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Example with a context (Wikipedia article):

A stack may be implemented to have a bounded capacity. If the stack is full and does not contain enough space to accept an entity to be pushed, the stack is then considered to be in an overflow state. The pop operation removes an item from the top of the stack. A pop either reveals previously concealed items or results in an empty stack, but, if the stack is empty, it goes into underflow state, which means no items are present in stack to be removed.

Could you please explain when we should use the word state with an article (countable noun) and when with no article (mass noun)? The meaning of the word state that I'm referring to here, of course, has to do with a particular condition that something is in at a specific time as in the usage example above that talks about software objects such as stacks that can change their state during their lifetime.

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"Into underflow state" here is simply a mistake. It should have an article: "into AN underflow state".

"State", in the sense of "condition or status", is a countable noun, period.

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You do not use articles before nouns if they describe a type and not an instance.

if the stack is empty, it goes into underflow state

The writer here is saying that underflow is a type of state that the stack can be in. In a programming context, this has the implication that the stack can be in one of several well-defined states, such as "underflow", "normal", "overflow", "disabled", for example.

Saying an underflow state would make underflow no longer a state that can be "labeled" upon the stack, and we are no longer making the implication there is an enumeration of states for the stack.

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