Mathematics & Logic Computer Science CSCI 341 Theory of Computation Introduction CSCI 341 - Bucknell University Fall 2013
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Mathematics & Logic Computer Science CSCI 341
Theory of ComputationIntroduction
CSCI 341 - Bucknell University
Fall 2013
Mathematics & Logic Computer Science CSCI 341
David Hilbert (1862-1943)
In 1900, Hilbert challenged the community of mathematicians with23 problems.
Mathematics & Logic Computer Science CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Description of the language of Mathematics (Formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove theorems)
Mathematics & Logic Computer Science CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Description of the language of Mathematics (Formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove theorems)
Mathematics & Logic Computer Science CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Description of the language of Mathematics (Formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove theorems)
Mathematics & Logic Computer Science CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Description of the language of Mathematics (Formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove theorems)
Mathematics & Logic Computer Science CSCI 341
Hilbert’s program
Secure the foundations of mathematics.
• Description of the language of Mathematics (Formal logic)
• Completeness (all true statements should be provable)
• Consistency (no contradiction should be found)
• Decidability (automatically prove theorems)
Mathematics & Logic Computer Science CSCI 341
Bertrand Russell (1872-1970)
Provide the first formal logic for mathematics in PrincipiaMathematica(joint work with Whitehead in 1910).
Mathematics & Logic Computer Science CSCI 341
Principia Mathematical 1910
.
Mathematics & Logic Computer Science CSCI 341
Kurt Godel (1906-1978)
• Godel’s Incompleteness Theorem (1st):In an effective logic system, there are statements that cannotbe proved nor disproved.
• Godel’s Incompleteness Theorem (2nd):In an effective logic system, the consistency of the system isnot provable within the system.
Mathematics & Logic Computer Science CSCI 341
Kurt Godel (1906-1978)
• Godel’s Incompleteness Theorem (1st):In an effective logic system, there are statements that cannotbe proved nor disproved.
• Godel’s Incompleteness Theorem (2nd):In an effective logic system, the consistency of the system isnot provable within the system.
Mathematics & Logic Computer Science CSCI 341
Godel is a VIP(the pizzeria slide)
Mathematics & Logic Computer Science CSCI 341
Alan Turing (1912-1954)
Break the enigma code and a pioneer in designing computers.
Undecidability: There are problems that cannot be solvedautomatically.
Mathematics & Logic Computer Science CSCI 341
Breaking Enigma Code
Machine used by Germany for encrypting information during WW2.
Mathematics & Logic Computer Science CSCI 341
Kleene, Rabin & Scott
In the 1950’s, pioneers of the Theory of Automata (Regularexpressions).
Mathematics & Logic Computer Science CSCI 341
Chomsky & Schutzenberger
In the 1960’s, pioneers in the study of Formal Grammar.
Mathematics & Logic Computer Science CSCI 341
Understanding language
Mathematics & Logic Computer Science CSCI 341
Stephen Cook
State the P vs NP problem in the 1970’s.
Mathematics & Logic Computer Science 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
Mathematics & Logic Computer Science 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
Mathematics & Logic Computer Science 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 90 different proofs listed onhttp://www.win.tue.nl/~gwoegi/P-versus-NP.htm
Mathematics & Logic Computer Science CSCI 341
Christos Papadimitriou
Major contributor of the Computational Complexity Theory.Author of your Textbook !
Mathematics & Logic Computer Science CSCI 341
The life of Russell in a Comic Book
Mathematics & Logic Computer Science CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of compution
• Practice formal proofs
• Implement the theory (Project)
• Critical thinking on article
Mathematics & Logic Computer Science CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of compution
• Practice formal proofs
• Implement the theory (Project)
• Critical thinking on article
Mathematics & Logic Computer Science CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of compution
• Practice formal proofs
• Implement the theory (Project)
• Critical thinking on article
Mathematics & Logic Computer Science CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of compution
• Practice formal proofs
• Implement the theory (Project)
• Critical thinking on article
Mathematics & Logic Computer Science CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of compution
• Practice formal proofs
• Implement the theory (Project)
• Critical thinking on article
Mathematics & Logic Computer Science CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of compution
• Practice formal proofs
• Implement the theory (Project)
• Critical thinking on article
Mathematics & Logic Computer Science CSCI 341
CSCI 341Theory of Computation
What are you going to learn during this semester?
• History
• Foundations
• Models of compution
• Practice formal proofs
• Implement the theory (Project)
• Critical thinking on article
Mathematics & Logic Computer Science CSCI 341
See you on Friday !
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