Alan Turing proved the halting problem is undecidable
Image: Oregon State University, CC BY-SA 2.0, via Wikimedia Commons
Halting problem
Alan Turing proved the halting problem is undecidable
Alan Turing's 1937 proof established that no general algorithm can decide if every program halts. This result is fundamental in computability theory, showing the limits of what can be computed. The proof demonstrates that some functions, while mathematically definable, cannot be computed by any algorithm.
Example
Consider a program designed to determine if another program halts. If this program were to exist, it would contradict Turing's proof, as it would imply that the halting problem is decidable.
Understanding Turing's proof is crucial for recognizing the inherent limitations of computational systems and the boundaries of what can be algorithmically determined.
Related concepts
Kolmogorov complexity
Kolmogorov complexity is uncomputable
Overlapping subproblems
Dynamic programming solves overlapping subproblems by storing results of subproblems to avoid redundant calculations
the optional stopping theorem says about martingales and stopping times
The optional stopping theorem states that for a martingale, stopping at a stopping time with finite expectation preserves the martingale property
approximation algorithms guarantee: solution within factor α of optimal
Approximation algorithms guarantee a solution within a factor α of the optimal solution
the Y combinator does: enables recursion in languages without named functions
The Y combinator allows anonymous functions to call themselves recursively
merge sort: O(n log n) always
Merge sort consistently performs at O(n log n) time complexity for any input size
One email a day: 5 concepts + the 5 stories that matter →
Swipe through 100 ML concepts daily
Open TickerNews