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I am learning about finite automata and trying to create a machine that matches

{w ∈ Σ∗| w does not contain the substring 10}

I created a DFA where it either starts with 1 or 0. I know it's not the most concise FA, but is it accurate?

user168855
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    Hello , your diagram seems to be incomplete, the initial state isn't indicated. Also, and this might be some convention I'm simply not aware of, but the transitions in a FSA usually only contain one character. – Knogger Mar 14 '24 at 12:25
  • https://cs.stackexchange.com/q/1331/755 – D.W. Mar 14 '24 at 18:30

2 Answers2

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Your automaton (automata is the plural word) is wrong:

  • as @Knogger stated, there is no initial state
  • finite automata (unless generalized) can only have one-letter transitions, so transitions $01$ are not allowed
  • even if $01$ transitions are allowed, this would create $10$ as substring, since $0101$ contains $10$.

You should try again, noting the fact that a word $w\in \{0,1\}^*$ does not contain $10$ as a substring if and only if $w = 0^p1^q$, for some $p, q\geqslant 0$.

Nathaniel
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This one is a classic textbook problem. You can build your required DFA as follows:

step 1: build a complete DFA $M$ that accepts all strings containing 10 as a substring

step 2: just swap the status of final/non-final states in this DFA to get another DFA $M'$

step 3: argue that $M'$ has to be your required one

Read more about DFA complementation here.

codeR
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