Formal Languages & Automata Theory Course

1. Course Overview • Why this course “formal languages and automata theory?” • What do computers really do? • What are the practical benefits/application of formal languages and automata theory? • Will you ever be using these facts / skill?

2. What do computers really do? • Take an input, process, and produce correct output. • What is input? In abstract way? • A finite string. On a finite alphabet – a set of characters. • Output? • “Input processed successfully,” Or “not”. • In other words: True / False. • Everything else in output is just the “detail”! • Don’t forget the “finite time” for processing! Definition of algorithm.

3. Practical aspects of Formal Lang. Auto. Theory? • Difference between a “programmer” and a “computer scientist” • Software system design. • Many hardware design. • Actual algorithms, e.g. string processing. • And of course, new language and compiler design. • Anything else?

4. Formal Language Finite set of alphabets Σ: e.g., {0, 1}, {a, b, c}, { ‘{‘, ‘}’ } Language L is a subset of strings on Σ, e.g., {00, 110, 01} a finite language, or, {strings starting with 0} an infinite language Σ* is a special language with all possible strings on Σ

5. Finite State Machine …….. • Tape, broken into cells • Tape head. • Finite control, i.e., a program, containing the position of the read head, current symbol being scanned, and the current “state”, etc. • A string is placed on the tape, read head is positioned at the left end, and the machine reads the string one symbol at a time until all symbols have been read, • and then either accepts or rejects the input (to be in the language or not) Finite Control

6. Non-Recursively Enumerable Languages Recursively Enumerable Languages Recursive Languages Context-Free Languages Regular Languages Hierarchy of languages

7. General Topics • Abstract machine models of computation (e.g., Deterministic Finite Automata, Non-deterministic FA, Turing Machine) • Classes of grammars (e.g., regular grammars, context-free grammars, context sensitive grammars, unrestricted grammars) • Classes of formal languages (e.g., regular, context-free, context-sensitive, recursive, recursively enumerable) • formal languages evolve to programming languages, and related to natural languages • Limitation of computational power of the abstract machine models • not about speed in this class, but about ultimate computability • Relationships between the languages, grammars and machines.

8. Language: Definitions • Symbol – An atomic unit, such as a digit, character, lower-case letter, etc. Sometimes a word. [Formal language does not deal with the “meaning” of the symbols.] • Alphabet – A finite set of symbols, usually denoted by Σ. Σ = {0, 1} Σ = {0, a, , 4} Σ = {a, b, c, d} • String – A finite length sequence of symbols, presumably from some alphabet. u=ε w = 0110 y = 0aa x = aabcaa z = 111 special string: ε (also denoted by λ) concatenation: wz = 0110111 length: |w| = 4 |x| = 6 but |u| = 0 reversal: yR = aa0

9. Some special sets of strings: Σ* All strings of symbols from Σ Σ+ Σ* - {ε} • Example: Σ = {0, 1} Σ* = {ε, 0, 1, 00, 01, 10, 11, 000, 001,…} Σ+ = {0, 1, 00, 01, 10, 11, 000, 001,…} • A (formal) language is: 1) A set of strings from some alphabet (finite or infinite), in other words… 2) any subset L of Σ* • Some special languages: {} The empty set/language, containing no strings {ε} A language containing one string, the empty string.

10. Examples: Σ = {0, 1} L = {x | x is in Σ* and x contains an even number of 0’s} Σ = {0, 1, 2,…, 9, .} L = {x | x is in Σ* and x forms a finite length real number} = {0, 1.5, 9.326,…} Σ = {a, b, c,…, z, A, B,…, Z} L = {x | x is in Σ* and x is a Pascal reserved word} = {BEGIN, END, IF,…} Σ = {Pascal reserved words} U { (, ), ., :, ;,…} U {Legal Pascal identifiers} L = {x | x is in Σ* and x is a syntactically correct Pascal program} Σ = {English words} L = {x | x Σ* and x is a syntactically correct English sentence}