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Languages

Languages. Languages. A language is a set of strings String: A sequence of letters Examples: “cat”, “dog”, “house”, … Defined over an alphabet:. Alphabets and Strings. We will use small alphabets: Strings. String Operations. Concatenation. Reverse. String Length.

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Languages

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  1. Languages

  2. Languages • A language is a set of strings • String:A sequence of letters • Examples: “cat”, “dog”, “house”,… • Defined over an alphabet:

  3. Alphabets and Strings • We will use small alphabets: • Strings

  4. String Operations Concatenation

  5. Reverse

  6. String Length • Length: • Examples:

  7. Recursive Definition of Length • For any letter: • For any string : • Example:

  8. Length of Concatenation • Example:

  9. Proof of Concatenation Length • Claim: • Proof: By induction on the length • Induction basis: • From definition of length:

  10. Inductive hypothesis: • for • Inductive step: we will prove • for

  11. Inductive Step • Write , where • From definition of length: • From inductive hypothesis: • Thus:

  12. Empty String • A string with no letters: • Observations:

  13. Substring • Substring of string: • a subsequence of consecutive characters • String Substring

  14. Prefix and Suffix • Prefixes Suffixes prefix suffix

  15. Another Operation • Example: • Definition:

  16. The * Operation • : the set of all possible strings from • alphabet

  17. The + Operation : the set of all possible strings from alphabet except

  18. Language • A language is any subset of • Example: • Languages:

  19. Another Example • An infinite language

  20. Operations on Languages • The usual set operations • Complement:

  21. Reverse • Definition: • Examples:

  22. Concatenation • Definition: • Example:

  23. Another Operation • Definition: • Special case:

  24. More Examples

  25. Star-Closure (Kleene *) • Definition: • Example:

  26. Positive Closure • Definition:

  27. Finite Automata

  28. Finite Automaton Input String Output Finite Automaton String

  29. Finite Accepter Input String Output “Accept” or “Reject” Finite Automaton

  30. Transition Graph Abba -Finite Accepter initial state final state “accept” transition state

  31. Initial Configuration Input String

  32. Reading the Input

  33. Input finished Output: “accept”

  34. Rejection

  35. Input finished Output: “reject”

  36. Another Example

  37. Input finished Output: “accept”

  38. Rejection

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