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In this paper, we describe a simple language for parallel programming. Its semantics is studied thoroughly. The desirable properties of this language and its deficiencies are exhibited by this theoretical study. Basic results on parallel program schemata are given. We hope in this way to make a case for more formal (i.e. mathematical) approach to the design of languages for systems programming and the design of operating systems. There is a wide disagreement among systems designers as to what are the best primitives for writing systems programs. In this paper, we describe a simple language for parallel programming and study its mathematical properties. 1. A SIMPLE LANGUAGE FOR PARALLEL PROGRAMMING The features of our mini-language are exhibited on the sample program S on Figure 1. The conventions are close to Algol1 and we only insist upon the new features. The program S consists of a set of declarations and a body. Variables of type integer channel are declared at line (1), and for any simple type σ (boolean, real, etc. . . ) we could have declared a σ channel. Then processes f , g and h are declared, much like procedures. Aside from usual parameters (passed by value in this example, like INIT at line (3)), we can declare in the heading of the process how it is linked to other processes : at line (2) f is stated to communicate via two input lines that can carry integers, and one similar output line. The body of a process is an usual Algol program except for invocation of wait until something on an input line (e.g. at (4)) or send a variable on a line of compatible type (e.g. at (5)). The process stays blocked on a wait until something is being sent on this line by another process, but nothing can prevent a process from performing a send on a line. In others words, processes communicate via first-in first-out (fifo) queues. Calling instances of the processes is done in the body of the main program at line (6) where the actual names of he channels are bound to the formal parameters of the processes. The infix operator par initiates the concurrent activation of the processes. Such a style of programming is close to may systems using EVENT mechanisms ([1, 2, 3, 4]). A pictorial representation of the program is the schema P on Figure 2, where the nodes represent processes and the arcs communication channels between these processes. What sort of things would we like to prove on a program like S? Firstly, that all processes in S run forever. Secondly, Begin (1) In t eg e r channel X, Y, Z , T1 , T2 ; (2 ) Process f ( i n t e r g e r in U,V; i n t e r g e r out W) ; Begin i n t e g e r I ; l o g i c a l B; B := true ; Repeat Begin (4 ) I := i f B then wait (U) e l s e wait (V) ; (7 ) p r in t ( I ) ; (5 ) send I on W; B := not B; End ; End ; Process g ( i n t e g e r in U ; i n t e g e r out V, W) ; Begin i n t e g e r I ; l o g i c a l B; B := true ; Repeat Begin I := wait (U) ; i f B then send I on V e l s e send I on W : B := not B; End ; End ; (3 ) Process h( i n t e g e r in U; i n t e g e r out V; i n t e g e r INIT ) ; Begin i n t e g e r I ; send INIT on V; Repeat Begin I := wait (U) ; send I on V; End ; End ; Comment : body o f mainprogram ; (6 ) f (X,Y,Z) par g (X,T1 ,T2) par h(T1 ,Y, 0 ) par h(T2 , Z , 1 ) ; End ; Figure 1: Sample parallel program S. more precisely, that S prints out (at line (7)) an alternating sequence of 0’s and 1’s forever. Third, that if one of the processes were to stop at some time for an extraneous reason, the whole systems would stop. The ability to state formally this kind of property of a parallel program and to prove them within a formal logical framework is the central motivation for the theoretical study of the next sections. 2. PARALLEL COMPUTATION Informally speaking, a parallel computation is organized in the following way: some autonomous computing stations are connected to each other in a network by communication lines. Computing stations exchange information through these lines. A given station computes on data coming along