theory of computation introduction - bucknell universitycsci341/2016-fall/notes/history.pdf ·...
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![Page 1: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/1.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Theory of ComputationIntroduction
CSCI 341 - Bucknell University
Fall 2016
![Page 2: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/2.jpg)
Mathematics & Logic Theory of Computation CSCI 341
David Hilbert (1862-1943)
• In 1900, Hilbert challenged the community of mathematicianswith 23 problems.
• Some problems specifically related to the so called Hilbert’sprogram.
![Page 3: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/3.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Formal language for mathematics (formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove/disprove statements)
![Page 4: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/4.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Formal language for mathematics (formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove/disprove statements)
![Page 5: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/5.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Formal language for mathematics (formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove/disprove statements)
![Page 6: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/6.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Formal language for mathematics (formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove/disprove statements)
![Page 7: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/7.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Formal language for mathematics (formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove/disprove statements)
![Page 8: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/8.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Bertrand Russell (1872-1970)
Provide the first formal logic for mathematics inPrincipia Mathematica(joint work with Whitehead in 1910).
![Page 9: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/9.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Principia Mathematical 1910
.
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Mathematics & Logic Theory of Computation CSCI 341
Kurt Godel (1906-1978)
• Godel’s Incompleteness Theorem (1st):In an effective logic system, there exist statements thatcannot be proved nor disproved. There is a true statementthat cannot be proved.
• Godel’s Incompleteness Theorem (2nd):In an effective logic system, the consistency of the system isnot provable within the system.
![Page 11: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/11.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Kurt Godel (1906-1978)
• Godel’s Incompleteness Theorem (1st):In an effective logic system, there exist statements thatcannot be proved nor disproved. There is a true statementthat cannot be proved.
• Godel’s Incompleteness Theorem (2nd):In an effective logic system, the consistency of the system isnot provable within the system.
![Page 12: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/12.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Godel is a VIP(the pizzeria slide)
![Page 13: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/13.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Alan Turing (1912-1954)
Turing machines, break the enigma code and a pioneer incomputer design.
Undecidability: There are problems that cannot be solvedautomatically.
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Mathematics & Logic Theory of Computation CSCI 341
Breaking Enigma Code
Machine used by Germany for encrypting information during WW2.
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Mathematics & Logic Theory of Computation CSCI 341
Kleene, Rabin & Scott
In the 1950’s, pioneers of the Theory of Automata (Regularexpressions).
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Mathematics & Logic Theory of Computation CSCI 341
Chomsky & Schutzenberger
In the 1960’s, pioneers in the study of Formal Languages.
![Page 17: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/17.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Understanding language
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Mathematics & Logic Theory of Computation CSCI 341
Cook, Papadimitriou and Sipser
• Major contributors to the Complexity Theory, the theory oftime-space resources necessay to solve problems.
• Cook: first stated the P vs NP problem in the 1970’s.
![Page 19: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/19.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Reward
The Clay Mathematics Institute is offering a
US$1 million reward
to anyone who has a formal proof that P = NP or that P 6= NP.
There are already more than 70 different proofs listed onhttp://www.win.tue.nl/~gwoegi/P-versus-NP.htm
![Page 20: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/20.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Reward
The Clay Mathematics Institute is offering a
US$1 million reward
to anyone who has a formal proof that P = NP or that P 6= NP.
There are already more than 70 different proofs listed onhttp://www.win.tue.nl/~gwoegi/P-versus-NP.htm
![Page 21: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/21.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Reward
The Clay Mathematics Institute is offering a
US$1 million reward
to anyone who has a formal proof that P = NP or that P 6= NP.
There are already more than 115 different proofs listed onhttp://www.win.tue.nl/~gwoegi/P-versus-NP.htm
![Page 22: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/22.jpg)
Mathematics & Logic Theory of Computation CSCI 341
The life of Russell in a Comic Book
![Page 23: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/23.jpg)
Mathematics & Logic Theory of Computation CSCI 341
Introduction to the Theory of ComputationSipser, 3rd Edition
• Ch 0. Mathematical background
• Ch 1-2. Automata and Languages
• Ch 3-4-5. Computability Theory
• Ch 7-8-9. Complexity Theory
![Page 24: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/24.jpg)
Mathematics & Logic Theory of Computation CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of computation
• Practice formal reasoning
• Implement the theory
![Page 25: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/25.jpg)
Mathematics & Logic Theory of Computation CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of computation
• Practice formal reasoning
• Implement the theory
![Page 26: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/26.jpg)
Mathematics & Logic Theory of Computation CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of computation
• Practice formal reasoning
• Implement the theory
![Page 27: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/27.jpg)
Mathematics & Logic Theory of Computation CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of computation
• Practice formal reasoning
• Implement the theory
![Page 28: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/28.jpg)
Mathematics & Logic Theory of Computation CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of computation
• Practice formal reasoning
• Implement the theory
![Page 29: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/29.jpg)
Mathematics & Logic Theory of Computation CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of computation
• Practice formal reasoning
• Implement the theory
![Page 30: Theory of Computation Introduction - Bucknell Universitycsci341/2016-fall/notes/history.pdf · Mathematics & LogicTheory of ComputationCSCI 341 Theory of Computation Introduction](https://reader031.vdocuments.mx/reader031/viewer/2022030805/5b1427037f8b9a487c8b8851/html5/thumbnails/30.jpg)
Mathematics & Logic Theory of Computation CSCI 341
See you tomorrow!