2024-01-26 13:20:09

.obsidian/workspace.json

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@@ -125,7 +125,7 @@
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@@ -158,8 +158,9 @@
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UNB/Year 4/Semester 2/CS2333/2024-01-26.md

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