computability theory i - sjtubasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf ·...
TRANSCRIPT
![Page 1: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/1.jpg)
Computability Theory IIntroduction
Guoqiang Li
Shanghai Jiao Tong University
Sep. 19, 2014
![Page 2: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/2.jpg)
Instructor and Teaching Assistant
• Guoqiang LI
• Homepage: http://basics.sjtu.edu.cn/∼liguoqiang• Course page:
http://basics.sjtu.edu.cn/∼liguoqiang/teaching/comp14/index.htm• Email: [email protected]• Office: Rm. 1212, Building of Software• Phone: 3420-4167• YSSY: many IDs…• weibo: http://www.weibo.com/flyinsail
• TA:
• Mingzhang HUANG: [email protected]• Xiuting TAO: [email protected]
• Office hour: Wed. 14:00-17:00 @ SEIEE 3-327
![Page 3: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/3.jpg)
Instructor and Teaching Assistant
• Guoqiang LI• Homepage: http://basics.sjtu.edu.cn/∼liguoqiang• Course page:
http://basics.sjtu.edu.cn/∼liguoqiang/teaching/comp14/index.htm• Email: [email protected]• Office: Rm. 1212, Building of Software• Phone: 3420-4167
• YSSY: many IDs…• weibo: http://www.weibo.com/flyinsail
• TA:
• Mingzhang HUANG: [email protected]• Xiuting TAO: [email protected]
• Office hour: Wed. 14:00-17:00 @ SEIEE 3-327
![Page 4: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/4.jpg)
Instructor and Teaching Assistant
• Guoqiang LI• Homepage: http://basics.sjtu.edu.cn/∼liguoqiang• Course page:
http://basics.sjtu.edu.cn/∼liguoqiang/teaching/comp14/index.htm• Email: [email protected]• Office: Rm. 1212, Building of Software• Phone: 3420-4167• YSSY: many IDs…• weibo: http://www.weibo.com/flyinsail
• TA:
• Mingzhang HUANG: [email protected]• Xiuting TAO: [email protected]
• Office hour: Wed. 14:00-17:00 @ SEIEE 3-327
![Page 5: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/5.jpg)
Instructor and Teaching Assistant
• Guoqiang LI• Homepage: http://basics.sjtu.edu.cn/∼liguoqiang• Course page:
http://basics.sjtu.edu.cn/∼liguoqiang/teaching/comp14/index.htm• Email: [email protected]• Office: Rm. 1212, Building of Software• Phone: 3420-4167• YSSY: many IDs…• weibo: http://www.weibo.com/flyinsail
• TA:• Mingzhang HUANG: [email protected]• Xiuting TAO: [email protected]
• Office hour: Wed. 14:00-17:00 @ SEIEE 3-327
![Page 6: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/6.jpg)
What do you think you can learn from this course?
![Page 7: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/7.jpg)
Aim of the Course
• Q: Can the course improve the skill of programming?
• A: Nope!• Q: Can the course improve the ability of algorithms?
• A: Perhaps, seldom.
• The course may provide a view of computation, an overlook ofwhat we are doing in computer science, and a basic study oftheoretical computer science.
• It is rather a philosophy than a technique, although some partsare quite technically.
![Page 8: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/8.jpg)
Aim of the Course
• Q: Can the course improve the skill of programming?• A: Nope!
• Q: Can the course improve the ability of algorithms?
• A: Perhaps, seldom.
• The course may provide a view of computation, an overlook ofwhat we are doing in computer science, and a basic study oftheoretical computer science.
• It is rather a philosophy than a technique, although some partsare quite technically.
![Page 9: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/9.jpg)
Aim of the Course
• Q: Can the course improve the skill of programming?• A: Nope!
• Q: Can the course improve the ability of algorithms?
• A: Perhaps, seldom.
• The course may provide a view of computation, an overlook ofwhat we are doing in computer science, and a basic study oftheoretical computer science.
• It is rather a philosophy than a technique, although some partsare quite technically.
