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http://www.cs.virginia.edu/~evans".P"MenuReview: Gdel s Theorem, Proof in Axiomatic Systems
Are there some problems that it is impossible to write a program to solve?" @w`
@ReviewAxiomatic 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 `);o);oAProofA 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?
S~&3' : %>I>>,BProof Checking ProblemInput: 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
n2N
C#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.`bN/V
DProof Finding Problem(rInput: an axiomatic system, a statement S
Output:
A valid proof of S if S is true. If there is no proof, false.N3@(
9EQuiz Answers
FWhat is Computer Science?B 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. L))GWhat 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.:[[HWhat is Computer Science?4My answer would be:
Study of ways to describe procedures and reason about the processes they produce?
My alternate answer:
Playing with procedures. PVVIReading 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&ddJExam 2LSimilar to Exam 1 225
Like Exam 1, but allow DrScheme 111222
In class, open notes
In class, closed notes 5555
There shouldn t be another Exam 1112%3lKHWhat does it mean for an axiomatic system to be complete and consistent?II$
LHWhat does it mean for an axiomatic system to be complete and consistent?II$
M
ComputabilityN
ComputabilityIs 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 problemLM7M7O#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.
^P@D9PAny others?QUndecidable Problems
We can prove a problem is undecidable by showing it is at least as hard as the proof-finding problem
Here s a famous one:
Halting Problem
Input: a procedure P (described by a Scheme program)
Output: true if P always halts (finished execution), false otherwise.
\zz#15RHalting 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) & )j"5,& SExamples
!TCan 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?6
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find-proofu(define (find-proof S axioms rules)
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(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.Dvs$#VAnother Informal Proof
$W"Russell s ParadoxS: 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.
O8,-%XUndecidable ProblemsIf 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.R3+>ChargeFriday
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!LWpWp/ ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@ " @ ` n?" dd@ @@``PR @ ` `p>>(
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http://www.cs.virginia.edu/~evans".P"MenuReview: Gdel s Theorem, Proof in Axiomatic Systems
Are there some problems that it is impossible to write a program to solve?" @w`
@ReviewAxiomatic 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 `);o);oAProofA 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?
S~&3' : %>I>>,BProof Checking ProblemInput: 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
n2N
C#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.`bN/V
DProof Finding Problem(rInput: an axiomatic system, a statement S
Output:
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9EQuiz Answers
FWhat is Computer Science?B 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. L))GWhat 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.:[[HWhat is Computer Science?4My answer would be:
Study of ways to describe procedures and reason about the processes they produce?
My alternate answer:
Playing with procedures. PVVIReading 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&ddJExam 2LSimilar to Exam 1 225
Like Exam 1, but allow DrScheme 111222
In class, open notes
In class, closed notes 5555
There shouldn t be another Exam 1112%3lKHWhat does it mean for an axiomatic system to be complete and consistent?II$
LHWhat does it mean for an axiomatic system to be complete and consistent?II$
M
ComputabilityN
ComputabilityIs 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 problemLM7M7O#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.
^P@D9PAny others?QUndecidable Problems
We can prove a problem is undecidable by showing it is at least as hard as the proof-finding 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.
\zz#15 RHalting 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) & )j"5,& SExamples
!TCan we define halts??$We could try for a really long time, get something to work for simple examples, but could we solve the problem? (Make it work for all possible inputs.)
Could we compute find-proof if we had always-halts?6
"U
find-proofu(define (find-proof S axioms rules)
;; If S is provable, evaluates to a proof of S.
;; Otherwise, evaluates to #f.
(if (always-halts?
(find-proof-exhaustive S axioms rules))
((find-proof-exhaustive S axioms rules)))
#f))
Where (find-proof-exhaustive S axioms rules) evaluates to a procedure that tries all possible proofs starting from the axioms.Dvs$#VAnother Informal Proof
$W"Russell s ParadoxS: set of all sets that are not members of themselves
Is S a member of itself?
If S is an element of S, then S is a member of itself and should not be in S.
If S is not an element of S, then S is not a member of itself, and should be in S.
O8,-%XUndecidable ProblemsIf solving a problem P would allow us to solve the halting problem, then P is undecidable there is no solution to P, since we have proved there is no solution to the halting problem!
There are lots of practical problems like this& we ll practice on them Friday.R3+>ChargeFriday
Practice determining if problems are decidable (in P, in NP, not in NP) or undecidable
PS 6
Even if you take into account Hofstadter s Law and Byrd s Law, it may be longer than you think so get cracking!LWpWp/}
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Quiz AnswersWhat is Computer Science?What is Computer Science?What is Computer Science?
