The **Stream X-machine** (**SXM**) is a model of computation introduced by Gilbert Laycock in his 1993 PhD thesis, *The Theory and Practice of Specification Based Software Testing*.^{[1]} Based on Samuel Eilenberg’s X-machine, an extended finite state machine for processing data of the type *X*,^{[2]} the Stream X-Machine is a kind of X-machine for processing a memory data type *Mem* with associated input and output streams *In** and *Out**, that is, where *X* = *Out** × *Mem* × *In**. The transitions of a Stream X-Machine are labelled by functions of the form φ: *Mem* × *In* → *Out* × *Mem*, that is, which compute an output value and update the memory, from the current memory and an input value.

Although the general X-machine had been identified in the 1980s as a potentially useful formal model for specifying software systems,^{[3]} it was not until the emergence of the Stream X-Machine that this idea could be fully exploited. Florentin Ipate and Mike Holcombe went on to develop a theory of *complete* functional testing,^{[4]} in which complex software systems with hundreds of thousands of states and millions of transitions could be decomposed into separate SXMs that could be tested exhaustively, with a guaranteed proof of correct integration.^{[5]}

Because of the intuitive interpretation of Stream X-Machines as “processing agents with inputs and outputs”, they have attracted increasing interest, because of their utility in modelling real-world phenomena. The SXM model has important applications in fields as diverse as computational biology, software testing and agent-based computational economics.

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 φ_{i} 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 φ_{i}, which consumes one input, possibly modifies the memory and produces one output. Each recognised path through the machine therefore generates a list φ_{1} … φ_{n} of functions, and the SXM composes these functions together to generate a relation on the fundamental data type |φ_{1} … φ_{n}|: *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 φ_{i} 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 φ_{SXM}: *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* = φ_{1} … φ_{n} 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.^{[1]}

Holcombe and Ipate developed this into a practical theory of software testing^{[4]} which was fully compositional, scaling up to very large systems.^{[6]} A proof of correct integration^{[5]} 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.

References

- ^ Jump up to:
^{a}Gilbert Laycock (1993)^{b}*The Theory and Practice of Specification Based Software Testing*. PhD Thesis, University of Sheffield, Dept of Computer Science. Abstract Archived 2007-11-05 at the Wayback Machine **^**Samuel Eilenberg (1974)*Automata, Languages and Machines, Vol. A*. London: Academic Press.**^**Holcombe (1988) ‘X-machines as a basis for dynamic system specification’.*Software Engineering Journal***3***(2)*, pp. 69-76.- ^ Jump up to:
^{a}Mike Holcombe and Florentin Ipate (1998)^{b}*Correct systems – building a business process solution*. Applied Computing Series. Berlin: Springer-Verlag. - ^ Jump up to:
^{a}Ipate and W. M. L. Holcombe (1997) ‘An integration testing method which is proved to find all faults’.^{b}*Int. J. Comp. Math.*,**63**, pp. 159-178. **^**Ipate and M. Holcombe (1998) ‘A method for refining and testing generalised machine specifications’.*Int. J. Comp. Math.***68**, pp. 197-219.

Ofer Abarbanel – Executive Profile

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