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?

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 4

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 01100

6 Hierarchy of languages Regular Languages Context-Free Languages Recursive Languages Recursively Enumerable Languages Non-Recursively Enumerable 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 = 0aax = aabcaaz = 111 special string:ε (also denoted by λ) concatenation:wz = length:|w| = 4 |x| = 6 but |u| = 0 reversal:y R = 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}