ITEC 420 2022fall ibarland

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Definitions
sets; FSMs

READING: Chpt.03

Previously, on itec420…

Examples of regular-expressions: For b(ba*b)* :

• What are some strings that match this pattern?
• What are some strings that don't match?
• What is the shortest string that matches?
• What is the shortest string that doesn't match?

def'n: A Language is: a

Draw FSM for b(ba*b)* (and minimize it, informally).

Defining Finite State Machines

Generalize: 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, ... )

Def'n of FSM "accepts" a string: (informal for now): Feed a string w into the FSM M (starting at s); if after transitioning on all characters of w, M is in a state in A, then we say "M accepts w". (more formal def'n will use terms "configuration of M" and a sequencek-of-configurations, "a computation".)
Def'n: For a FSM M, we say "L(M)" is the language it *accepts*; we also say M "recognizes" L. E.g. We just gave a FSM which recognized b(ba*b)* (and even: that machine computes ).

Sets

Common Sets: ∅, {false,true}, ℕ, ℤ, ℚ, ℝ, ℂ; Σ*

Sample sets to work with:

L₀ = {}
L₁ = {17}
L₂ = {vw, saab, bmw}
L₃ = {2,3,5,7}
P  = prime numbers
L₄ = state-abbreviations

L₅ = strings over {a,b}* where every 'a' is followed by a 'b'
L₆ = b*aa* = {a, aa, ba, aaa, baa, bba,     ... bbbaa, ... }
L₇ = b*a*b* = { ε,   a, b,    aa, ab, ba, bb,    aaa, aab, abb, baa, bab, bba, bbb,       ...}
L₈ = reg-exp-example

How to make new sets out of old ones:

• union: A ∪ B. x∈A∪B means x∈A ∨ x∈B
• intersection: A ∩ B x∈A∩B means x∈A ^ x∈B
• set-difference: A ∖ B x∈A∖B means x∈A and ¬x∈B
• cross-product: A x B means: x∈AxB means x=(a,b) where a∈A and b∈B

Exercise (challenge): Find two sets-of-strings A,B such that |AB| < |A| * |B| Answers:

• {a,b,ab}
  {b,bb}
  =
  {ab,abb,bb,bbb,abbb}
  In this case, the string
  abb
  can be created via two different concatenations:
• Challenge: give a smaller pair of languages which also has |AB| < |A|⋅|B|.                                                                                      Can you give the smallest such pair of languages? (How even to best define smallest?)                                     

Define: lexicographic order
   Sort by length, and within length alphabetically.

   E.g. {a,b}* = { 
ε
, 
a
, 
b
, 
aa
, 
ab
, 
ba
, 
bb
, 
aaa
, 
aab
, 
aba
, … }

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