![Page 10: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/10.jpg)
Aim of the Course
• Q: Can the course improve the skill of programming?• A: Nope!
• Q: Can the course improve the ability of algorithms?• A: Perhaps, seldom.
• The course may provide a view of computation, an overlook ofwhat we are doing in computer science, and a basic study oftheoretical computer science.
• It is rather a philosophy than a technique, although some partsare quite technically.
![Page 11: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/11.jpg)
Aim of the Course
• Q: Can the course improve the skill of programming?• A: Nope!
• Q: Can the course improve the ability of algorithms?• A: Perhaps, seldom.
• The course may provide a view of computation, an overlook ofwhat we are doing in computer science, and a basic study oftheoretical computer science.
• It is rather a philosophy than a technique, although some partsare quite technically.
![Page 12: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/12.jpg)
Aim of the Course
• Q: Can the course improve the skill of programming?• A: Nope!
• Q: Can the course improve the ability of algorithms?• A: Perhaps, seldom.
• The course may provide a view of computation, an overlook ofwhat we are doing in computer science, and a basic study oftheoretical computer science.
• It is rather a philosophy than a technique, although some partsare quite technically.
![Page 13: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/13.jpg)
It May Answer
• A software company that is developing a compiler capable ofchecking if a program contains a loop.
• A hardware company that is determined to design a computerthat can solve problems that no existing computers can solve.
• A service provider that is working on a theorem prover that issupposed to answer every question about numbers.
![Page 14: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/14.jpg)
It May Answer
• A software company that is developing a compiler capable ofchecking if a program contains a loop.
• A hardware company that is determined to design a computerthat can solve problems that no existing computers can solve.
• A service provider that is working on a theorem prover that issupposed to answer every question about numbers.
![Page 15: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/15.jpg)
It May Answer
• A software company that is developing a compiler capable ofchecking if a program contains a loop.
• A hardware company that is determined to design a computerthat can solve problems that no existing computers can solve.
• A service provider that is working on a theorem prover that issupposed to answer every question about numbers.
![Page 16: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/16.jpg)
It May Answer
• A software company that is developing a compiler capable ofchecking if a program contains a loop.
• A hardware company that is determined to design a computerthat can solve problems that no existing computers can solve.
• A service provider that is working on a theorem prover that issupposed to answer every question about numbers.
![Page 17: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/17.jpg)
History of Velocity
Human beings are keen on speed, and cannot stop the step to chasemoving as fast as possible.
• Wheels: Mid-4th millennium BC.• Automobiles: 1762• Trains: 1807• Airplanes: 1903• Supersonic: 1947 343.2m/s• Circular velocity 1957 7.9km/s• Earth escape velocity 1959 11.2km/s• Solar system escape velocity 1977 16.7km/s• Q: Can we achieve in arbitrarily fast velocity?
• Grandfather paradox
![Page 18: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/18.jpg)
History of Velocity
Human beings are keen on speed, and cannot stop the step to chasemoving as fast as possible.
• Wheels: Mid-4th millennium BC.
• Automobiles: 1762• Trains: 1807• Airplanes: 1903• Supersonic: 1947 343.2m/s• Circular velocity 1957 7.9km/s• Earth escape velocity 1959 11.2km/s• Solar system escape velocity 1977 16.7km/s• Q: Can we achieve in arbitrarily fast velocity?
• Grandfather paradox
![Page 19: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/19.jpg)
History of Velocity
Human beings are keen on speed, and cannot stop the step to chasemoving as fast as possible.
• Wheels: Mid-4th millennium BC.• Automobiles: 1762
• Trains: 1807• Airplanes: 1903• Supersonic: 1947 343.2m/s• Circular velocity 1957 7.9km/s• Earth escape velocity 1959 11.2km/s• Solar system escape velocity 1977 16.7km/s• Q: Can we achieve in arbitrarily fast velocity?
• Grandfather paradox
![Page 20: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/20.jpg)
History of Velocity
Human beings are keen on speed, and cannot stop the step to chasemoving as fast as possible.
