![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
ÎŁ.
ÎŁ = {0, 1} ÎŁ = {0, a, 4} ÎŁ = {a, b, c, d}
• String – A finite length sequence of symbols,](https://image.slidesharecdn.com/01-240115041441-e60c61c9/85/01-ppt-3-320.jpg)
01.ppt
- 1. Automata and complexity Theory
- 3. 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,
- 4. Alphabets and strings • A common way to talk about words, number, pairs of words, etc. is by representing them as strings • To define strings, we start with an alphabet • Examples An alphabet is a finite set of symbols. S1 = {a, b, c, d, …, z}: the set of letters in English S2 = {0, 1, …, 9}: the set of (base 10) digits S3 = {a, b, …, z, #}: the set of letters plus the special symbol # S4 = {(, )}: the set of open and closed brackets
- 5. Strings • The empty string will be denoted by e (epsilon) • Examples A string over alphabet S is a finite sequence of symbols in S. abfbz is a string over S1 = {a, b, c, d, …, z} 9021 is a string over S2 = {0, 1, …, 9} ab#bc is a string over S3 = {a, b, …, z, #} ))()(() is a string over S4 = {(, )}
- 6. Let: u=ε w = 0110 y = 0aa x = aabcaa z = 111 concatenation: wz = 0110111 length: |w| = 4 |x| = 6 but |u| = 0 reversal: yR = aa0 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,…} Strings
- 7. What is Formal Language ? i.e. any subset L of Σ* • Classes of formal languages (e.g., regular, context-free, context-sensitive, recursive, recursively enumerable). • Formal languages are related to programming languages and natural languages A (formal) language is a set of strings over an alphabet.
- 8. Formal Language 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 CPP reserved word}
- 9. Formal Language Examples: Σ = {CPP reserved words} U { (, ), ., :, ;,…} U {Legal CPP identifiers} L = {x | x is in Σ* and x is a syntactically correct CPP program} Σ = {English words} L = {x | x Σ* and x is a syntactically correct English sentence}
- 10. What is automata theory ? • Automata theory is the study of abstract computational devices • Abstract devices are (simplified) models of real computations • Computations happen everywhere: On your laptop, on your cell phone, in nature, … • Why do we need abstract models?
- 11. A simple “computer” BATTERY input: switch output: light bulb actions: flip switch states: on, off
- 12. A simple “computer” BATTERY off on start f f input: switch output: light bulb actions: f for “flip switch” states: on, off bulb is on if and only if there was an odd number of flips
- 13. Another “computer” BATTERY off off start 1 inputs: switches 1 and 2 actions: 1 for “flip switch 1” actions: 2 for “flip switch 2” states: on, off bulb is on if and only if both switches were flipped an odd number of times 1 2 1 off on 1 1 2 2 2 2
- 14. Another Example • A state diagram for a simple Game Enemy model Wander - player is with in sight - Attack Attack - player is out of sight - Wander Attack - player is attacking - Evade Evade - player is idle - Attack
- 15. Different kinds of automata • This was only some examples of a computational device, and there are others • We will look at different devices, and look at the following questions: – What can a given type of device compute, and what are its limitations? – Is one type of device more powerful than another?
- 16. Some devices we will see finite automata Devices with a finite amount of memory. Used to model “small” computers. push-down automata Devices with infinite memory that can be accessed in a restricted way. Used to model parsers, etc. Turing Machines Devices with infinite memory. Used to model any computer.
- 17. Preliminaries of automata theory • How do we formalize the question • First, we need a formal way of describing the problems that we are interested in solving Can device A solve problem B?
- 18. Problems • Examples of problems we will consider – Given a word s, does it contain the subword “fool”? – Given a number n, is it divisible by 7? – Given a pair of words s and t, are they the same? – Given an expression with brackets, e.g. (()()), does every left bracket match with a subsequent right bracket? • All of these have “yes/no” answers. • There are other types of problems, that ask “Find this” or “How many of that” but we won’t look at those.
- 19. Reading Assignment • Mathematical Preliminaries – sets, – relations, – functions, – sequences, – graphs, – trees.