Reading GEB?Exam 2IWhat does it mean for an axiomatic system to be complete and consistent?IWhat does it mean for an axiomatic system to be complete and consistent?ComputabilityComputability$Are there any undecidable problems?Any others?Undecidable ProblemsAlan Turing (1912-1954)Halting Problem ExamplesCan we define halts??find-proofAnother Informal ProofRussells ParadoxUndecidable ProblemsChargeFonts UsedDesign Template
Slide Titles6_V~5Department of Computer ScienceDepartment of Computer ScienceȠPk"EU9}IGo'zU,ZKA,7z}w}m/PPOOU 6A\̾kSֶ|MzKN\Ҳbc1ǥhA
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L___PPT9.&J?Z-18 March 2002 $CS 200 Spring 2002O
=5
-David Evans
http://www.cs.virginia.edu/~evans".P"MenuReview: Gdel s Theorem, Proof in Axiomatic Systems
Are there some problems that it is impossible to write a program to solve?" @w`
@ReviewAxiomatic 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 `);o);oAProofA 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?
S~&3' : %>I>>,BProof Checking ProblemInput: 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
n2N
C#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.`bN/V
DProof Finding Problem(rInput: an axiomatic system, a statement S
Output:
A valid proof of S if S is true. If there is no proof, false.N3@(
9EQuiz Answers
FWhat is Computer Science?B 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. L))GWhat 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.:[[HWhat is Computer Science?4My answer would be:
Study of ways to describe procedures and reason about the processes they produce?
My alternate answer:
Playing with procedures. PVVIReading 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&ddJExam 2LSimilar to Exam 1 225
Like Exam 1, but allow DrScheme 111222
In class, open notes
In class, closed notes 5555
There shouldn t be another Exam 1112%3lKHWhat does it mean for an axiomatic system to be complete and consistent?II$
LHWhat does it mean for an axiomatic system to be complete and consistent?II$
M
ComputabilityN
ComputabilityIs 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 problemLM7M7O#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.
^P@D9PAny others?QUndecidable Problems
We can prove a problem is undecidable by showing it is at least as hard as the proof-finding 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.
\zz#15'YAlan Turing (1912-1954)Published On Computable Numbers & (1936)
Introduced the Halting Problem
Also: formal model of computation
WWII: codebreaker at Bletchley Park (broke Enigma Cipher)
Probably even more important than Lorenz Cipher (PS4)
After the war: convicted of homosexuality (then a crime in Britian), commited suicide eating cyanide apple)B:6k
":6k>qRHalting 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) & )j"5,& SExamples
!TCan 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?6
"U
find-proofu(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.Dvs$#VAnother Informal Proof
$W"Russell s ParadoxS: 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.
O8,-%XUndecidable ProblemsIf 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.R3+>ChargeFriday
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!LWpWp/
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-David Evans
http://www.cs.virginia.edu/~evans".P"MenuReview: Gdel s Theorem, Proof in Axiomatic Systems
Are there some problems that it is impossible to write a program to solve?" @w`
@ReviewAxiomatic 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 `);o);oAProofA 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?
S~&3' : %>I>>,BProof Checking ProblemInput: 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
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C#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.`bN/V
DProof Finding Problem(rInput: an axiomatic system, a statement S
Output:
A valid proof of S if S is true. If there is no proof, false.N3@(
9EQuiz Answers
FWhat is Computer Science?B 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. L))GWhat 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.:[[HWhat is Computer Science?4My answer would be:
Study of ways to describe procedures and reason about the processes they produce?
My alternate answer:
Playing with procedures. PVVIReading 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&ddJExam 2LSimilar to Exam 1 225
Like Exam 1, but allow DrScheme 111222
In class, open notes
In class, closed notes 5555
There shouldn t be another Exam 1112%3lKHWhat does it mean for an axiomatic system to be complete and consistent?II$
LHWhat does it mean for an axiomatic system to be complete and consistent?II$
M
ComputabilityN
ComputabilityIs 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 problemLM7M7O#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.
^P@D9PAny others?QUndecidable Problems
We can prove a problem is undecidable by showing it is at least as hard as the proof-finding 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.
\zz#15'YAlan 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)[: k
;: k>RHalting 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) & )j"5,& SExamples
!TCan we define halts??$We could try for a really long time, get something to work for simple examples, but could we solve the problem? (Make it work for all possible inputs.)
Could we compute find-proof if we had always-halts?6
"U
find-proofu(define (find-proof S axioms rules)
;; If S is provable, evaluates to a proof of S.
;; Otherwise, evaluates to #f.
(if (always-halts?
(find-proof-exhaustive S axioms rules))
((find-proof-exhaustive S axioms rules)))
#f))
Where (find-proof-exhaustive S axioms rules) evaluates to a procedure that tries all possible proofs starting from the axioms.Dvs$#VAnother Informal Proof
$W"Russell s ParadoxS: 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.
O8,-%XUndecidable ProblemsIf 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.R3+>ChargeFriday
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!LWpWp/
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