Transition System

Integer transition system

Has button, stores an integer.
When you press the button the integer changes.

Transition system:

• Set $S$ (of states)
• transition funciton $\zeta :S\to \mathbb{Z}\times S$
• Initial state $s_0$

Adding a boolean

• Set $S$ (of states)
• Family of transition funciton
  • $\zeta :color\to S\to \mathbb{Z}\times S$
• Initial state $s_0$

Passive state

Nothing happens until an input is provided.
Also called reactive transition systems.

Active State

State where it spontaneously changes state.

Mixed system

Set X of passive states
Set Y of active sets

$$\begin{array}{r}\zeta :X\to Y^{\mathbb{Z}}\\ \xi :X\to \mathbb{Z}\times X\end{array}$$

These then create a pairing so forms a co-algebra in Set^2.

$$\left(\begin{array}{c}\zeta \\ \xi \end{array}\right):\left(\begin{array}{c}X\to Y^{\mathbb{Z}}\\ X\to \mathbb{Z}\times X\end{array}\right)$$