Weighted automaton
A weighted automaton is like a special calculator that can read words and give you a value, like a reward, for every word it reads. Imagine you have a machine that reads the word "apple" and gives you a reward of 3 stickers. Now imagine another machine that reads the word "banana" and gives you a reward of 5 stickers. A weighted automaton is like combining these machines into one that can read both "apple" and "banana" and give you the correct reward for each word.
But how does it work? The machine has a bunch of states it can be in, kind of like different rooms in a house. When you give it a word to read, it starts in a special state called the "start state". Then, it looks at each letter in the word and follows an arrow from one state to another based on the letter. Each arrow has a "weight" which is like a value that tells you what reward you get for taking that arrow. At the end of the word, the machine ends up in one of these states and gives you the total reward you earned by following the arrows in the machine.
So let's say you give the machine the word "applebanana". It starts in the start state, reads the letter 'a' and follows an arrow to a new state with weight 3 (because 'a' means 3 stickers in the "apple" machine). It then reads 'p' and follows another arrow to a new state, still with weight 3. It does this for each letter until it reaches the end of "apple". At that point, it's in a new state and has earned a total of 3 stickers. Then, it starts again in the start state with the "banana" part of the word. This time it follows arrows with weight 5 for each letter until it reaches the end and ends up in a new state with a total reward of 5 stickers. Finally, the machine combines these rewards to give you a final total of 8 stickers earned from the word "applebanana".
This is just the basics of how a weighted automaton works, but it can get more complicated with different types of automata and different ways of calculating rewards. Overall though, you can think of it like a special calculator that reads words and gives you rewards for them.
But how does it work? The machine has a bunch of states it can be in, kind of like different rooms in a house. When you give it a word to read, it starts in a special state called the "start state". Then, it looks at each letter in the word and follows an arrow from one state to another based on the letter. Each arrow has a "weight" which is like a value that tells you what reward you get for taking that arrow. At the end of the word, the machine ends up in one of these states and gives you the total reward you earned by following the arrows in the machine.
So let's say you give the machine the word "applebanana". It starts in the start state, reads the letter 'a' and follows an arrow to a new state with weight 3 (because 'a' means 3 stickers in the "apple" machine). It then reads 'p' and follows another arrow to a new state, still with weight 3. It does this for each letter until it reaches the end of "apple". At that point, it's in a new state and has earned a total of 3 stickers. Then, it starts again in the start state with the "banana" part of the word. This time it follows arrows with weight 5 for each letter until it reaches the end and ends up in a new state with a total reward of 5 stickers. Finally, the machine combines these rewards to give you a final total of 8 stickers earned from the word "applebanana".
This is just the basics of how a weighted automaton works, but it can get more complicated with different types of automata and different ways of calculating rewards. Overall though, you can think of it like a special calculator that reads words and gives you rewards for them.
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