• Wheels: Mid-4th millennium BC.• Automobiles: 1762• Trains: 1807• Airplanes: 1903
• Supersonic: 1947 343.2m/s• Circular velocity 1957 7.9km/s• Earth escape velocity 1959 11.2km/s• Solar system escape velocity 1977 16.7km/s• Q: Can we achieve in arbitrarily fast velocity?
• Grandfather paradox
![Page 21: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/21.jpg)
History of Velocity
Human beings are keen on speed, and cannot stop the step to chasemoving as fast as possible.
• Wheels: Mid-4th millennium BC.• Automobiles: 1762• Trains: 1807• Airplanes: 1903• Supersonic: 1947 343.2m/s
• Circular velocity 1957 7.9km/s• Earth escape velocity 1959 11.2km/s• Solar system escape velocity 1977 16.7km/s• Q: Can we achieve in arbitrarily fast velocity?
• Grandfather paradox
![Page 22: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/22.jpg)
History of Velocity
Human beings are keen on speed, and cannot stop the step to chasemoving as fast as possible.
• Wheels: Mid-4th millennium BC.• Automobiles: 1762• Trains: 1807• Airplanes: 1903• Supersonic: 1947 343.2m/s• Circular velocity 1957 7.9km/s• Earth escape velocity 1959 11.2km/s• Solar system escape velocity 1977 16.7km/s
• Q: Can we achieve in arbitrarily fast velocity?
• Grandfather paradox
![Page 23: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/23.jpg)
History of Velocity
Human beings are keen on speed, and cannot stop the step to chasemoving as fast as possible.
• Wheels: Mid-4th millennium BC.• Automobiles: 1762• Trains: 1807• Airplanes: 1903• Supersonic: 1947 343.2m/s• Circular velocity 1957 7.9km/s• Earth escape velocity 1959 11.2km/s• Solar system escape velocity 1977 16.7km/s• Q: Can we achieve in arbitrarily fast velocity?
• Grandfather paradox
![Page 24: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/24.jpg)
History of Velocity
Human beings are keen on speed, and cannot stop the step to chasemoving as fast as possible.
• Wheels: Mid-4th millennium BC.• Automobiles: 1762• Trains: 1807• Airplanes: 1903• Supersonic: 1947 343.2m/s• Circular velocity 1957 7.9km/s• Earth escape velocity 1959 11.2km/s• Solar system escape velocity 1977 16.7km/s• Q: Can we achieve in arbitrarily fast velocity?
• Grandfather paradox
![Page 25: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/25.jpg)
Computation
Human beings are also keen on computation, and cannot stop the stepto chase computing as complex as possible.
• Decimal system: AD 600• Basic arithmetic: Al Khwarizmi (780 - 850)• ENIAC: 1946• NP problem• The curse of exponential time• Advanced algorithms: simplex, DPLL, antichain.• Q: Can we achieve in arbitrarily complex computation?
![Page 26: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/26.jpg)
Computation
Human beings are also keen on computation, and cannot stop the stepto chase computing as complex as possible.
• Decimal system: AD 600
• Basic arithmetic: Al Khwarizmi (780 - 850)• ENIAC: 1946• NP problem• The curse of exponential time• Advanced algorithms: simplex, DPLL, antichain.• Q: Can we achieve in arbitrarily complex computation?
![Page 27: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/27.jpg)
Computation
Human beings are also keen on computation, and cannot stop the stepto chase computing as complex as possible.
• Decimal system: AD 600• Basic arithmetic: Al Khwarizmi (780 - 850)
• ENIAC: 1946• NP problem• The curse of exponential time• Advanced algorithms: simplex, DPLL, antichain.• Q: Can we achieve in arbitrarily complex computation?
![Page 28: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/28.jpg)
Computation
Human beings are also keen on computation, and cannot stop the stepto chase computing as complex as possible.
