When empty it responds to the signal empty at all

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(10) A process which behaves like a stack of messages. When empty, it responds to the signal empty. At all times it is ready to input a new message from the left and put it on top of the stack; and whenever nonempty, it is prepared to output and remove the top element of the stack STACK = P hi where, P hi = ( empty P hi | left ? x P h x i ) P h x i _ s = ( right ! x P s | left ? y P h y i _ h x i _ s ) 11
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& $ % 5. Specifications (1) Some Notations: If c is a channel name, we define (see Section 1.9.6) tr c = message * ( tr αc ) It is convenient just to omit the tr , and write right left instead of tr right tr left . Another useful definition places a lower bound on the length of a prefix s n t = ( s t # t # s + N ) Properties: (a) s 0 t ( s = t ) (b) s n t t m u ) s n + m u (c) s t ≡ ∃ n s n t 12
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& $ % (2) Examples X1 COPY sat right 1 left X2 DOUBLE sat right 1 double * ( left ) X3 UNPACK sat right _ /left where, _ / h s 0 , s 1 , ..., s n - 1 i = s _ 0 s _ 1 ...s n - 1 The specification here states that the output on the right is obtained by flattening the sequence of sequences input on the left. X4 PACK sat (( _ /right 125 left ) (# * right ∈ { 125 } * )) 13
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& $ % Communication 1. Laws Thus output may be regarded as a specialized case of the prefix operator, and input a special case of choice; and this leads to the law L1 ( c ! v P ) || ( c ? x Q ( x )) = c ! v ( P || Q ( v )) If desired, such internal communications can be concealed by applying the concealment operator described in Section 3.5 outside the parallel composition of the two processes which communicate on the same channel, as shown by the law L2 (( c ! v P ) || ( c ? x Q ( x ))) \ C = ( P || Q ( v )) \ C where C = { c.v | v αc } 14
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& $ % 2. Examples We use the examples below to illustrate the communication sequence between two parallel branches X1 Let P = ( left ? x mid !( x × x ) P ) Q = ( mid ? y right !(173 × y ) Q ) Clearly P sat ( mid 1 square * ( left )) Q sat ( right 1 173 × mid ) where (173 × mid ) multiples each message of mid by 173. It follows that ( P || Q ) sat ( right 1 173 × mid ) ( mid 1 square * ( left )) The specification here implies right 173 × square * ( left ) which was presumably the original intention. 15
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& $ % Pipes 1. Definition of Pipes (a) Processes: with only two channels, namely an input channel left and an output channel right (b) Connection: The processes P and Q may be joined together so that the right channel of P
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