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Knowledge
Computable and uncomputable functions are explored, emphasizing that certain functions cannot be computed regardless of available resources. The proof of existence of uncomputable functions involves comparing the cardinality of valid programs to functions. This culminates in a fundamental understanding that not all computational tasks can be resolved using algorithms.
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Sections
References
ch31.pdfClass Notes
Memorization
What we have learnt
- There exist functions that ...
- The set of all valid progra...
- Uncomputable functions are ...
Final Test
Revision Tests
What we have learnt
- There exist functions that cannot be computed by any program.
- The set of all valid programs is countable, whereas the set of all functions from positive integers to a finite set is uncountable.
- Uncomputable functions are fundamental in understanding the limitations of computation.
Key Concepts
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Term: Computable Function
Definition: A function for which there exists a program that can compute its value for every input from its domain.
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Term: Uncomputable Function
Definition: A function for which no program can compute its value for every input, regardless of the resources available.
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Term: Cardinality
Definition: A measure of the 'size' of a set, particularly regarding the countability of sets.
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Term: Nonconstructive Proof
Definition: A type of proof that demonstrates the existence of a case without constructing an example of it.