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What is the difference between transition function delta ($\delta$) and extended transition function delta hat ($\hat{\delta}$) in finite automata?

Both of them, when started at a state $q$ for a string $w$, will lead to the same state $p$. So what is the difference?

See, for example, page 15 of the third edition of Dexter Kozen's Automata and Computability (https://doi.org/10.1007/978-3-642-85706-5)

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    $\begingroup$ Please define the "extended transition function" -- it's not a concept I'm familiar with. And, I suggest that, if you write out the definition, the difference from the standard transition function should become obvious. $\endgroup$ Jun 22, 2019 at 10:23
  • $\begingroup$ Please edit the question to add a reference to the textbook or article you are talking about. $\endgroup$
    – John L.
    Jun 22, 2019 at 12:15

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I read some references in order to answer your question.

  • Transition function: takes as arguments a state and an input symbol and returns a state, denoted by $\delta$.

  • Extended transition function: Describes what happens when we start in any state and follow any sequence of inputs. It is a function that takes a state $q$ and a string $w$ and returns a state $p$ (the state that the automaton reaches when starting in state $q$ and processing the sequence of inputs $w$). It is denoted by $\hat{\delta}$.

In simple terms: transition function takes two parameters: a state and a symbol, while extended transition function takes a state and a string.

Example of transition function

Example 1

Example of extended transition function

Example 2

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