• Decimal system: AD 600• Basic arithmetic: Al Khwarizmi (780 - 850)• ENIAC: 1946
• NP problem• The curse of exponential time• Advanced algorithms: simplex, DPLL, antichain.• Q: Can we achieve in arbitrarily complex computation?
![Page 29: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/29.jpg)
Computation
Human beings are also keen on computation, and cannot stop the stepto chase computing as complex as possible.
• Decimal system: AD 600• Basic arithmetic: Al Khwarizmi (780 - 850)• ENIAC: 1946• NP problem• The curse of exponential time
• Advanced algorithms: simplex, DPLL, antichain.• Q: Can we achieve in arbitrarily complex computation?
![Page 30: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/30.jpg)
Computation
Human beings are also keen on computation, and cannot stop the stepto chase computing as complex as possible.
• Decimal system: AD 600• Basic arithmetic: Al Khwarizmi (780 - 850)• ENIAC: 1946• NP problem• The curse of exponential time• Advanced algorithms: simplex, DPLL, antichain.
• Q: Can we achieve in arbitrarily complex computation?
![Page 31: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/31.jpg)
Computation
Human beings are also keen on computation, and cannot stop the stepto chase computing as complex as possible.
• Decimal system: AD 600• Basic arithmetic: Al Khwarizmi (780 - 850)• ENIAC: 1946• NP problem• The curse of exponential time• Advanced algorithms: simplex, DPLL, antichain.• Q: Can we achieve in arbitrarily complex computation?
![Page 32: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/32.jpg)
What problems can be solved by computers?
![Page 33: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/33.jpg)
Computer science is no more about computers thanastronomy is about telescopes.
Edsger Dijkstra
![Page 34: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/34.jpg)
Let us begin to learn some basic astronomical phenomena!
![Page 35: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/35.jpg)
The technique part is quite similar to puzzles of wise men.
So, please have a fun!
![Page 36: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/36.jpg)
Intuition is extremely important!
![Page 37: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/37.jpg)
Reference Book
• Computability: An Introduction to RecursiveFunction Theory.
• Nigel J. Cutland
• plus extra reading materials.
![Page 38: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/38.jpg)
Reference Book
• Computability: An Introduction to RecursiveFunction Theory.
• Nigel J. Cutland
• plus extra reading materials.
![Page 39: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/39.jpg)
Scoring Policy
• 10% Attendance.• 20% Assignments.• 70% Final exam.
• There are also several homework. The answer may be given inexercise lectures, two or three times.
![Page 40: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/40.jpg)
Scoring Policy
• 10% Attendance.• 20% Assignments.
• Four assignments.
• Each one is 5 pts.• Work out individually.
• 70% Final exam.
• There are also several homework. The answer may be given inexercise lectures, two or three times.
![Page 41: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/41.jpg)
Scoring Policy
• 10% Attendance.• 20% Assignments.
• Four assignments.• Each one is 5 pts.
• Work out individually.
• 70% Final exam.
• There are also several homework. The answer may be given inexercise lectures, two or three times.
![Page 42: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/42.jpg)
Scoring Policy
• 10% Attendance.• 20% Assignments.
• Four assignments.• Each one is 5 pts.• Work out individually.
• 70% Final exam.
• There are also several homework. The answer may be given inexercise lectures, two or three times.
![Page 43: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/43.jpg)
Scoring Policy
• 10% Attendance.• 20% Assignments.
• Four assignments.• Each one is 5 pts.• Work out individually.
• 70% Final exam.• There are also several homework. The answer may be given in
exercise lectures, two or three times.
![Page 44: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/44.jpg)
Special Requirements
A notebook and a pen.
![Page 45: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/45.jpg)
Any questions?
![Page 46: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/46.jpg)
0. Prologue
![Page 47: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/47.jpg)
Effective Solutions
![Page 48: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/48.jpg)
What problems can be solved by computers?
![Page 49: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/49.jpg)
Famous Problems
• Diophantine equations• Shortest path problem• Travelling salesman problem (TSP)• Graph isomorphism problem (GI)
![Page 50: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/50.jpg)
Intuition
An effective procedure consists of a finite set of instructions which,given an input from some set of possible inputs, enables us to obtainan output through a systematic execution of the instructions thatterminates in a finite number of steps.
![Page 51: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/51.jpg)
Intuition
Theorem proving is in general not effective.
Proof verification is effective.
Unbounded search is in general not effective.
Bounded search is effective.
![Page 52: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/52.jpg)
Intuition
Theorem proving is in general not effective.
Proof verification is effective.
Unbounded search is in general not effective.
Bounded search is effective.
![Page 53: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/53.jpg)
Intuition
Theorem proving is in general not effective.
Proof verification is effective.
Unbounded search is in general not effective.
Bounded search is effective.
![Page 54: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/54.jpg)
Representation of Problem
• How does a computer solve the GI problem or the TSP Problem?
• How is a problem instance (a graph) represented in a computer?• How is the answer to a problem instance represented?• How is an effective procedure formalized?
• Can every function from N to N be calculated by a C program?
• Negative.
![Page 55: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/55.jpg)
Representation of Problem
• How does a computer solve the GI problem or the TSP Problem?• How is a problem instance (a graph) represented in a computer?
• How is the answer to a problem instance represented?• How is an effective procedure formalized?
• Can every function from N to N be calculated by a C program?
• Negative.
![Page 56: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/56.jpg)
Representation of Problem
• How does a computer solve the GI problem or the TSP Problem?• How is a problem instance (a graph) represented in a computer?• How is the answer to a problem instance represented?
• How is an effective procedure formalized?
• Can every function from N to N be calculated by a C program?
• Negative.
![Page 57: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/57.jpg)
Representation of Problem
• How does a computer solve the GI problem or the TSP Problem?• How is a problem instance (a graph) represented in a computer?• How is the answer to a problem instance represented?• How is an effective procedure formalized?
• Can every function from N to N be calculated by a C program?
• Negative.
![Page 58: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/58.jpg)
Representation of Problem
• How does a computer solve the GI problem or the TSP Problem?• How is a problem instance (a graph) represented in a computer?• How is the answer to a problem instance represented?• How is an effective procedure formalized?
• Can every function from N to N be calculated by a C program?
• Negative.
![Page 59: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/59.jpg)
Representation of Problem
• How does a computer solve the GI problem or the TSP Problem?• How is a problem instance (a graph) represented in a computer?• How is the answer to a problem instance represented?• How is an effective procedure formalized?
• Can every function from N to N be calculated by a C program?• Negative.
![Page 60: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/60.jpg)
Punchline
• In a formal theory of computability, every problem instance canbe represented by a number and every number represents aproblem instance.
• A problem is a function f : N→ N from numbers to numbers.• A problem is computable if it can be calculated by a program.
![Page 61: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/61.jpg)
Punchline
• In a formal theory of computability, every problem instance canbe represented by a number and every number represents aproblem instance.
• A problem is a function f : N→ N from numbers to numbers.
• A problem is computable if it can be calculated by a program.
![Page 62: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/62.jpg)
Punchline
• In a formal theory of computability, every problem instance canbe represented by a number and every number represents aproblem instance.
• A problem is a function f : N→ N from numbers to numbers.• A problem is computable if it can be calculated by a program.
![Page 63: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/63.jpg)
Everything is number!
Pythagoras
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Decision Problem
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Decision Problem
A problem f : N→ N is a decision problem if the range ran(f ) of f is{0, 1}, where 1 denotes a ‘yes’ answer and 0 a ‘no’ answer.
A decision problem g can be identified with the set {n | g(n) = 1}.
Conversely a subset A of N can be seen as a decision problem via thecharacteristic function of A:
cA(n) =
{1, if x ∈ A,0, otherwise.
![Page 66: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/66.jpg)
Decision Problem as Predicate
A decision problem can be stated as a predicate P(x) on number.
