IITBombay Math Proofs Computing 1 Mathematics Proofs and Computation Madhu Sudan Harvard Logic Mathematics Proofs Reasoning Start with body of knowledge Add to body of knowledge by new observations and new deductions ID: 562591
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January 4, 2016
IIT-Bombay: Math, Proofs, Computing
1
Mathematics, Proofs and Computation
Madhu
Sudan
HarvardSlide2
Logic, Mathematics, Proofs
Reasoning:Start with body of knowledge.Add to body of knowledge by new observations, and new deductionsProcess susceptible to errors:
One erroneous observation may propagate.Constant process of “consistency checking”.Mathematics = Language of PrecisionCaptures (subset of) knowledge precisely.
Proofs: Enable checking of consistency of precisely stated facts.January 4, 2016IIT-Bombay: Math, Proofs, Computing2Slide3
In this talk: Proofs and Computation
“Computer Assisted Proofs ?”[Appel-Haken] – 4-color theorem
[Hales] – Kepler Conjecture[Petkovsky,Wilf,Zeilberger] – “A=B”
January 4, 2016IIT-Bombay: Math, Proofs, Computing3No!
Mathematics
Computing
ProofsSlide4
Formal Logic
Attempts to convert reasoning to symbolic manipulation.Remarkably powerful.Originated independently, and with different levels of impact, in different civilizations …
January 4, 2016
IIT-Bombay: Math, Proofs, Computing4"Aristotle Altemps Inv8575" by Copy of Lysippus - Jastrow (2006). Licensed under Public Domain via Commons - https://commons.wikimedia.org/wiki/File:Aristotle_Altemps_Inv8575.jpg#/media/File:Aristotle_Altemps_Inv8575.jpg
Mathematics
ProofsSlide5
George Boole (1815-1864)
The strange math of
Typical Derivation:
Axiom: Repitition does not add knowledgeFormally: Example: Object is Good and Good Object is GoodConsequence: Principle of Contradiction“… it is impossible for any being to possess a quality and at the same time to not possess it.”Proof:
or
or
does not
hold
January 4, 2016
IIT-Bombay: Math, Proofs, Computing
5
(page 34)
Mathematics
ProofsSlide6
vs.
Boole’s Mathematics:
Focus on tiny part of mathematical universe. January 4, 2016IIT-Bombay: Math, Proofs, Computing6
: Algebra/Calculus
: optimization
: number theory
{0,1}
Progress
In MathSlide7
Boole’s “modest” ambition
“The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind
.” [G.Boole, “On the laws of thought …” p.1]
January 4, 2016IIT-Bombay: Math, Proofs, Computing7
: Algebra/Calculus
: optimization
: number theory
{0,1}
Mathematics
All of reasoning
Slide8
Whither Computing?
How well does the logic capture mathematics?January 4, 2016
IIT-Bombay: Math, Proofs, Computing
8Cantor‘1890: Logic may face some problems?Hilbert ‘1900:Should capture everything!
Godel
‘1920s:
Incompleteness
Church-Turing 1930s: Incompleteness holds for any effective reasoning procedure.
This statement
is not
true
provableSlide9
January 4, 2016
IIT-Bombay: Math, Proofs, Computing
9Turing’s Machine
Model of computerFinite State Control
R/W
Universal
Machine
Encodings of other machines
One machine to rule them all!
→ von Neumann architecture
CPU
RAM
-
Universal!
Mathematics
Computing
ProofsSlide10
Proofs: Story so far
Proof: Has to be mechanically verifiable.Theorem:
Statement with a proof. Incompleteness: There exist statements consistent with the system of logic that do not admit a proof.
Unaddressed: What difference does proof make?January 4, 2016IIT-Bombay: Math, Proofs, Computing10Theorem: Proof:
Has
lines
Theorem:
Proof:
Both mechanically verifiable!
#steps
Slide11
Origins of Modern Complexity
[Gödel 1956] in letter to von Neumann: “Is there a more “effective” procedure to find proof of length
if one exists?” (in
steps? ?)[Cobham, Edmonds, Hartmanis, Stearns – 60s]:Time Complexity is a (coarse) measure.
But
.
problems solvable in time
for constant
Edmonds Conjecture: Travelling Salesman Problem is not solvable in
January 4, 2016
IIT-Bombay: Math, Proofs, Computing
11Slide12
Proofs, Complexity & Optimization!
January 4, 2016
IIT-Bombay: Math, Proofs, Computing
12[Cook ’70] Complexity of Theorem Proving[Levin ’71] Universal Search problems
Formalized Edmond’s Conjecture:
Problems w. efficiently verifiable solutions
-complete
= Hardest problem in NP
Theorem-Proving
NP-Complete
SAT
(simple format of proofs) NP-complete
Domino tiling
NP-Complete
Godel’s
question
“Is
?”
Slide13
Proofs, Complexity & Optimization - 2
Showed central importance of
. Nineteen problems
-Complete!Cover optimization, logic, combinatorics, graph theory, chip design. January 4, 2016IIT-Bombay: Math, Proofs, Computing13[Karp ‘72] Reducibility among combinatorial optimization problemsSlide14
Some NP-complete Problems
Map Coloring: Can you color a given map with 3-colors, s.t. bordering states have diff. colors?
