Categorical Combinators, Sequential Algorithms, and by Pierre-Louis Curien

By Pierre-Louis Curien

This publication is a revised version of the monograph which seemed below a similar name within the sequence examine Notes in Theoretical machine technology, Pit­ guy, in 1986. as well as a basic attempt to enhance typography, English, and presentation, the most novelty of this moment variation is the combination of a few new fabric. a part of it's mine (mostly together with coauthors). here's short advisor to those additions. i've got augmented the account of express combinatory common sense with an outline of the confluence homes of rewriting structures of categor­ ical combinators (Hardin, Yokouchi), and of the newly built cal­ culi of particular substitutions (Abadi, Cardelli, Curien, Hardin, Levy, and Rios), that are related in spirit to the explicit combinatory common sense, yet are towards the syntax of A-calculus (Section 1.2). The examine of the total abstraction challenge for PCF and extensions of it's been enriched with a brand new complete abstraction consequence: the version of sequential algorithms is absolutely summary with appreciate to an extension of PCF with a keep watch over operator (Cartwright, Felleisen, Curien). An order­ extensional version of error-sensitive sequential algorithms can also be absolutely summary for a corresponding extension of PCF with a keep watch over operator and mistakes (Sections 2.6 and 4.1). I recommend that sequential algorithms lend themselves to a decomposition of the functionality areas that ends up in versions of linear common sense (Lamarche, Curien), and that connects sequentiality with video games (Joyal, Blass, Abramsky) (Sections 2.1 and 2.6).

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Xn}, there exists a unique closed term A* modulo eeL ~llSP such that: [ A =CCLIJIl SP A * (xn ,.. e ,DB (x ,xn ,.. 2 A-CALCULUS (x,xn ,··,xO) =CCLPllSP 1t(X ,(Xn ,··,xo». Then A • = A(A 0 1t)ld fits. Uniqueness is by ext. 0 Intuitively this result means that categorical combinatory logic is its own meta-language: in the above statement, A may be seen as the specification of a function, and A· as the code for that function. One may also say that A· "internalizes" the "function" (xn ,.. ,xo) ~ A.

XnpNDB(xo-··,xn» (MN)DB(xo-.. ,xn) = Snd 0 MDB(xo, .. 2) MDB(xO, .. ,xn) is closed. 1. Exercise Show that the following properties hold: MDB(xo-"'xn) = NDB(xo-.. y,xi+t ... xi_t,x ,xi+t,xn) =MDB(xo-.. 16), M[xof-No, .. st m+! :: {Yo,oo,Ym}, and where z is any variable. -calculus's substitution. More technically, the presence of z in the equation corresponds to the intuition that the environment is made of its useful part, and of "the rest", which z represents. Now we come to the simulation of substitution and lifting.

Xjw = A. )). We just have to notice: A =IdR,AI,DA A(App 0

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