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Languages
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Languages A language is a set of strings String: A sequence of letters
Examples: “cat”, “dog”, “house”, … Defined over an alphabet:
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Alphabets and Strings We will use small alphabets: Strings
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String Operations Concatenation
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Reverse
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String Length Length: Examples:
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Recursive Definition of Length
For any letter: For any string : Example:
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Length of Concatenation
Example:
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Proof of Concatenation Length
Claim: Proof: By induction on the length Induction basis: From definition of length:
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Inductive hypothesis:
for Inductive step: we will prove
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Inductive Step Write , where From definition of length:
From inductive hypothesis: Thus:
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Empty String A string with no letters: Observations:
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Substring Substring of string: a subsequence of consecutive characters
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Prefix and Suffix Prefixes Suffixes prefix suffix
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Another Operation Example: Definition:
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The * Operation : the set of all possible strings from alphabet
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The + Operation : the set of all possible strings from alphabet except
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Language A language is any subset of Example: Languages:
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Another Example An infinite language
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Operations on Languages
The usual set operations Complement:
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Reverse Definition: Examples:
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Concatenation Definition: Example:
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Another Operation Definition: Special case:
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More Examples
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Star-Closure (Kleene *)
Definition: Example:
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Positive Closure Definition:
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Finite Automata
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Finite Automaton Input String Output Finite Automaton String
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Finite Accepter Input String Output “Accept” or Finite “Reject”
Automaton
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Transition Graph Abba -Finite Accepter initial state final state
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Initial Configuration
Input String
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Reading the Input
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Input finished Output: “accept”
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Rejection
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Input finished Output: “reject”
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Another Example
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Input finished Output: “accept”
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Rejection
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Input finished Output: “reject”
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Formalities Deterministic Finite Accepter (DFA) : set of states
: input alphabet : transition function : initial state : set of final states
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Input Alphabet
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Set of States
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Initial State
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Set of Final States
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Transition Function
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Transition Function
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Extended Transition Function
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Observation: There is a walk from to
with label
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Recursive Definition
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Languages Accepted by DFAs
Take DFA Definition: The language contains all input strings accepted by = { strings that drive to a final state}
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Example accept
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Another Example accept accept accept
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Formally For a DFA Language accepted by : alphabet transition function
initial state final states
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Observation Language accepted by : Language rejected by :
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More Examples trap state accept
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= { all substrings with prefix }
accept
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= { all strings without substring }
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Regular Languages A language is regular if there is a DFA such that
All regular languages form a language family
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Example The language is regular:
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