CS 420 2024fall ibarland

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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.58pdf=43book).

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 ⊆ KxΣxK, such that this relation is a function¹ — we'll call it the machine's transition function. (We could also write δ ∈ KxΣ → K, though this function-as-relation phrasing sets us up for non-determinstic automata later.)

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)
    Map< Pair<State,Character>, State> δ; // @pre States are all in K; Characters are all in Σ.
               // We could also use java.function.BiFunction
               // Or we could use a Set<ThreeTuple<State,Character,State>>, so long as that set (relation) represents a function.
    Set<State> A; // @pre: K.containsAll(A)
    }

Example: All equivalent:

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

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

• Define configuration: an element of KxΣ^* (think: in a current state, w/ the remaining input)
• Define ⊢, reaches in exactly one step:
  〈q₁,cw〉 ⊢ 〈q₂,w〉 iff (q₁,c,q₂)  ∈  δ (or equiv., δ(q₁,c) = q₂, since δ is a function)
  We see that ⊢ ⊆ Configurations x Configurations, so ⊢ is a relation (I.e. some configurations are related to others via the reaches in exactly one step relation).
• Define a computation of a FSM M on the string w: A sequence of configurations C₀, C₁, …, C_n where
  1. C₀ = <s,w> where s is M's start-state,
  2. C_i ⊢ C_i+1 for all i  ∈  [0,n),
     Also written C₀ ⊢^* C_n. That is, the relation ⊢^* is the transitive closure of ⊢.
  and
  3. C_n = <q,ε>.
• Define accepts: a DFSM M accepts w iff there is a computation 〈s,w〉 ⊢^* 〈q,ε〉 where q∈A (and of course, s is M's start-state).
• Define language of a machine L(M): {w | M accepts w }
• Define regular languages: { L(M) | M is a DFSM }

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