1. Costas Busch - LSU1 Languages

2. Costas Busch - LSU2 Language: a set of strings String: a sequence of symbols from some alphabet Example: Strings: cat, dog, house Language: {cat, dog, house} Alphabet:

3. Costas Busch - LSU3 Languages are used to describe computation problems: Alphabet:

4. Costas Busch - LSU4 Computation is translated to set membership Is number prime? Example computation problem: Equivalent set membership problem:

5. Costas Busch - LSU5 Alphabets and Strings Example Strings Example Alphabet: An alphabet is a set of symbols A string is a sequence of symbols from the alphabet String variables

6. Costas Busch - LSU6 Decimal numbers alphabet Binary numbers alphabet

7. Costas Busch - LSU7 Unary numbers alphabet Unary number: Decimal number:

8. Costas Busch - LSU8 String Operations Concatenation

9. Costas Busch - LSU9 Reverse

10. Costas Busch - LSU10 String Length Length: Examples:

11. Costas Busch - LSU11 Length of Concatenation Example:

12. Costas Busch - LSU12 A string with no letters is denoted: Acts as a neutral element Observations: Empty String

13. Costas Busch - LSU13 Substring Substring of string: a subsequence of consecutive characters String Substring

14. Costas Busch - LSU14 Prefix and Suffix Prefixes Suffixes prefix suffix string

15. Costas Busch - LSU15 Exponent Operation Example: Definition:

16. Costas Busch - LSU16 The * Operation : the set of all possible strings from alphabet

17. Costas Busch - LSU17 The + Operation : the set of all possible strings from alphabet except

18. Costas Busch - LSU18 Languages A language over alphabet is any subset of Example: Language:

19. Costas Busch - LSU19 More Language Examples An infinite language Alphabet

20. Costas Busch - LSU20 Prime numbers Alphabet Language:

21. Costas Busch - LSU21 Even and odd numbers Alphabet Languages:

22. Costas Busch - LSU22 Addition (of unary numbers) Alphabet: Language:

23. Costas Busch - LSU23 Squares (of unary numbers) Alphabet: Language:

24. Costas Busch - LSU24 Two special languages Size of a language (number of elements): Empty language Language with empty string

25. Costas Busch - LSU25 Note that: Sets Set size String length

26. Costas Busch - LSU26 Operations on Languages The usual set operations Complement:

27. Costas Busch - LSU27 Reverse Definition: Examples:

28. Costas Busch - LSU28 Concatenation Definition: Example:

29. Costas Busch - LSU29 Another Operation Definition: Special case:

30. Costas Busch - LSU30 Example

31. Costas Busch - LSU31 Star-Closure (Kleene *) All strings that can be constructed from Definition: Example:

32. Costas Busch - LSU32 Positive Closure Definition: