85 lines
2.8 KiB
Markdown
85 lines
2.8 KiB
Markdown
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# Finite Automata
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Simple machines that take in strings (sequences of symbols) as input and recognize whether or note each input string satisfies some condition(s)
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Finite automata use states to keep track of important information about symbols
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## Mathematical definition of a finite automaton
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$M = (Q, \sum, \delta, q, F)$
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- $Q$ is a finite set of states
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- $\sum$ is a finite set of input symbols called the input alphabet
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- $\delta$ is the transition function
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- Takes every pair consisting of a state and an input smbol and returns the next state, this tells us everything we need to know about what the machine does in one computation step
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- $q \in Q$ is the start state
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- $F \subseteq Q$ is the set of accept states
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## Acceptance by a finite automaton
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Let M be a finite automaton
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Let $w = w_1, w_2, ..., w_n$ be n input string over \sum
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As input string w is process by $M$, define the sequence of visited states $r_0, r_1, ..., r_n$ as follows:
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- $r_0 = q$
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- $\forall i = 0,1, ... n-1, r_{i+1} = \delta(r_i, w_{i+1})$
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If $r_n \in F$ then $M$ accepts $w$, otherwise $M$ rejects $w$
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Note: The empty string $\epsilon$ has length $0$, it is accepted by $M$ if any only if the start state is an accept state
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For a given finite automaton $M$, the set of stings accepted by $M$ is the language of $M$ and is denoted $L(M)$
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A language $A$ is called regular if there exists a finite automaton $M$ such that $A = L(M)$
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## Example of finite automata
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A finite automation $M_1$ that accepts
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$$\{w \in \{0,1\}^* \ | \ n_1(w) \geq 2\}$$
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(State diagram on board)
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A is the start sate and C is the accept state
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A -> B (On 1)
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B -> C (On 1)
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A -> A (On 0)
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B -> B (On 0)
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C -> C (On 0, 1)
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A: we have seen no ones yet
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B: we have seen exactly one 1
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C: we have seen two ore more 1's
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## Example 2 of finite automata
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A finite automation $M_2$ that accepts
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$$\{w \in \{0,1\}^* \ | \ n_1(w) = 2\}$$
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(State diagram on board)
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A is the start state and C is the accept state
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A -> B (On 1)
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B -> C (On 1)
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C -> D (On 1)
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A -> A (On 0)
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B -> B (On 0)
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C -> C (On 0)
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D -> D (On 0, 1)
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Note: D is an example of a dead state, in which you cannot escape after
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## Example 3 of finite automata
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A finite automation $M_3$ that accepts
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$$\{w \in \{a,b\}^* \ | \ \text{w starts with abb}\}$$
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(State diagram on board)
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A is the start state and D is the accept state
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A -> B (On a)
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B -> C (On b)
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C -> D (On b)
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D -> D (On a, b)
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A -> X (On b)
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B -> X (On a)
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C -> X (On a)
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X -> X (On a ,b)
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## Example 4 of finite automata
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A finite automation $M_4$ that accepts
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(Didn't catch the definition)
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$$\{w \in \{0,1\}^* \ | \ n_1(w) = 2\}$$
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(State diagram on board)
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A is the start state and D is the accept state
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(state transitions would go here)
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A: the most recent symbol was 1 or we have seen no symbols
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B: the last symbol was 0, but we have noon seen 010 yet
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C: the last two symbols were 01, but we have not seen 010
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D: we have seen the pattern 010
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