Fall 2006Costas Busch - RPI1 Languages

Fall 2006Costas Busch - RPI2 Language: a set of strings String: a sequence of symbols from some alphabet Example: Strings: cat, dog, house Language: {cat, dog, house} Alphabet:

Fall 2006Costas Busch - RPI3 Languages are used to describe computation problems: Alphabet:

Fall 2006Costas Busch - RPI4 Alphabets and Strings Example Strings Example Alphabet: An alphabet is a set of symbols A string is a sequence of symbols from the alphabet

Fall 2006Costas Busch - RPI5 Decimal numbers alphabet Binary numbers alphabet

Fall 2006Costas Busch - RPI6 Unary numbers alphabet Unary number: Decimal number:

Fall 2006Costas Busch - RPI7 String Operations Concatenation

Fall 2006Costas Busch - RPI8 Reverse

Fall 2006Costas Busch - RPI9 String Length Length: Examples:

Fall 2006Costas Busch - RPI10 Length of Concatenation Example:

Fall 2006Costas Busch - RPI11 Empty String A string with no letters is denoted: Observations:

Fall 2006Costas Busch - RPI12 Substring Substring of string: a subsequence of consecutive characters String Substring

Fall 2006Costas Busch - RPI13 Prefix and Suffix Prefixes Suffixes prefix suffix

Fall 2006Costas Busch - RPI14 Another Operation Example: Definition:

Fall 2006Costas Busch - RPI15 The * Operation : the set of all possible strings from alphabet

Fall 2006Costas Busch - RPI16 The + Operation : the set of all possible strings from alphabet except

Fall 2006Costas Busch - RPI17 Languages A language over alphabet is any subset of Examples: Language:

Fall 2006Costas Busch - RPI18 More Language Examples An infinite language Alphabet

Fall 2006Costas Busch - RPI19 Prime numbers Alphabet Language:

Fall 2006Costas Busch - RPI20 Even and odd numbers Alphabet

Fall 2006Costas Busch - RPI21 Unary Addition Alphabet: Language:

Fall 2006Costas Busch - RPI22 Squares Alphabet: Language:

Fall 2006Costas Busch - RPI23 Note that: Sets Set size String length

Fall 2006Costas Busch - RPI24 Operations on Languages The usual set operations Complement:

Fall 2006Costas Busch - RPI25 Reverse Definition: Examples:

Fall 2006Costas Busch - RPI26 Concatenation Definition: Example:

Fall 2006Costas Busch - RPI27 Another Operation Definition: Special case:

Fall 2006Costas Busch - RPI28

Fall 2006Costas Busch - RPI29 Star-Closure (Kleene *) All strings that can be constructed from Definition: Example:

Fall 2006Costas Busch - RPI30 Positive Closure Definition: Same with but without the