1. Theory of Computation
Theory of computation is mainly concerned with the study of how problems can be solved using algorithms.  Therefore, we can infer that it is very relevant to the study of logic and mainly logic within mathematics. These studies are used to understand the way an algorithm is meant to work, and to actually prove it works through analyzing problems that may arise with the technique used and finding solutions to these problems. It is quiet huge field of study, but it is split up into three different parts: automata theory, computability theory, and computational complexity theory. 

2. Theory of Computation: areas
CSC 573 Theory of Computation
7/28/2018
Theory of Computation: areas
Formal Language Theory
language = set of sentences (strings)
grammar = rules for generating strings
production/deduction systems
capabilities & limitations
application: programming language specification
Automata Theory (Abstract Machines)
models for the computing process with various resources
characterize ``computable”
application: parsing algorithms
Complexity Theory
time/space resources
application: algorithm design

3. What is Automata Theory?
Cpt S 317: Spring 2009
What is Automata Theory?
Study of abstract computing devices, or “machines”
Automaton = an abstract computing device
Note: A “device” need not even be a physical hardware!
Computability vs. Complexity
School of EECS, WSU

4. Alan Turing (1912-1954) Father of Modern Computer Science
Cpt S 317: Spring 2009
Alan Turing ( )
(A pioneer of automata theory)
Father of Modern Computer Science
English mathematician
Studied abstract machines called Turing machines even before computers existed
Heard of the Turing test?
School of EECS, WSU

5. Theory of Computation: A Historical Perspective
Cpt S 317: Spring 2009
Theory of Computation: A Historical Perspective
1930s
Alan Turing studies Turing machines
Decidability
Halting problem s
“Finite automata” machines studied
Noam Chomsky proposes the “Chomsky Hierarchy” for formal languages
1969
Cook introduces “intractable” problems or “NP-Hard” problems
1970-
Modern computer science: compilers, computational & complexity theory evolve
School of EECS, WSU

6. Cpt S 317: Spring 2009
Languages & Grammars
Languages: “A language is a collection of sentences of finite length all constructed from a finite alphabet of symbols”
Grammars: “A grammar can be regarded as a device that enumerates the sentences of a language” - nothing more, nothing less
N. Chomsky, Information and Control, Vol 2, 1959
Or “words”
Image source: Nowak et al. Nature, vol 417, 2002
School of EECS, WSU

7. Cpt S 317: Spring 2009
The Chomsky Hierachy
A containment hierarchy of classes of formal languages
Recursively- enumerable (TM)
Context- sensitive (LBA)
Context- free
(PDA)
Regular
(DFA)
School of EECS, WSU

8. Language
In English, there are at least three different types of entities: letters, words, sentences.
Letters are from a finite alphabet { a, b, c, , z }
Words are made up of certain combinations of letters from the alphabet.
Not all combinations of letters lead to a valid English word.

9. Sentences are made up of certain combinations of words.
Not all combinations of words lead to a valid English sentence.
So we see that some basic units are combined to make bigger units.

10. Languages
How can you tell whether a given sentence belongs to a particular languages
Black is cat the
The tea is hot
I like chocolates two much
Rules give a clue to forming as well as validating sentences.

11. How to Communicate with machines ?
Need a language: what sort
Machines don’t have human mind though may have its partial imitation
Would fail on incorrect or ambiguous input
Some recovery or input corrections may be proposed but again very limited.
Thus need a precise, explicit and universal definition of communication language

12. Summary of Languages Three aspects/specifications Lexical Syntactic
Defines valid words/units of a language
Syntactic
Defines rules for combining the units to form valid sentences (computer programs in context of machines)
Semantic
Concerned with the interpretation or meaning of a sentence (what output to produce in context of machines)
Affected by ambiguity the most.

13. Formal languages
A language designed for use in situations in which natural language is unsuitable, as for example in mathematics, logic, or computer programming. The symbols and formulas of such languages stand in precisely specified syntactic and semantic relations to one another

14. Formal languages Rules defined explicitly and clearly No ambiguities
Universally uniform understanding
Lets the machine
Interpret an input uniformly every time. i.e. always produces same output for a particular input
Avoid crashes because of ambiguity.
Explicitly and categorically reject invalid input

15. Formal Languages Need uniformly understandable notation
Representations
Alphabet
Represents a finite set of fundamental units of lanauges, e.g. for English ={a,b,….z.A,…Z,}
∑ = {0,1}
∑ = {0,1,2,3,4,5,6,7,8,9}

16. Formal Languages
List of words
Set of all valid words of a given language, e.g., a language English_Words that contains all valid words of English would have a  = {all entries of the dictionary + punctuation marks and blank space}
Denoted by 
Is  Finite or Infinite set.
Strings: A string a finite sequence of symbols chosen from alphabet. For example
, , abbbcdeg etc.

17. String Variable: A letter used for denoting a string
String Variable: A letter used for denoting a string. The author uses w, x, y and z as string variable. For example
w = , x = , z = abbbcdeg
Length of String: The number of positions for symbols in the string. For simplicity we can say that it is the number of symbols in the string. For example
|w| = 7 , |x| = ? , |z| = ?

18. Finite vs. Infinite Languages
Countable set of words
Can be defined by rigorously listing the words in 
E.g. English-Words
Infinite Languages
Infinite set of valid words
Cant be listed completely
E.g. English_Sentences

19. Infinite Languages Most of the languages are infinite
How can u check whether a word belongs to a language if it is
Finite
Checking its entry in 
Infinite
Validating against rules

20. Defining Languages Define alphabet set
Define rules for forming valid words and sequences of words from 
Called grammar
Can be descriptive
Limitations of informalism
Can be mathematical
Can also define supporting functions e.g., length(X), reverse(x)

21. Defining languages Example ={a,b,…z}
L = {all words formed only of odd number of xs}
L = {xn | n is odd}
L = {all words of length less than or equal to 4}
PALINDROME ={Λ, all strings x such that reverse (x) = x}

22. Kleene Closure Set closure Kleene Closure (applied to )
A set of all the strings (finite) that can be formed by the elements of  where the elements may be repeated any number of times.
Denoted by *
Also called Kleene star.

23. ∑* : The set of all strings over an alphabet ∑ and called Kleene Star Closure of alphabet. So we have
∑* = ∑0 U ∑1 U ∑2 U ∑3 U……………
∑+ : The set of all strings over an alphabet ∑ excluding empty string, ε, and called plus operation. So we have
∑+ = ∑1 U ∑2 U ∑3 U……………

24. Some observations
Λ represents an empty string (not alphabet thus not a part of )
ε also represents the same
ε is not equivalent to 
If  =  then
* = {Λ}
Is S* == (S*)* and so on

25. Length:
Examples:
String Length

26. Length of Concatenation
Example:

27. Empty String
A string with no letters:
Observations:

28. Substring Substring of string: a subsequence of consecutive characters

29. Prefix and Suffix
Prefixes Suffixes
prefix
suffix

30. Another Operation
Example:
Definition:

31. The * Operation
: the set of all possible strings from
alphabet

32. The + Operation
: the set of all possible strings from
alphabet except

33. Languages
A language is any subset of
Example:
Languages:

34. Note that:
Sets
Set size
Set size
String length

35. Another Example
An infinite language

36. Operations on Languages
The usual set operations
Complement:

37. Reverse
Definition:
Examples:

38. Concatenation
Definition:
Example:

39. Another Operation
Definition:
Special case:

40. More Examples

41. Star-Closure (Kleene *)
Definition:
Example:

42. Positive Closure
Definition: