Languages Prof. Busch - LSU.

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Presentation transcript:

Languages Prof. Busch - LSU

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

Languages are used to describe computation problems: Alphabet: Prof. Busch - LSU

Computation is translated to set membership Example computation problem: Is number prime? Equivalent set membership problem: Prof. Busch - LSU

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

Decimal numbers alphabet Binary numbers alphabet Prof. Busch - LSU

Unary numbers alphabet Decimal number: Prof. Busch - LSU

String Operations Concatenation Prof. Busch - LSU

Reverse Prof. Busch - LSU

String Length Length: Examples: Prof. Busch - LSU

Length of Concatenation Example: Prof. Busch - LSU

Empty String A string with no letters is denoted: Acts as a neutral element Observations: Prof. Busch - LSU

Substring Substring of string: a subsequence of consecutive characters Prof. Busch - LSU

Prefix and Suffix string Prefixes Suffixes prefix suffix Prof. Busch - LSU

Exponent Operation Example: Definition: Prof. Busch - LSU

The * Operation : the set of all possible strings from alphabet Prof. Busch - LSU

The + Operation : the set of all possible strings from alphabet except Prof. Busch - LSU

Languages A language over alphabet is any subset of Example: Language: Prof. Busch - LSU

More Language Examples An infinite language Alphabet Prof. Busch - LSU

Prime numbers Alphabet Language: Prof. Busch - LSU

Even and odd numbers Alphabet Languages: Prof. Busch - LSU

Addition (of unary numbers) Alphabet: Language: Prof. Busch - LSU

Squares (of unary numbers) Alphabet: Language: Prof. Busch - LSU

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

Note that: Sets Set size Set size String length Prof. Busch - LSU

Operations on Languages The usual set operations Complement: Prof. Busch - LSU

Reverse Definition: Examples: Prof. Busch - LSU

Concatenation Definition: Example: Prof. Busch - LSU

Another Operation Definition: Special case: Prof. Busch - LSU

Example Prof. Busch - LSU

Star-Closure (Kleene *) All strings that can be constructed from Definition: Example: Prof. Busch - LSU

Positive Closure Definition: Prof. Busch - LSU