January 4, 2016
IIT-Bombay: Math, Proofs, Computing14Slide15
Some NP-Complete Problems
Travelling Salesman Problem: (TSP) – Find tour of minimum length visiting given set of cities.
January 4, 2016IIT-Bombay: Math, Proofs, Computing
15Image due to [Applegate, Bixby, Chvatal, Cook]. Optimal TSP visiting ~13000 most populated cities in US.Slide16
Some NP-Complete Problems
Biology: Fold DNA sequence so as to minimize energy.Economics: Finding optimal portfolio of stocks subject to budget constraint.
Industrial Engineering: Schedule tasks subject to precedence constraints to minimize completion time.…
January 4, 2016IIT-Bombay: Math, Proofs, Computing16Slide17
Consequences to Proof Checking
NP-Complete problem
Format for proofs.3-coloring is NP-complete
exists function Map with regions s.t. has proof of length Map is 3-colorable no proofs of length Map not 3-colorable
Format?
Rather than convention proof, can simply give coloring of map!
Advantage: Error is local (two improperly colored regions)
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IIT-Bombay: Math, Proofs, Computing
17
Verifier computes
and
verifies coloring is good
Slide18
Don’t know …
If P=NP …Cryptography might well be impossible (current systems all broken simultaneously)
All optimization problems become “easy”… You get whatever you wish … if you
can verify satisfaction.Mathematicians replaced by computers.If P≠NP … … Consistent with current thinking, so no radical changes.Proof would be very educational.Might provide sound cryptosystems.Independent of Peano’s axioms, Choice …?January 4, 2016IIT-Bombay: Math, Proofs, Computing18Is P=NP?“Of all the Clay Problems, this might be the one to find the shortest solution, by an amateur mathematician.”- Devlin
,
The
Millenium
Problems (Possibly
thinking
P=NP
)
“
If someone shows P=NP, then they prove any theorem they wish. So they would walk away not just with $1M, but $6M by solving all the Clay Problems
!”
- Lance
Fortnow
,
Complexity Blog
“P
= NP
?”
is
Mathematics-Complete !!Slide19
Post-Modern Complexity
Emphasis on Randomness.Randomness can potentially speed up algorithms.Essential for
Equilibrium behavior Coordination among multiple playersCryptographyBut it probably can’t help with Logic – right?
Actually – it does!!January 4, 2016IIT-Bombay: Math, Proofs, Computing19Slide20
Interactive Proofs
[
Goldwasser, Micali, Rackoff], [Babai
] ~1985Verifier asks questions and Prover responds:Space of questions exponentially large in the length!Prover has to be ready for all!Many striking examples:Pepsi Coke! (“Graphs not isomorphic”)Can prove “theorem has no short proof”.“IP = PSPACE” [LFKN, Shamir]“Zero Knowledge Protocols” – Foundations of Secure communication January 4, 2016IIT-Bombay: Math, Proofs, Computing20Slide21
Probabilistically Checkable Proofs
Do proofs have to be read in entirety to verify?
January 4, 2016IIT-Bombay: Math, Proofs, Computing
21Slide22
Probabilistically Checkable Proofs
Do proofs have to be read in entirety to verify?Conventional formats for proofs – YES!
But we can change the format!Format Verification Algorithm
Any verifier is ok, provided:If has proof of length in standard system, then should accept some proof of length poly If has no proofs, then should not accept any proof PCP Theorem [Arora, Lund, Motwani, Safra, Sudan, Szegedy ‘92]:A format exists where V reads onlyconstant number of bits of proof!
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IIT-Bombay: Math, Proofs, Computing
22
with
probability
X
Slide23
PCPs and Optimization
Classical connection: [Cook
Karp]:Solving optimization problems finding proofs
New Connection: [Feige et al., Arora et al.]Solving optimization problems approximately finding nearly valid proofs.Existence of nearly valid proofs Existence of perfectly valid proofs (due to PCPs)!Conclude: Solving (some/many) optimizations approximately is as hard as solving them exactly!1992-today: PCP-induced revolution in understanding approximability!! January 4, 2016IIT-Bombay: Math, Proofs, Computing23Slide24
Summary and Conclusions
Computing is a science:Goes to the very heart of scientific inquiry.What big implications follow from local steps?
Search for proofs captures essence of all search and optimization.“Is P=NP?” Central mathematical question.
Still open.But lots of progress …“Khot’s UGC” (Unique Games Conjecture): Cutting edge of optimization. Khot’s Unique Games Conjecture: Cutting edge of optimization. Sharp thresholds almost everywhere?Communication of Knowledge: Will we use PCPs to communicate math proofs?Will we use PCPs anywhere?Computing is a science!!What global changes can/can not be effected by sequence of local changes?January 4, 2016IIT-Bombay: Math, Proofs, Computing24Slide25
Thank You!
January 4, 2016
IIT-Bombay: Math, Proofs, Computing
25