Relationship To X-machines
The Stream X-Machine is a variant of X-machine in which the fundamental data type X = Out* × Mem × In*. In the original X-machine, the φi are general relations on X. In the Stream X-Machine, these are usually restricted to functions; however the SXM is still only deterministic if (at most) one transition is enabled in each state.
A general X-machine handles input and output using a prior encoding function α: Y → X for input, and a posterior decoding function β: X → Z for output, where Y and Z are respectively the input and output types. In a Stream X-Machine, these types are streams:Y = In* Z = Out*
and the encoding and decoding functions are defined as:α(ins) = (<>, mem0, ins) β(outs, memn, <>) = outs
where ins: In*, outs: Out* and memi: Mem. In other words, the machine is initialized with the whole of the input stream; and the decoded result is the whole of the output stream, provided the input stream is eventually consumed (otherwise the result is undefined).
Each processing function in a SXM is given the abbreviated type φSXM: Mem × In → Out × Mem. This can be mapped onto a general X-machine relation of the type φ: X → X if we treat this as computing:φ(outs, memi, in :: ins) = (outs :: out, memi+1, ins)
:: denotes concatenation of an element and a sequence. In other words, the relation extracts the head of the input stream, modifies memory and appends a value to the tail of the output stream.
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