# What is the difference between transition function (delta) and extended transition function (delta hat) in finite automata?

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)

• 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. Jun 22, 2019 at 10:23
• Please edit the question to add a reference to the textbook or article you are talking about. Jun 22, 2019 at 12:15

• 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 $$\hat{\delta}$$.