1. Automata, Formal Grammars, and Consequences
A Bit of Theory
Automata, Formal Grammars, and Consequences
John M. Hunt

2. What is Computer Science?
"Computer science is no more about computers than astronomy is about telescopes." Edsger Dijkstra

3. Hardware Independent
Works at the level of logic, physical implementation is not important
We can trade hardware for software
Virtual machines
Can be implemented
Electronics
Mechanically
Biologically – DNA and bacteria

4. Algorithm Like a Recipe, but . . . Strictly defines the order of steps
“In mathematics and computing, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state.” - Wikipedia
Like a Recipe, but . . .
Strictly defines the order of steps

5. Formal Language the way we describe problems
A set of finite length strings drawn from a finite set of symbols
We do not consider the meaning of the symbols
Grammar – rules on how the symbols may be fitted together
By using recursive rules infinitely large languages can be generated from a finite number of symbols
Key question is how to determine if a particular string belongs to a particular language

6. Two Approaches
Automata
Formal Grammars

7. Automata
Imaginary machines
Thought experiments
Models of computation

8. Formal Grammar Variables – capital letters
Terminals – small letters / numbers
Replacement rules
S -> aS |

9. Formal Grammar
One way to specify a language is to provide grammar rules, called productions
Example with letters a, b, c:
S -> aS
S -> bA
A ->
A -> cA
Which belong:
cab
aaaccc
aabc
abbcc

10. Machine and Grammar Two sides of a coin
Given a description of a machine we can predict what kind of language it can process and what grammar describes it
Given a language we can describe what sort of machine is required to process it.
Given an unknown machine we can try out different languages on it to asses its power.

11. Java is a formal language
Like most programming languages it has a
formal grammar specified using BNF notation
Which strings belong to the Java language?:
a = b + c;
b + c = a:
a = b # c;

12. Chomsky Hierarchy Increasingly Powerful Automata Grammar
Production Rule
Finite State
Regular
A->a, A->aB
Push Down
Context Free
A->γ
Turing Machine
Recursively Enumerable
Ω->β
Increasingly
Powerful

13. Finite State Automata
Finite number of states, transition, and conditions
No memory, no record of how we got to the current state
Computing today

14. FSA Example a Door Circles are states Lines are transitions Open
Closed

15. Example c b c S b a a q1 q2 exit q0 Example with letters a, b, c:
S -> aS
S -> bA
A ->
A -> cA c q1 b c q2 S b
exit a q0 a

16. . . . Push Down Automata FSA + Stack Stack provides memory
Stack accessed only at top
Stack is infinite
. . .

17. Turing Machines FSA + Memory Can move around in memory
Memory is infinite
Assume operations are infinitely fast
. . .
. . .

18. Turing Operations Move Left Move Right Set to 0 Set to 1 Compare Reset
Note: any set of discrete symbols can be represented by multiple 1’ & 0’s

19. Turing Machines (cont)
All problems that can be solved using an algorithm can be solved using a Turing machine
There are a number of other notations for expressing algorithms – Lambda calculus, cellular automata, Wang models, Markov algorithm – that are equivalent to TM, none are more powerful, although they maybe more covenant

20. Thoughts
Odd that such simple operations are capable of doing everything possible.
Odd that there are many Turing equivalent approaches but nothing better.
It would seem that Turing machines should be discouraging to AI proponents, but the have not been.

21. So What? Computing in Nature Genetics Ecological cycles Flock behavior
Linguistics
Syntax of human languages

22. Chomsky
Institute Professor and Professor (Emeritus) in the Dpt. of Linguistics & Philosophy at MIT
Father of modern linguistics
considered the "most cited living author“
eighth most cited source of all time
Author of over 100 books

23. Chomsky
Responsible for much of the development of formal languages and related machine models
His intention was to use language and more specifically grammar as a window into the human mind
His work was not only important to computer science, but also to cognitive psychology and forms the basis of modern linguistics

24. Chomsky’s Insights Grammar is creative – sentences are original
All human languages share an underlying meta-grammar
Children are able to learn languages, including grammar just by being exposed to them
Human grammar requires a Turing machine to process

25. Consequences
Basic AI tasks such as understanding a conversation or translating a document require a Turing machine to perform correctly
It has become common to refer to people as “wetware” or to suggest that people are a type of computer. Their ability to perform TM tasks suggests other wise

26. Consequences
The ability of children to learn grammar at an early age, without training, shows that it is an innate ability of the human mind not a result of the child’s environment
This refutes the idea of the mind as a blank slate promoted by such people as BF Skinner

27. Consequences
Complex Grammar is universal in human languages, even “primitive” ones, and innate to children
Animal “languages” (bird songs, etc.) show no evidence of grammar
Where then did human grammar come from?

28. Applications

29. Broad Application #1
Natural Man rejects that man is made in the image of God. To avoid this the argument is made that man is just a machine
However, Automata Theory, and the related linguistic and cognitive understandings, show man must be more than a machine as we can conceive it.

30. Broad Application #2
We are often tempted to have “rules to live by” that go beyond what Scripture gives us. Which becomes “legalism”.
Automata theory shows the limits of rules to solve problems.
Many places in Scripture refuse to provide us rules. Example: Luke 10:25-37

31. Conclusion
We started with two very abstract topics – formal grammar and automata
We have show they give us insight on:
Foundations of computer science
Nature of man
Understanding of our Faith