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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

(possibly infinite)

set

(finite)

strings

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.

In Java:

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
    }

The math people sure have more succinct notation, to convey the same info!

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 }

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

function-space: A → B means:
the set of all possible functions with domain A, and codomain B.

For example, Σ^* → ℕ is the set of all functions taking a string and returning a number; string-length is one member of this set.
We say that Σ^* → ℕ is string-length's signature. (When you write a function in Java, you must give its signature — albeit in Java, not math.)

What about the signature of substring?

Sigma* x ℕ x ℕ → Σ^*

(hint: we can say that its input-type is one thing: a tuple of size three.)

concatenation (for sets-of-strings): AB w∈AB means w=ab where a∈A and b∈B
examples: L₄L2 = { vavw, vasaab, utbmw, … }; there are 50*3 elements.
L₆L₇ = { a, aa, ba, ab, aaa, … }

Kleene Star, A*

Questions, for various sets created from the ones listed above: no. of elements? contains ""?

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
, … }

This page licensed CC-BY 4.0 Ian Barland
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(incl. typos, broken links)
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Definitions sets; FSMs

Previously, on itec420…

Defining Finite State Machines

Sets

Definitions
sets; FSMs