It relates to the problem-as-function viewpoint by the following characteristicfunction of P(x):
cP(n) =
{1, if P(n) is valid,0, otherwise.
![Page 67: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/67.jpg)
Decision Problem ⇔ Subset of N⇔ Predicate on N
![Page 68: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/68.jpg)
Several Problems
![Page 69: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/69.jpg)
Problem I
Is the function tower(x) defined below computable?
tower(x) = 22...2︸︷︷︸
x
Theoretically it is computable.
![Page 70: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/70.jpg)
Problem I
Is the function tower(x) defined below computable?
tower(x) = 22...2︸︷︷︸
x
Theoretically it is computable.
![Page 71: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/71.jpg)
Problem II
Consider the function f defined as follows:
f (n) =
{1, if n > 1 and 2n is the sum of 2 primes,0, otherwise.
The Goldbach Conjecture remains unsolved. Is f computable?
It is clearly computable even if we do not know what it is.
![Page 72: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/72.jpg)
Problem II
Consider the function f defined as follows:
f (n) =
{1, if n > 1 and 2n is the sum of 2 primes,0, otherwise.
The Goldbach Conjecture remains unsolved. Is f computable?
It is clearly computable even if we do not know what it is.
![Page 73: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/73.jpg)
Problem III
Consider the function g defined as follows:
g(n) =
1, if there is a run of exactly n consecutive 7′s
in the decimal expansion of π,0, otherwise.
It is known that π can be calculated by 4(1− 1
3 + 15 −
17 + . . .
).
Is g computable?
We do not know whether it is computable or not.
![Page 74: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/74.jpg)
Problem III
Consider the function g defined as follows:
g(n) =
1, if there is a run of exactly n consecutive 7′s
in the decimal expansion of π,0, otherwise.
It is known that π can be calculated by 4(1− 1
3 + 15 −
17 + . . .
).
Is g computable?
We do not know whether it is computable or not.
![Page 75: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/75.jpg)
Problem IV
Consider the function h defined as follows:
h(n) =
1, if n is the machine code of a C program that
terminates in all inputs,0, otherwise.
This is the Halting Problem, a well known undecidable problem. Inother words there does not exist any C program calculating h.
The only general approach to check if a function is defined on allnumbers is to calculate it on all inputs.
![Page 76: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/76.jpg)
Problem IV
Consider the function h defined as follows:
h(n) =
1, if n is the machine code of a C program that
terminates in all inputs,0, otherwise.
This is the Halting Problem, a well known undecidable problem. Inother words there does not exist any C program calculating h.
The only general approach to check if a function is defined on allnumbers is to calculate it on all inputs.
![Page 77: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/77.jpg)
Problem V
Consider the function i defined as follows:
i(x, n, t) =
1, if on input x, the machine coded by n
terminates in t steps,0, otherwise.
There could be a number of ways to interpret “t steps”.
The function i is intuitively computable.
![Page 78: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/78.jpg)
Problem V
Consider the function i defined as follows:
i(x, n, t) =
1, if on input x, the machine coded by n
terminates in t steps,0, otherwise.
There could be a number of ways to interpret “t steps”.
The function i is intuitively computable.
![Page 79: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/79.jpg)
Next Lecture
The examples try to suggest that in order to study computability onemight as well look for a theory of computable functions.
We will begin with a machine model, register machine.
![Page 80: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/80.jpg)
Next Lecture
The examples try to suggest that in order to study computability onemight as well look for a theory of computable functions.
We will begin with a machine model, register machine.
![Page 81: Computability Theory I - SJTUbasics.sjtu.edu.cn/~liguoqiang/teaching/comp/lectures/comp1.pdf · Computability Theory I Introduction Guoqiang Li Shanghai Jiao Tong University Sep](https://reader034.vdocument.in/reader034/viewer/2022042804/5f53225b9b57b969ac6912f2/html5/thumbnails/81.jpg)
Homework
• home reading: diagonal method.• home reading: Presburger arithmetic.