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M15, Limits to Computation - Coggle Diagram
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Limits to Computation
NP-Completeness Proofs
To prove a problem is NP-complete, one must show:
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Cook-Levin Theorem is a foundational result proving that the Boolean satisfiability problem (SAT) is NP-complete.
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Reductions
Reductions are transformations of one problem into another. They help in understanding the relative difficulty of problems. If problem A can be reduced to problem B, solving B would provide a solution to A.
Polynomial-time reduction is a specific type of reduction where the transformation takes polynomial time, essential in the context of NP-completeness.
The Halting Problem
The Halting Problem: Determining whether a given program will halt (stop running) or continue to run indefinitely.
Alan Turing proved that there is no general algorithm to solve the halting problem for all possible program-input pairs, establishing a fundamental limit to computation.
Hard Problems
Hard Problems refer to those problems for which no efficient algorithms are known. These problems often require exploring a vast number of possibilities.
17.3 Impossible Problems
Impossible Problems are those for which no algorithm can ever be devised to solve them for all possible inputs.
Uncountability
Some problems are uncountable, meaning there are more possible inputs than can be listed in any infinite sequence. These problems often arise in the context of real numbers.