Languages 1 Languages A language is a set
- Slides: 78
Languages 1
Languages A language is a set of strings String: A sequence of letters Examples: “cat”, “dog”, “house”, … Defined over an alphabet: 2
Alphabets and Strings We will use small alphabets: Strings 3
String Operations Concatenation 4
Reverse 5
String Length: Examples: 6
Recursive Definition of Length For any letter: For any string : Example: 7
Length of Concatenation Example: 8
Proof of Concatenation Length Claim: Proof: By induction on the length Induction basis: From definition of length: 9
Inductive hypothesis: for Inductive step: we will prove for 10
Inductive Step Write , where From definition of length: From inductive hypothesis: Thus: 11
Empty String A string with no letters: Observations: 12
Substring of string: a subsequence of consecutive characters String Substring 13
Prefix and Suffix Prefixes Suffixes prefix suffix 14
Another Operation Example: Definition: 15
The * Operation : the set of all possible strings from alphabet 16
The + Operation : the set of all possible strings from alphabet except 17
Language A language is any subset of Example: Languages: 18
Another Example An infinite language 19
Operations on Languages The usual set operations Complement: 20
Reverse Definition: Examples: 21
Concatenation Definition: Example: 22
Another Operation Definition: Special case: 23
More Examples 24
Star-Closure (Kleene *) Definition: Example: 25
Positive Closure Definition: 26
Finite Automata 27
Finite Automaton Input String Output Finite Automaton String 28
Finite Accepter Input String Output Finite Automaton “Accept” or “Reject” 29
Transition Graph Abba -Finite Accepter initial state transition final state “accept” 30
Initial Configuration Input String 31
Reading the Input 32
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Input finished Output: “accept” 36
Rejection 37
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Input finished Output: “reject” 41
Another Example 42
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Input finished Output: “accept” 46
Rejection 47
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Input finished Output: “reject” 51
Formalities Deterministic Finite Accepter (DFA) : set of states : input alphabet : transition function : initial state : set of final states 52
Input Alphabet 53
Set of States 54
Initial State 55
Set of Final States 56
Transition Function 57
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Transition Function 61
Extended Transition Function 62
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Observation: There is a walk from with label to 66
Recursive Definition 67
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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} 69
Example accept 70
Another Example accept 71
Formally For a DFA Language accepted by alphabet transition function : initial state final states 72
Observation Language accepted by : Language rejected by : 73
More Examples accept trap state 74
= { all substrings with prefix } accept 75
= { all strings without substring } 76
Regular Languages A language is regular if there is a DFA such that All regular languages form a language family 77
Example The language is regular: 78