Automata

1
Automata

• Chapter 1

2
Automaton

• A machine, robot, or formal system designed
  to follow a precise sequence of instructions.
• Automata theory, the invention and study of
  automata, includes the study of the capabilities
  and limitations of computing processes, the
  manner in which systems receive input, process
  it, and produce output, and the relationships
  between behavioral theories and the operation and
  use of automated devices.
• Automata is the plural of automaton

3
computation

• Computation is the movement, and perhaps the
  alteration during transit, of data.
• Examples
• Vending machine
• ATM machine

4
Math Preliminaries

• N natural numbers 0, 1, 2, 3,
• Can be defined using the concept of a collection.
• collection of nothing, can represent 0
• collect the 0, represents 1
• , collect the 0 and 1 together to get
  2
• This is an inductive definition
• Base object specified (in this case the empty
  object)
• And an operation to produce new objects from
  previously defined objects (in this case,
  collection)

5
Sets

• Can form sets from the Natural numbers
• Example
• Prime numbers lt10 are 2, 3, 5, 7
• ? Set membership
• ? set exclusion
• ? subset
• 2,3,5,7 ? 1,2,3,,100

6
Sets continued

• For a finite set of numbers the max element of
  the set is denoted by max
• Infinite sets, like E (set of Evens) can be
  represented as
• E 0,2,4, or
• Closed form E 2n n ? N

7
Set Difference

• A common way to form a new set from another is to
  remove some elements
• Example
• N-0 is the set of all positive natural numbers
• N-E is the set of all odd numbers

8
Union and Intersection

• Denoted by ? and n
• Example
• A 1,2,3 B4,5
• A ? B 1,2,3,4,5
• C 1,3,5 D1,2,3
• C n D 1,3

9
Cartesian Product

• Can form new sets from a pair of sets by taking
  all combinations of elements from both
• Given 2 sets, R and S, the Cartesian product of R
  and S, denoted R x S, is the set of pairs that
  includes all combinations of something from R and
  something from S.

10
Cartesian Product

• R 0,1 S0,1
• R x S (0,0),(0,1),(1,0),(1,1)

11
Strings

• Numbers stored in binary as sequence of 0s and
  1s.
• Alphabetic symbols fixed length sequence of 0s
  and 1s
• Words are sequences of symbols representing
  letters of alphabet
• Sentences are sequences of symbols that alternate
  between the symbols for a word and a blank
  symbol, or some punctuation symbol

12
Strings

• Strings are finite sequences of symbols where
  each symbol is chose from a fixed alphabet.
• Strings are sometimes referred to as words.
• An alphabet is a finite set of symbols
• The empty word, unique string with no symbols is
  denoted by e.

13
Language

• The length of word w is w
• e 0
• If w is a word, then wR is the reverse of w
• A word w such that w wR is a palindrome
• The inputs and outputs of an automaton are
  strings.
• A set of words is called a language.

14
Pigeonhole Principle

• If try to place n items into m slots and ngtm,
  then at least one slot will have more than one
  item in it.

15
Relations

• The lt relation can be defined formally as
• R (x,y) x ? N, y ? N and xlty
• (2,3) ? R, but (3,2) ? R
• Common Properties of Relations
• Reflexive
• If R is a relation on S x S, then R is reflexive
  if (a, a) ? R for each a ? S
• Symmetric
• R is symmetric if (a, b) ? R whenever (b, a) ? R
• Transitive
• If (a, b) ? R and (b, c) ? R then (a, c) ?R

16
Relation Examples

• S a, b, c
• Any subset of S x S is a relation
• R1 (a, a), (a, b), (b, b)
• R1 is not reflexive but R1 ? (c,c) is
  reflexive
• R1 is not symmetric, but R1 ? (b, a) is
• R1 is transitive, but R1 ? (b, c) is not

17
Equivalence Relation

• An equivalence relation is a relation which is
• Reflexive
• Symmetric
• transitive
• Give rise to Equivalence classes
• Example 40 watt bulbs

18
Function

• Suppose R is a relation over a pair of sets S and
  T such that for each a ? S there is at most one b
  ? T such that (a, b) ? R.
• We say that R is a function.
• More familiar notation
• f S ? T to indicate a function that take inputs
  from set S and returns elements from set T.
• More formal Definition
• f ? S x T and for each x ? S there is at most one
  y ? T such that f(x) y, that is, (x,y) is in
  the relation.
• S is called the Domain and can be denoted as
  Dom(f)

19
Bijection

• A function f is 1-to-1 and onto, also called a
  bijection, if for any x ? Dom(f) and y ? Dom(f)
  if x ? y then f(x) ? f(y) and for each y in the
  range of f, there is an x ? Dom(f) with f(x) y.