CS150 Notes 24 (19 October 2005)

Upcoming Schedule

• 2pm Today: Marc Levoy, Digital Michaelangelo Project (Newcomb Hall, South Meeting Room, 2pm)
• Thursday, 20 October: Problem Set 6 partner requests
• Before Friday, 21 October: Read rest of GEB part I (Chapters 2-4 and 6-9, in addition to Chapters 1 and 5 you have already read). See the study guide questions on Notes 18. I recommend reading a chapter every few days rather than trying to read it all just before October 21.
• (Ideally as soon as possible) Read GEB, Aria with Diverse Variations and Chapter 13. This chapter proves that the halting problem is not decidable, and introduces the Church-Turing Thesis (which we will explore more in later classes). You will not be assigned to read Chapter XIV, but it goes into more depth on Gödel's proof and is recommended.
• Monday, 31 October: Problem Set 6

Alrogithm: A procedure that always terminates.

Proof: A proof of S in an axiomatic system is a sequence of strings, T₀, T₁, ..., T_n where:

• The first string is the axioms
• For all i from 1 to n, T_n is the result of applying one of the inference rules to T_n-1
• T_n is S

Input: an axiomatic system (a set of axioms and inference rules), a statement S, and a proof P containing n steps of S.

Output: true if P is a valid proof of S; false otherwise.

How much work is a proof checking procedure?

Input: an axiomatic system (a set of axioms and inference rules), a statement S, n (the maximum number of proof steps)

Output: A valid proof of S with no more then n steps if there is one. If there is no proof of S with <= n steps, unprovable.

How much work is a straightforward proof finding procedure?

n — maximum number of steps
r — number of inference rules

At worst, we can try all possible rules at every step:

Proof Finding Problem

Input: an axiomatic system, a statement S.

Output: If S is true, output is a valid proof. If S is not true, output is false.

Can we write an algorithm to solve the proof finding problem?

Computability

Is there an algorithm (a procedure that always terminates) that solves the problem?

A problem is 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 decidable, only to know that some algorithm to solve it must exist. For example, chess is a decidable problem, even if we do not yet know an algorithm that solves it.

A problem is undecidable is there is no algorithm that can solve the problem. There might be a procedure, but it is not guaranteed to terminate.

Is the proof finding problem decidable?

Halting Problem

Input: a procedure P (described by a Scheme program), and the input to that procedure
Output: true if applying P to input halts (finishes execution), false otherwise.

(define (halts? procedure input) ...?...)

(define (find-proof S axioms rules)
  ;; If S is provable, evaluates to a proof of S.
  ;; Otherwise, evaluates to #f.
  (if (halts? find-proof-exhaustive 
	      S axioms rules)
      (find-proof-exhaustive S axioms rules)
      #f))

Where (find-proof-exhaustive S axioms rules) is a procedure that tries all possible proofs starting from the axioms that evaluates to a proof if it finds one, and keeps working if it doesn't.

(define (contradict-halts input)
  (if (halts? contradict-halts null) 
      (infinite-loop) 
      150))

If solving a problem P would allow us to solve the halting problem, then P is undecidable — there is no solution to P, since we have proved there is no solution to the halting problem!

• Alan Turing
• On Computable Numbers, with an Application to the Entscheidungsproblem — Turing's original (1936) paper that introduced the Halting Problem (tougher to read than Gödel's)
• Breaking the Code — movie about Alan Turing
• Halting Problem D-League Hockey Team (Statistics)
• Time 100: Alan Turing

There is a remarkably close parallel between the problems of the physicist and those of the cryptographer. The system on which a message is enciphered corresponds to the laws of the universe, the intercepted messages to the evidence available, the keys for a day or a message to important constants which have yet to be determined. The correspondence is very close, but the subject matter of cryptography is very easily dealt with by discrete machinery, physics not so easily.
Alan Turing