Transcript Lecture 23 - University of Virginia, Department of Computer Science

Lecture 23:
Computability
CS200: Computer Science
University of Virginia
Computer Science
David Evans
http://www.cs.virginia.edu/~evans
Menu
• Review: Gödel’s Theorem, Proof in
Axiomatic Systems
• Are there some problems that it is
impossible to write a program to solve?
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Review
• Axiomatic System
– Set of axioms
– Set of inference rules
• Example: MIU-System from GEB
– Axioms: MI
– Inference rules: 4 rules for making new strings
• An axiomatic system is a formal system
where the string we can generate are
meant to represent “true theorems”
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Proof
• A proof of S in an axiomatic system is a
sequence of strings, T0, T1, …, Tn where:
– The first string is the axioms
– For all i from 1 to n, Tn is the result of applying
one of the inference rules to Tn-1
– Tn is S
• What is the proof-checking problem?
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Proof Checking Problem
• Input: an axiomatic system (a set of
axioms and inference rules), a statement
S, and a proof P of S
• Output:
true if P is a valid proof of S
false otherwise
How much work is the proof-checking problem?
n = length of the proof (number of steps)
Checking a proof is O(n)
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Finite-Length Proof Finding Problem
• 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, unprovable.
How much work? At worst, we can try all possible proofs:
Finite-Length ProofFinding Problem is
NP-Complete.
O(rn) and (n)
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r inference rules
0 - n steps
~ rn possible proofs
Checking each proof is O(max-steps)
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Proof Finding Problem
• Input: an axiomatic system, a statement S
• Output:
A valid proof of S if S is true. If there is no
proof, false.
How much work?
It is impossible!
Gödel’s theorem says it cannot be done.
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Quiz Answers
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What is Computer Science?
• “Correct” answers:
– “The most wonderful thing in the world!”
• Okay answers:
– “a liberal art (the only legitimate one) that
incorporates how into figuring things out. It is
the only class I have that can hurt my head.”
– “Study of systems and their actions through
the use of language systems.”
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What is Computer Science?
• More Okay answers:
– “A combination of logic, math, and other
disciplines to create systems.”
– “Study of language, math, logic, and all kinds of
good stuff… (music, cognition, etc.)”
– Neither “about” computers nor a science, more
of a liberal art.
– “Complicated, but nothing to with computers or
science.”
• Actually, it has a lot to do with computers. Like
chemistry has a lot to do with beekers.
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What is Computer Science?
• My answer would be:
“Study of ways to describe procedures and
reason about the processes they produce?”
• My alternate answer:
“Playing with procedures.”
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Reading GEB?
– Don’t remember: 1 (?)
– Through Ch 5 or less: 4
– All of Part I: 1
– All of Part I and some in Part II: 2
• Reading GEB is probably not necessary to
get a good grade in this class, but I really
hope you will read it and enjoy reading it!
• Ch 13 is the last assigned reading in it for
this class
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Exam 2
─ Similar to Exam 1
─ Like Exam 1, but allow DrScheme
─ In class, open notes
─ In class, closed notes
─ There shouldn’t be another Exam
225
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Exam 2 will involve:
- Questions about axiomatic systems, complexity and computability
(some practice questions on Friday)
- Questions about evaluation models (environments, evaluation rules)
- Writing a program that uses everything up through and including PS7
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What does it mean for an axiomatic
system to be complete and consistent?
Derives all true
statements, and no false
statements starting from a
finite number of axioms
and following mechanical
inference rules.
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What does it mean for an axiomatic
system to be complete and consistent?
It means the axiomatic system is
weak.
Its is so weak, it cannot express
“This statement has no proof.”
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Computability
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Computability
• Is there a procedure that solves a problem?
• Decidable (computable) problems:
– There is a procedure that solves the problem
– Make a photomosaic, sorting, drug discovery,
who will win NCAA tournament (it doesn’t mean
we know the procedure, but there is one)
• Undecidable problems:
– There is no possible procedure that solves the
problem
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Are there any undecidable
problems?
The Proof-Finding Problem:
Input: an axiomatic system, a statement S
Output:
A valid proof of S if S is true. If there is no
proof, false.
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Any others?
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Undecidable Problems
• We can prove a problem is undecidable by
showing it is at least as hard as the prooffinding problem
• Here’s a famous one:
Halting Problem
Input: a procedure P (described by a Scheme
program)
Output: true if P always halts (finishes
execution), false otherwise.
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Alan Turing (1912-1954)
• Published On Computable
Numbers … (1936)
– Introduced the Halting Problem
– Also: formal model of computation
and design for computers
• WWII: codebreaker at Bletchley
Park (broke Enigma Cipher)
– Even more important than Lorenz
• After the war: convicted of
homosexuality (then a crime in Britian),
commited suicide eating cyanide apple
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Halting Problem
Can we define a procedure always-halts
that takes the code for a procedure and
evaluates to #t if the procedure always
terminates, and to #f if it may not
terminate?
(define (always-halts procedure) … )
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Examples
> (always-halts ‘(lambda (x) (+ x x)))
#t
> (always-halts ‘(lambda (x)
(define (f x) (f x))
(f x)))
#f
> (always-halts ‘(lambda (x)
(define (fact n)
(if (= n 1) 1 (* n (fact (- n 1)))))
(fact x)))
#f
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Can we define halts??
• We could try for a really long time, get
something to work for simple examples,
but could we solve the problem? (Make it
work for all possible inputs.)
• Could we compute find-proof if we had
always-halts?
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find-proof
I cheated a little here –
we only know we can’t
do this for “true”.
(define (find-proof S axioms rules)
;; If S is provable, evaluates to a proof of S.
;; Otherwise, evaluates to #f.
(if (always-halts?
(find-proof-exhaustive S axioms rules))
((find-proof-exhaustive S axioms rules)))
#f))
Where (find-proof-exhaustive S axioms rules) evaluates to a procedure
that tries all possible proofs starting from the axioms.
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Another Informal Proof
(define (contradict-halts)
(if (always-halts contradict-halts)
(loop-forever)
#t))
If contradict-halts halts, the if test is true and
it evaluates to (loop-forever) - it doesn’t halt!
If contradict-halts doesn’t halt, the if test if false,
and it evaluates to #t. It halts!
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This should remind you of…
Russell’s Paradox
• S: set of all sets that are not members of
themselves
• Is S a member of itself?
– If S is an element of S, then S is a member of
itself and should not be in S.
– If S is not an element of S, then S is not a
member of itself, and should be in S.
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Undecidable Problems
• 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!
• There are lots of practical problems like
this…we’ll practice on them Friday.
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Charge
• Friday
– Practice determining if problems are
decidable (in P, in NP, not in NP) or
undecidable
• PS 6
– Even if you take into account Hofstadter’s
Law and Byrd’s Law, it may be longer than
you think so get cracking!
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