What Is State?

Definition

State is the current description of a system at a given moment, relative to what the system can store, observe, or use.

Examples:

• x = 5
• (x = 5, y = 10, flag = true)
• a webpage: content, form inputs, selected tab, scroll position

State does not mean every physical detail of reality.

It means the relevant current condition of the system being described.

State = the current snapshot of a system's relevant information

State vs Input

Input is the information given to a system.

State is the system's current situation during computation.

Example:

Input: 3
Initial state: x = 3
Rule: add 2
New state: x = 5

Input often helps create the first state.

After that, computation proceeds by changing state step by step.

Input can start computation, but state is what changes during computation.

Initial State

The initial state is the starting condition of the system before the first transition.

Example:

Initial state: x = 3

If the system applies +2, the next states are:

3 -> 5 -> 7 -> 9

The input may determine the initial state, but they are not always identical.

Example:

Input: user clicks submit
Initial state after input: form_submitted = true

Initial state = where the system begins before rules start changing it

State Transition

A state transition is the movement from one state to another by applying a rule.

Example:

Current state: x = 3
Rule: add 2
Next state: x = 5

In general:

State(t + 1) = Rule(State(t))

Computation can be understood as a sequence of state transitions.

State Changes Over Time

State is not fixed.

Example:

Initial: 3
+2 -> 5
+2 -> 7
+2 -> 9

Each step:

• reads the current state
• applies a rule
• produces a new state

Computation = rule-governed change of state over time

Relationship Between States

States are connected through rules.

The next state is not random if the rule is defined.

Example:

Rule: add 2

3 -> 5
5 -> 7
7 -> 9

The relationship between states is structured.

Rules define how one state leads to another.

Relevant vs Irrelevant Parts Of State

A system may contain more information than a rule uses at a particular step.

Example:

State: x = 5, y = 10
Rule: add 1 to x

The rule only uses x, but y can still be part of the system state.

Why?

Because a later rule may use y.

Better distinction:

• system state = information the system currently has
• active information = the part used by the current rule

A rule may inspect only part of the state, but the unused part can still matter later.

State vs Memory

State is the current condition of the system.

Memory is the mechanism that preserves information across steps.

Example:

Step 1: x = 5
Step 2: x = 6
Step 3: x = 7

For x to persist across steps, the system must have some way to store it.

That storage is memory.

State describes what the system is like now. Memory helps carry state across time.

Internal vs External State

Some state is internal to the system.

Some state is externally visible.

Examples:

• internal state: variables, hidden flags, memory contents
• external state: printed output, screen display, file contents

Example:

Internal state: counter = 3
External state: screen shows "3"

These are related, but not identical.

The internal state may change before the external state is updated.

Internal state affects how the system behaves. External state is what the outside world can observe.

State vs Configuration

State is the current data or condition.

Configuration is state plus execution context.

Example:

(x = 5, at step 1)
(x = 5, at step 2)

The data value is the same: x = 5.

But the system is in a different position in the process.

That difference can change what happens next.

Configuration = state + where the system is in execution

State Space

State space is the set of all possible states a system can be in.

Example:

x can be 1, 2, or 3

Then the state space is:

{1, 2, 3}

For two binary values:

x in {0, 1}
y in {0, 1}

The state space is:

(0, 0)
(0, 1)
(1, 0)
(1, 1)

Computation moves within a state space.

Fixed Rule vs Rate Of Change

A fixed rule creates consistent structure, but not necessarily constant change.

Examples:

Rule: add 2
3 -> 5 -> 7 -> 9

The amount of change is constant.

Rule: multiply by 2
3 -> 6 -> 12 -> 24

The rule is fixed, but the amount of change grows.

Fixed rule does not mean fixed rate of change.

State Can Stabilize, Repeat, Or Stop

State does not always keep changing in a progressive direction.

Fixed Point

A fixed point is a state that maps back to itself.

10 -> 10 -> 10

Cycle

A cycle repeats previous states.

0 -> 1 -> 0 -> 1

Termination

Termination means the computation stops.

running -> halted

State evolution does not guarantee progress.

Reversibility

Some state transitions can be reversed.

Others cannot.

Reversible

x -> x + 2

If the result is 7, the previous value can be recovered by subtracting 2.

Irreversible

x -> floor(x / 2)

Both 4 and 5 become 2.

The original value is lost.

Reversibility depends on whether information is preserved.

Deterministic vs Non-Deterministic

In a deterministic system, one state has one next state.

Example:

5 -> 6

In a non-deterministic system, one state may have multiple possible next states.

Example:

5 -> 6
5 -> 4

This does not mean the system is meaningless.

It means the transition rule allows more than one possible next state.

State may or may not uniquely determine what happens next.

Final Summary

• State is the current description of a system.
• Input can initialize state.
• Computation can be understood as a sequence of state transitions.
• Rules connect one state to another.
• A rule may use only part of the state.
• Memory preserves information across steps.
• Configuration adds execution context.
• State space defines all possible states.
• State can stabilize, repeat, or stop.
• Some transitions are reversible, others are not.
• Behavior can be deterministic or non-deterministic.