Automata and Formal Languages (CS 350/550) Thomas Gilray TA: Jiaqing “Jordan” Yuan gilray.org/classes/fall2018/cs350/

What are Automata?

What are Automata? Formal systems---abstract machines---that model (usually restricted) notions of computation. HMU says “Automata theory is the study of abstract computing devices...before there were computers...” + Finite automata originally proposed to model brain function. + Turing machines preceded, and lead to, the development of modern computers.

David Gelertner Elegance = power + simplicity

David Gelertner Elegance = power + simplicity Expressivity

Class of Automata Class of Formal Languages Finite Automata (Deterministic, Nondeterministic) Pushdown Automata (Automata with an unbounded stack.) Turing Machines (Automata with an unbounded memory.) Regular Languages (Specified using regular expressions) Context-Free Languages (Specified using context-free grammars) Recursively Enumerable Languages (Specified using computable functions; e.g., Python,C)

Class of Automata Class of Formal Languages Regular Languages (Specified using regular expressions) (ab)*c

Class of Automata Class of Formal Languages Regular Languages (Specified using regular expressions) (ab)*c Requires interpretation as a formal language.

Class of Automata Class of Formal Languages Regular Languages (Specified using regular expressions) (ab)*c ...or modeling by a finite automata! Requires interpretation as a formal language.

Syntax vs. Semantics Syntax: Rules governing the order of letters/words/terms in a sentence, independent its meaning. Semantics: The meaning (operation, denotation) of the sentence. For example: 1+2/(1-1) is a syntactically valid arithmetic expression, but is semantically meaningless.

What are Formal Languages?

What are Formal Languages? Languages, defined formally, as (finite or infinite) sets of of strings (over a finite alphabet). E.g., [1-9][0-9]* is a regular expression we interpret as the formal language {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21, ...}

What are formal systems? Roughly speaking: A system that precisely specifies its forms and rules that manipulate them, without recourse to intuition or interpretation into another, otherwise understood, system.

What are formal systems? Roughly speaking: A system that precisely specifies its forms and rules that manipulate/relate them, without recourse to intuition or interpretation into another, otherwise understood, system.

Bertrand Russell “Pure mathematics consists entirely of asseverations to the extent that, if such and such a proposition is true of anything, then such and such another proposition is true of anything. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true ... If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

Conway’s game of life:

Isomorphism?

{ Isomorphism A structure-preserving bijection. Consider: f(x) = Is f a bijection for the domain ℤ and codomain ℕ? { -2x-1 0 2x x < 0 x = 0 x > 0

Isomorphism 1+1=0 (mod 2) 0+1=1 (mod 2) 1+0=1 (mod 2) 0+0=0 (mod 2) T xor T = F F xor T = T T xor F = T F xor F = F

Is meaning anything other than isomorphism…?

Our textbook: Hopcroft, Motwani, Ullman

Other suggestions...

Syllabus discussion, Pre-test