To achieve the maximum benefits from formal techniques, integration of approaches is required. In this chapter we present the integration of X-machine models and Z notation. The X-machine models sued to give the relationship between Z and X-machine are: (a) nondeterministic X-machine, (b) deterministic X-machine, (c) nondeterministic stream X-machine, (d) deterministic stream X-machine, (e) communicating stream X-machine, and (f) communicating stream X-machine system. The informal definitions of all these machines are taken from , , .
A Nondeterministic X-Machine is 10-tuple NXM = (X, Y, Z, α, β, Q, Φ, F, I, T) where:
1. X is a fundamental dataset on that the machine operates.
2. Y is a finite set of input alphabets.
3. Z is a finite set of output alphabets.
4. α and β are input and output partial functions, used to convert the input into the output sets from the fundamental sets, i. e., α: Y X and β: X Z.
5. Q is a finite nonempty set of states.
6. Φ is type of M, a set of relations on X, i. e., Φ: P (X X). The notation (XX) denotes a set of all possible partial functions from X to X.
7. F is a next state partial function, a transition function, i. e., F: Q x φ P Q, which takes a state a partial function and produces a new set of states.
8. I is a set of initial states, a subset of Q, and T is a set of terminal states, a subset of Q.
alpha: (SigmaIn Memory)
beta: (SigmaOut Memory)
function: Memory Memory
trans: Q Memory Memory Q
q, q1: Q; m, m1: Memory; i: SigmaIn; o: SigmaOut
q state q1 state m memory m1 memory i alphaIn
o alphaOut m m1 function i m alpha o m1 beta
s, s1: Q s state s1 state q m m1 s trans
q1 m m1 s1 trans q m m1 = q1 m m1 s = s1
a) The set of states states is a nonempty set.
b) The set of initial states I is a subset of states.
c) The set of final states T is a subset of states.
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