1. Deterministic FA/ PDA Sequential Machine Theory Prof. K. J. Hintz
Department of Electrical and Computer Engineering
Lecture 4
Updated by Marek Perkowski

2. Moret, Theory of Computation
Numerical Acceptor
A Non-intuitive Concept of a Language Is One Which Can Accept Binary Arithmetic Values, e.g. 1 5 3 2 4
Moret, Theory of Computation
Check all strings, from shortest that are accepted. What is their language?

3. Languages Accepted by FA
The Class of Languages Accepted by a Deterministic or Nondeterministic FA Is Closed Under
Union
Concatenation
Kleene Star
Complementation
Intersection

4. Union of Languages M1 M2 e
>

5. Concatenation of LanguagesM1 M2 e
> F2 S1 F1 S1

6. Kleene Star of LanguagesM1
> F1 q0

7. Difference of Languages

8. Intersection of Languages

9. Give examples of languages that are not regular
Languages and FA
Kleene’s Theorem
A Language Is Regular iff It Is Accepted by a Deterministic or Nondeterministic Finite Automata
A Regular Language Is One Which Can Be Defined by a Regular Expression
Not all languages are regular
Give examples of languages that are not regular

10. Context Free Languages
Up Until Now we learned the following:
But This Is a Limited Class of Languages
There Are Other Context-Free Languages Which Cannot Be Recognized by a DFA
There Are Other Types of Machines Which Can Accept More Context-Free Languages

11. A Non-Regular Language
Since a Language, L, Is a Subset of the Set of All Strings, I*, Are There Some Strings Which Cannot Be Produced by a Regular Expression or Recognized by a DFA? Yes, e.g.,

12. 1 Lewis & Papadimitriou, Elements of the Theory of Computation
an bn Regular?
Theorem1: Let L be an infinite regular language. Then there are strings x, y, and z such that y  e and x yn z  L for each n  0
Proof based on pigeonhole principle
If there are n + 1 pigeons and n pigeonholes, then at least two pigeons must be in the same hole
If a string has more characters than there are states in the language recognizer, then some state must be entered more than once
1 Lewis & Papadimitriou, Elements of the Theory of Computation

13. an bn Regular? Language Is Infinite, Therefore Theorem Applies
Is This Language Regular? If So, the Theorem Must Apply
Three Cases to Study see next slides

14. an bn Regular?
Case 1: y consists entirely of as

15. an bn Regular? Case 2: y consists entirely of bs
Similar argument to Case 1

16. an bn Regular? Case 3: y contains both as and bs
For n > 1, x yn z has an occurrence of b preceding an occurrence of a and therefore cannot be in L

17. Context Free Grammars: idea
A More General Way to Describe and/or Generate Languages
Context-Free
Replacement Rules Can Be Applied Independently of the Preceding or Following Elements.

18. Context Free Grammar: definition
A Quadruple, ( I, , R, s) where
I An alphabet
 A set of terminals,
R A finite set of rules, a subset of the crossproduct of the set of non-terminals and all strings,

19. Context Free Grammar: definition cont
s The start symbol, a non-terminal
The set of non-terminals
AND ...

20. Context Free Grammar: definition cont
For All Non-terminals,
And Strings
The Grammar, G, Maps the Non-Terminals to Strings
for strings u, v, x, y, v’  I*
and non-terminals

21. Context Free Grammar: definition of G-related
u is G-related to v
iff
u = x A y
v = x v’ y
and

22. Context Free Grammar: the set of all possible relations
The relation, , the * meaning the set of
all possible relations, is
reflexive
transitive
has closure I* G

23. Context Free Language
The Language Generated by the Context Free Grammar, G, Is the Set of Strings Which Map From the Start Symbol, s, Under the Reflexive, Transitive, and Closed Relation,

24. Context Free Language
A Context Free Language Is One Which Can Be Generated by a Context Free Grammar, and,
The Context Free Grammar (CFG) Can Be Used to Generate Strings of the Language

25. A Derivation Example
The Steps in Applying the Rules From the Start Symbol to Any String Is Called a Derivation in G, e.g.,

26. Pushdown Automata
All Context-Free Languages can Be Recognized by a PDA
{ an bn } Is Context-Free, but Not Regular
Problem is same number of as and bs
PDA Assumes
An unlimited memory in the form of a stack, LIFO
The machine only has access to the top of the stack.

27. Pushdown Automata: formal definition
A PDA is a Sextuple,

28. Pushdown Relation where “top of stack” is replaced by
“new top of stack.”

29. Pushdown Relation Example
s=p
s=q x x
s=q
s=p

30. PDA Properties
One Can Have the Same State Before and After a Transition, but Have Different Stack Contents Which Makes It Non-Deterministic
State Changes May Occur in the Absence of an Input, but Only If the Stack Is Not Empty. i.e., non-causal behavior.

31. PDA Properties: what is the state of PDA?
The “State” of the PDA Is Determined Not Only by S, and Not Only by the Top-of-Stack, but Rather by Both the Current State and the Complete Contents of the Stack, x
This “State of the PDA” Is Called the “Configuration of the PDA”

32. PDA Notation
The Input May Cause the PDA to Change From Configuration 1 to Configuration 2

33. PDA Recognizer
A string w  I* is accepted by a PDA

34. PDA Example* A Machine to Recognize Strings of the Form wcwR
* Lewis & Papadimitriou, pg. 114

35. PDA Example
wcwR Can Be Represented by a Context Free Grammar,

36. Equivalent NPDA
wcwR can be represented by a NPDA where

37. Is abccba Accepted?

38. Is abccba Accepted?

39. Is abccba Accepted?
The machine halts with unread inputs since there is no
to be executed.

40. Is bacab Accepted?

41. Is bacab Accepted?

42. Is bacab Accepted?
The machine halts with no unread input and nothing on the stack, therefore,
bacab  language wcwR.