13 KiB
| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Computability | Obsidian::STEM | computability |
|
Overview
An alphabet is any nonempty finite set. The members of an alphabet are called symbols of the alphabet. A string over an alphabet is a finite sequence of symbols from that alphabet, usually written next to one another without separation.
Then length of a string w over an alphabet, denoted \lvert w \rvert, is the number of symbols w contains. The empty string, denoted \epsilon, is the string of length 0. The reverse of w is the string obtained by writing w in the opposite order.
String z is a substring of w if z appears consecutively within w. The concatenation of strings x and y, written xy, is the string obtained by appending y to the end of x. We say string x is a prefix of string y if a string z exists where xz = y. We say x is a proper prefix of y if x \neq y.
A language is a set of strings. A language is prefix-free if no member is a proper prefix of another member.
%%ANKI Basic What is an alphabet? Back: A nonempty finite set. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
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%%ANKI Basic An alphabet is a set satisfying what two properties? Back: It is nonempty and finite. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic What is a symbol of an alphabet? Back: A member of the alphabet. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic What name is given to members of an alphabet? Back: Symbols. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
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%%ANKI Cloze A {symbol} is a {member} of an {alphabet}. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic What is a string over an alphabet? Back: A finite sequence of symbols from that alphabet. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic What kind of mathematical entity is a string over an alphabet? Back: A finite sequence. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
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%%ANKI
Basic
01001 is a string over what minimal alphabet?
Back: \{0, 1\}
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
hello is a string over what minimal alphabet?
Back: \{e, h, l, o\}
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
How is the length of string s over alphabet \Sigma denoted?
Back: As \lvert s \rvert.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic How is the length of a string over an alphabet defined? Back: As the number of symbols in the string. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
Let w be a string over some alphabet. How is the length of w denoted?
Back: \lvert w \rvert
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
What is the empty string over an alphabet?
Back: The string of length 0.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
How is the empty string over an alphabet typically denoted?
Back: \epsilon
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
What is the reverse of string w over some alphabet?
Back: The string obtained by writing w in opposite order.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
How is the reverse of string w over some alphabet typically denoted?
Back: w^R
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic How does the length of a string relate to the length of its reverse? Back: They are equal. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic How does the length of a string relate to the length of a substring? Back: The latter is less than or equal to the former. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
What is a substring of string w over some alphabet?
Back: A string that appears consecutively within w.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
Let x and y be strings over some alphabet. How is their concatenation denoted?
Back: xy
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
Let x and y be strings over some alphabet. What is the length of \lvert x y \rvert?
Back: \lvert x \rvert + \lvert y \rvert
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
Let x be a string over some alphabet. What does xx denote?
Back: The concatenation of x with itself.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Cloze
Let x be a string over some alphabet. Then {xx} is equivalently denoted as {x^2}.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
Let x be a string over some alphabet. What does x^k denote?
Back: The concatenation of x with itself k times.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
What does it mean for string x to be a prefix of string y over some alphabet?
Back: There exists some string z such that xz = y.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
What does it mean for string x to be a proper prefix of string y over some alphabet?
Back: There exists some string z such that xz = y and x \neq y.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
What does it mean for string x to be a suffix of string y over some alphabet?
Back: There exists some string z such that zx = y.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI
Basic
What does it mean for string x to be a proper suffix of string y over some alphabet?
Back: There exists some string z such that zx = y and x \neq y.
Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic What is a language? Back: A set of strings. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic A language is a set satisfying what? Back: It contains strings over some alphabet. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Cloze {1:Symbols} are to {2:alphabets} whereas {2:strings} are to {1:languages}. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic What does it mean for a language to be prefix-free? Back: No member is a proper prefix of another member. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
END%%
%%ANKI Basic What does it mean for a language to be suffix-free? Back: No member is a proper suffix of another member. Reference: Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
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Bibliography
- Michael Sipser, Introduction to the Theory of Computation, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).