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 Length: Examples:

Recursive Definition of Length For any letter: For any string : Example:

Length of Concatenation Example:

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

Inductive hypothesis: for Inductive step: we will prove

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

Empty String A string with no letters: Observations:

Substring Substring of string: a subsequence of consecutive characters

Prefix and Suffix Prefixes Suffixes prefix suffix

Another Operation Example: Definition:

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

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

Language A language is any subset of Example: Languages:

Another Example An infinite language

Operations on Languages The usual set operations Complement:

Reverse Definition: Examples:

Concatenation Definition: Example:

Another Operation Definition: Special case:

More Examples

Star-Closure (Kleene *) Definition: Example:

Positive Closure Definition:

Finite Automata

Finite Automaton Input String Output Finite Automaton String

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

Transition Graph Abba -Finite Accepter initial state final state

Initial Configuration Input String

Reading the Input

Input finished Output: “accept”

Rejection

Input finished Output: “reject”

Another Example

Input finished Output: “accept”

Rejection

Input finished Output: “reject”

Formalities Deterministic Finite Accepter (DFA) : set of states : input alphabet : transition function : initial state : set of final states

Input Alphabet

Set of States

Initial State

Set of Final States

Transition Function

Transition Function

Extended Transition Function

Observation: There is a walk from to with label

Recursive Definition

Languages Accepted by DFAs Take DFA Definition: The language contains all input strings accepted by = { strings that drive to a final state}

Example accept

Another Example accept accept accept

Formally For a DFA Language accepted by : alphabet transition function initial state final states

Observation Language accepted by : Language rejected by :

More Examples trap state accept

= { all substrings with prefix } accept

= { all strings without substring }

Regular Languages A language is regular if there is a DFA such that All regular languages form a language family

Example The language is regular: