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Programming in the Geometric Machine 1 Introduction.

Programming in the Geometric Machine 1 Introduction.

Programming in the Geometric Machine Renata H.S.Reiser,Antˆo nio Carlos R.Costa∗and Grac¸aliz P.Dimuro

{reiser,rocha,liz}@atlas.ucpel.tche.br

Universidade Cat´o lica de Pelotas,Escola de Inform´a tica

Pelotas,CP402,Brazil

Abstract.This paper presents the programming language induced by the ordered

structure of the Geometric Machine Model(GMM).The GMM is an abstract ma-

chine model,based on Girard’s coherence space,capable of modelling sequential,

alternative,parallel(synchronous)and non-deterministic computations on a(possi-

bly infinite)shared memory.The processes of the GMM are inductively constructed

in a Coherence Space of Processes.The memory of the GMM,supporting a coher-

ence space of states,is conceived as the set of points of a three dimensional euclidian

http://doc.xuehai.neting the programming language induced by such structure,simple recursive

equations and related algorithms are presented,and their domain-theoretic semantics

in the machine model is given.

1Introduction.

Inspired by the work of Scott[7],a constructive and intuitive machine model,with infinite memory,called Geometric Machine Model(GMM)is introduced in[5].Over the GMM or-dered structure,we obtain representations of(non-)deterministic processes,labelled by posi-tions of a geometric space,including two special types of parallelism–the temporal and the spacial parallelism.Its most basic notion is that of a coherence relation representing the ad-missibility of parallelism between two or more basic operations(elementary processes).That relation defines a web over which a coherence space[3]of parallel processes is step-wise and systematically built.In the dual construction,the incoherence relation interprets the condition that models non-determinism.The sequential product and the deterministic sum of parallel and non-deterministic processes are endofunctors in the category of such coherence spaces (with linear functions as morphisms).Following this approach,aspects of Domain Theory[1] and Concurrency Theory[4]are connected to obtain a programming language derived from ordered structure of the GMM,denoted by L(D∞).In this sense,the aim of this work is the development of examples of the recursive equations defined in the language L(D∞)and over that it is possible to sample the semantic analysis of concurrent and distributed systems in-terpreted in such model.Some(possibly recursive)definitions of constructors,the functional composition,the parallel and non-deterministic computations related to synchronization of processes andfinally,the spatial definition of a computational process based on a geometric space is presented.This paper is structured as follows.The Section2summarizes the ordered structure for the memory states and possible programs in the GMM.A computational state is defined as a linear function from the coherence space of labels to coherence space of values.

∗PPPGC/UFRGS,Porto Alegre,CP15064,Brazil

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