Outputs are functions ofa.present and next stateb.next statec.previous stated.present state

It appears that you're asking about the relationships between present states, next states, and previous states in a function context, likely within the domain of finite state machines or similar systems. Here's a detailed breakdown of the concepts involved:

Understanding State Functions

1. Definitions:

   • Present State: The current status or condition of the system.
   • Next State: The condition the system will transition into after a certain event occurs or a decision is made.
   • Previous State: The condition the system was in before the current present state.
   • Output: The result produced by the system based on the current inputs and states.

2. Relationship between States:

   • Functions can be defined, typically represented as:
     • $f: \text{Present State} \rightarrow \text{Next State}$
     • $g: \text{Present State, Input} \rightarrow \text{Output}$
     • Sometimes, the relationship might also involve the previous state, represented as:
       • $h: \text{Previous State, Present State} \rightarrow \text{Output}$

3. Example:

   • In a finite state machine, the transition from the present state to the next state can be triggered by an input:
     • If the present state is $S_1$, and the input is $x$, the next state can be defined by a transition function such as: $\text{Next State} = T(S_1, x)$

4. Use Cases:

   • In programming or digital logic design, state transition diagrams or tables often illustrate how states interact, incorporating conditions that lead to state changes and the associated outputs.

Understanding these functions is crucial for modeling systems in computational theory, automata, digital circuit design, and more. If you need further elaboration or specific examples, feel free to ask!