Gödel's Statement:
G: This statment does not have any proof in the system.
What would it mean if G is true?
What would it mean if G is false?
Proof: A proof of S in an axiomatic system is a sequence of strings,
T_{0}, T_{1}, ..., T_{n} where:
Procedure: A precise (mechanizable) description of a process.
Alrogithm: A procedure that always terminates.
Is there an algorithm (a procedure that always terminates) that solves the problem?
A problem is computable (decidable) is there exists an algorithm that can solve the problem for all possible inputs. It is not necessary to know what that algorithm is to say a problem is computable, only to know that some algorithm to solve it must exist. For example, chess is a computable problem, even if we do not yet know a practical algorithm that solves it (yes).
A problem is uncomputable if there is no algorithm that can solve the problem. There might be a procedure, but it is not guaranteed to terminate.
What is the Halting Problem?
Is it possible to define a procedure that solves the Halting Problem?
(define (contradict-halts) (if (halts? contradict-halts) (infinite-loop)
Alan Turing Links
There are thousands of statues of lousy generals and blowhard statesmen and enormous temples erected for the worship of presidents, and not much recognition of people such as Mr. Backus who did the work that actually made life better.
Much of my work has come from being lazy.
John Backus
(not a good approach for current UVa students!)You need the willingness to fail all the time. You have to generate many ideas and then you have to work very hard only to discover that they don't work. And you keep doing that over and over until you find one that does work.
John Backus
(better approach for current UVa students!)