By Rendong G., Zunquan x., Jinzhi W.

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2. ,f' has no return-test branches. If lv[' accepts an input x, then there is an accepting computation such that all calls return. Proof As Af can contain return-test branches we can assume that no processor is ever blocked waiting for a return. The machine Af will be simulated by a machine M'. In order to organize the communication between the processors, every active processor of the machine M' keeps a local clock set equal to the elapsed simulated time of the corresponding processor of Af. Let us explain how this clock is used.

2 and using replication and transposition of matrices we can construct the following vector: J-l = '" (iom-n'-' ° < n'-l > OSI·· ·Iom-n'-' < 1 > OSlom-n'-' < > OSJ)m' The vector / can be constructed as / = j3 V J-l. Assuming / constructed by replication we can obtain /e and by transposition we can obtain /T from /e' The dimension of the matrices is o (2(S(n)+log n». As recursive doubling, transposition and replication can be done in O(S(n) + log n) steps, we obtain the result 0 To construct the matrix Move we need to "present" at every pair of possible configurations a "copy" of the input word in the part of the configurations corresponding to the input head.

In a tree of processors each processor is a general purpose machine with unbounded memory capabilities. All processors execute the same program. But different processors can execute different instructions. It is a MIMD (multiple instruction stream, multiple data stream machine). The model has no global memory and there does not exist any processor with global control. The communication between processors is hierarchically organized and synchronized. The notion of k-offspring parallel random access machine, k-PRAM for short, gives a precise meaning to the concept of a tree of processors.

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