Foundations of the X-machine Theory for Testing Research Report CS-02-06

The X-machine (also known as Eilenberg X-machine) is an abstract model similar to the finite-state machine (FSM), which contains a data set. The edges between control-states are labelled with relations (functions) that operate on this data. In this way the machine integrates the control structure and the data processing under the same formalism, and moreover, it allows them to be specified separately. The Xmachine (EXM) is a general model of computation, in the sense that it can simulate other machine models including the Turing machine. Several classes of EXM have been proposed by means of restrictions on the data set or on the form of the relations (functions). The stream X-machine (SXM) is a subclass of EXM with capabilities for handling sequences (streams) of inputs and outputs. Even more important is that the Holcombe-Ipate testing approach (SXMT) corroborates the functional equivalence between SXM behaviours (finds all the faults). Some recent investigations have shown that the SXMT can be extended to some other classes of EXM, and to deal with non-determinism. This report presents a comprehensive survey of the theoretical foundations on which the testing approaches for these machines are based. The applicability of the SXMT to systems of communicating EXM has been studied elsewhere. This report covers two models related to these systems, namely the straight-move stream X-machine (Sm-SXM), and the multiple-stream X-machine (MSXM). The M-SXM is formalised and analysed here and it is proved that any MSXM can be reduced to a SXM that computes the same relation.

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