright and Felleisen on one hand, and of Lamarche on the other hand gave new, decisive impulses to the study of sequentiality. Cartwright and Felleisen observed that sequential algorithms give a direct semantics to control operators like call-cc and proposed to include explicit errors both in the syntax and in the semantics of the language PCF. Lamarche (unpublished) connected sequential algorithms to linear logic and games. The successful program of games semantics has spanned over the nineties until now, starting with syntax-independent characterizations of the term model of PCF by Abramsky, Jagadeesan, and Malacaria on one hand, and by Hyland and Ong on the other hand. Only a basic acquaintance with λ-calculus, domains and linear logic is assumed in sections 1 through 3. 1 Prologue: playing with Böhm trees We first make some preparations. For self-containedness, we briefly recall the relevant notions. The syntax of the untyped λ-calculus (λ-calculus for short) is given by the following three constructions: a variable x is a λ-term, if M and N are λ-terms, then the application MN is a λ-term, and if M is a term then the abstraction λx.M is a term. Usual abbreviations are λx1x2.M for λx1.(λx2.M), and MN1N2 for (MN1)N2, and similarly for nary abstraction and application. A more macroscopic view is quite useful: it 1 is easy to check that any λ-term has exactly one of the following two forms: (n ≥ 1, p ≥ 1) λx1 · · · xn.xM1 · · ·Mp (n ≥ 0, p ≥ 1) λx1 · · · xn.(λx.M)M1 · · ·Mp . The first form is called a head normal form (hnf), while the second exhibits the head redex (λx.M)M1. The following easy property justifies the name of head normal form: any reduction sequence starting from a hnf λx1 · · · xn.xM1 · · ·Mp consists of an interleaving of independent reductions of M1, . . . ,Mp. More precisely, we have: (λx1 · · · xn.xM1 · · ·Mp → ∗ P )⇒ ∃N1, . . . Np { P = λx1 · · · xn.xN1 · · ·Np and ∀ i ≤ p Mi → ∗ Ni . Here, reduction means the replacement in any term of a sub-expression of the form (λx.M)N , called a β-redex, by M [x ← N ]. A normal form is a term that contains no β-redex, or equivalently that contains no head redex. Hence the syntax of normal forms is given by the following two constructions: a variable x is a normal form, and if M1, . . . ,Mp are normal forms, then λx1 · · · xn.xM1 · · ·Mp is a normal form. Now, we are ready to play. Consider the following two normal forms: M = zM1M2(λz1z2.z1M3M4) N = λx1x2x3.x3(λy1y2.y1N1)N2 . The term M [z ← N ] = NM1M2(λz1z2.z1M3M4) is not a normal form anymore, and can be β-reduced as follows: NM1M2(λz1z2.z1M3M4) → (λz1z2.z1M3M4)(λy1y2.y1N ′ 1)N ′ 2 → (λy1y2.y1N ′ 1)M ′ 3M ′ 4 → M ′ 3N ′′ 1 , where N ′ 1 , etc... are suitable substitution instances of N1 etc... But there is a more geometric way of describing the interaction of M and N . First, we represent M and N explicitly as trees (read from left to right), as follows: