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Definitions
sets; FSMs

READING:

• previously Chpt.3 (specifically: 3.0–3.2, and skim 3.3)
• Chpt.05 §5.0–5.4 today: 5.0–5.3 (deterministic); tomorrow §5.4 (non-deterministic)

Chapter 5

1. Come up with: Example 5.3, every "a" region in w is even-length
2. Come up with: Example 5.4, odd-parity
3. Look at Example 5.1, vending-machine
4. Look at Example 5.2 (p.43+15=58(pdf)).

Defining Finite State Machines

A FSM = ⟨K, Σ, δ, s, A⟩. In math:

• K is the set-of-states.
• Σ is the alphabet – the set of letters we'll see of the input.
• s ∈ K, the initial state
• A ⊆ K, the accept-states
• δ ∈ KxΣ → K, the transition function.

typedef State String; // okay, this isn't legal Java, but fine: we're representing States by Strings (their name); using `int` is common too. (Really: we should have `class FSM<StateType>`.)

class FSM {
    Set<State> K;
    CharSet Σ;
    State s;      // @pre: K.contains(s)
    Set<State> A; // @pre: K.containsAll(A)
    Map< Pair<State,Character>, State> δ; // @pre States are all in K; Characters are all in Σ.
               // We could also use java.function.BiFunction
    }

Example: All equivalent:

• FSM as picture
• FSM as math: M = < {0,1,2,3}, {a,b}, δ, 0, {1} > where δ = { <0,a> ↦ 1, <0,b> ↦ 3, <1,a> ↦ 3, <1,b> ↦ 2, <2,a> ↦ 2, <2,b> ↦ 1, <3,b> ↦ 3, <3,b> ↦ 3 }
• FSM as Java object:

  new FSM( Set.of( 0,1,2,3 ), new CharSet("ab"), 0, Set.of(1), Map.of( new Pair(0,'a'), 1, new Pair(0,'b'), 3, ... ) )

• Define configuration: element of KxΣ^* (think: in a current state, w/ the remaining input)
• Define step: 〈q₁,cw〉 ⊢ 〈q₂,w〉 iff δ(q₁,c) = q₂
  ⊢ ⊆ Configurations x Configurations
• Define computation (see book)
• Define accepts: a DFSM M accepts w iff 〈s,w〉 ⊢ ^* 〈q,ε〉 where q∈A
• Define language of a machine L(M): {w | M accepts w }
• Define regular languages: { L(M) : M is a DFSM }

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