The Stream X-Machine
A Stream X-Machine (SXM) is an extended finite-state machine with auxiliary memory, inputs and outputs. It is a variant of the general X-machine, in which the fundamental data type ''X'' = ''Out''* × ''Mem'' × ''In''*, that is, a tuple consisting of an output stream, the memory and an input stream. A SXM separates the ''control flow'' of a system from the ''processing'' carried out by the system. The control is modelled by a finite-state machine (known as the ''associated automaton'') whose transitions are labelled with processing functions chosen from a set Φ (known as the ''type'' of the machine), which act upon the fundamental data type. Each processing function in Φ is a partial function, and can be considered to have the type φ: ''Mem'' × ''In'' → ''Out'' × ''Mem'', where ''Mem'' is the memory type, and ''In'' and ''Out'' are respectively the input and output types. In any given state, a transition is ''enabled'' if the domain of the associated function φi includes the next input value and the current memory state. If (at most) one transition is enabled in a given state, the machine is ''deterministic''. Crossing a transition is equivalent to applying the associated function φi, which consumes one input, possibly modifies the memory and produces one output. Each recognised path through the machine therefore generates a list φ1 ... φn of functions, and the SXM composes these functions together to generate a relation on the fundamental data type , φ1 ... φn, : ''X'' → ''X''.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'') = (<>, ''mem''0, ''ins'') β(''outs'', ''mem''n, <>) = ''outs'' where ''ins: In''*, ''outs: Out''* and ''mem''i: ''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'', ''mem''i, ''in'' :: ''ins'') = (''outs'' :: ''out'', ''mem''i+1, ''ins'') where:: 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.
Processing and Testable Properties
Because of the above equivalence, attention may focus on the way a Stream X-Machine processes inputs into outputs, using an auxiliary memory. Given an initial memory state ''mem''0 and an input stream ''ins'', the machine executes in a step-wise fashion, consuming one input at a time, and generating one output at a time. Provided that (at least) one recognised path ''path'' = φ1 ... φn exists leading to a state in which the input has been consumed, the machine yields a final memory state ''mem''n and an output stream ''outs''. In general, we can think of this as the relation computed by all recognised paths: , ''path'' , : ''In''* → ''Out''*. This is often called the ''behaviour'' of the Stream X-Machine. The behaviour is deterministic, if (at most) one transition is enabled in each state. This property, and the ability to control how the machine behaves in a step-wise fashion in response to inputs and memory, makes it an ideal model for the specification of software systems. If the specification and implementation are both assumed to be Stream X-Machines, then the implementation may be tested for conformance to the specification machine, by observing the inputs and outputs at each step. Laycock first highlighted the utility of single-step processing with observations for testing purposes. Holcombe and Ipate developed this into a practical theory of software testing which was fully compositional, scaling up to very large systems.F. Ipate and M. Holcombe (1998) 'A method for refining and testing generalised machine specifications'. ''Int. J. Comp. Math.'' 68, pp. 197-219. A proof of correct integration guarantees that testing each component and each integration layer separately corresponds to testing the whole system. This divide-and-conquer approach makes ''exhaustive'' testing feasible for large systems. The testing method is described in a separate article on the Stream X-Machine testing methodology.See also
* X-machines, a general description of the X-machine model, including a simple example. * The Stream X-Machine Testing Methodology, a ''complete functional testing'' technique. Using this methodology, it is possible to identify a ''finite'' set of tests that exhaustively determine whether an implementation matches its specification. The technique overcomes formal undecidability limitations by insisting that users apply carefully specified ''design for test'' principles during implementation. * Communicating Stream X-Machines (CSXMs), a concurrent version of the SXM model, with applications in fields ranging from social insects to economics.External links
References