# Nondeterministic X-Machine

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### 5.1 Overview

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 [1], [2], [3].

### 5.2 Design of Nondeterministic X-Machine

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 (XX) 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.

NXM
state:  Q
alphaIn:  SigmaIn
alphaOut:  SigmaOut
memory:  Memory
alpha:  (SigmaIn  Memory)
beta:  (SigmaOut  Memory)
function:  Memory  Memory
trans: Q  Memory  Memory   Q
I:  Q
T:  Q

state  
I  state
T  state
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


### Invariants:

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.