Impagliazzos Personal View of Five possible worlds Definitional - PowerPoint PPT Presentation

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Impagliazzo’s Personal View of Five possible worlds Definitional issues Average-Case Complexity summarized Hörður Helgi Helgason University of Iceland, Faculty of Industrial Engineering, Mechanical Engineering and Computer Science November 10, 2008

Impagliazzo’s Personal Levin and Impagliazzo View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

Impagliazzo’s Personal Levin and Impagliazzo View of Average-Case Complexity H.H.Helgason Professor Leonid A. Levin, BU CS Abstract Introduction ◮ Average Case Complete Problems, SIAM J. Comput. (1986) Five possible worlds 15(1):285-286 Definitional issues ◮ A structural theory of average-case complexity ◮ Many NP-complete problems fast, on average ◮ Distinguishing difficult-on-average problems is beneficial ◮ Save positive efforts ◮ Verify hardness where required, e.g. cryptography

Impagliazzo’s Personal Levin and Impagliazzo View of Average-Case Complexity H.H.Helgason Professor Leonid A. Levin, BU CS Abstract Introduction ◮ Average Case Complete Problems, SIAM J. Comput. (1986) Five possible worlds 15(1):285-286 Definitional issues ◮ A structural theory of average-case complexity ◮ Many NP-complete problems fast, on average ◮ Distinguishing difficult-on-average problems is beneficial ◮ Save positive efforts ◮ Verify hardness where required, e.g. cryptography Professor Russell Impagliazzo, UCSD CS ◮ A Personal View of Average-Case Complexity, Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT’95): 134-147 ◮ Summarize state of knowledge ◮ Motivate more research

Impagliazzo’s Personal What is needed? View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

Impagliazzo’s Personal What is needed? View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Conventional completeness results can be relatively Definitional issues meaningless

Impagliazzo’s Personal What is needed? View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Conventional completeness results can be relatively Definitional issues meaningless . . . but average-run-time arguments are also unenlightening:

Impagliazzo’s Personal What is needed? View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Conventional completeness results can be relatively Definitional issues meaningless . . . but average-run-time arguments are also unenlightening: A structural theory of distributional complexity is needed

Impagliazzo’s Personal What is needed? View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Conventional completeness results can be relatively Definitional issues meaningless . . . but average-run-time arguments are also unenlightening: A structural theory of distributional complexity is needed ◮ to allow comparisons of the inherent intractability of distributional problems ◮ to provide meaningful results from arbitrary distributions

Impagliazzo’s Personal The five worlds View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Algorithmica Heuristica Pessiland Minicrypt Cryptomania Definitional issues

Impagliazzo’s Personal The five worlds View of Average-Case Complexity H.H.Helgason Abstract Questions regarding the average case complexity of Introduction problems in NP have five possible answers Five possible worlds Algorithmica Heuristica Pessiland Minicrypt Cryptomania Definitional issues

Impagliazzo’s Personal The five worlds View of Average-Case Complexity H.H.Helgason Abstract Questions regarding the average case complexity of Introduction problems in NP have five possible answers Five possible worlds Algorithmica Heuristica Pessiland Benefits for AI, VLSI, Cryptography, Comp Sec, Minicrypt Cryptomania Derandomization of algorithms Definitional issues

Impagliazzo’s Personal The five worlds View of Average-Case Complexity H.H.Helgason Abstract Questions regarding the average case complexity of Introduction problems in NP have five possible answers Five possible worlds Algorithmica Heuristica Pessiland Benefits for AI, VLSI, Cryptography, Comp Sec, Minicrypt Cryptomania Derandomization of algorithms Definitional issues Carl Friedrich Gauß, and his teacher ◮ B¨ uttner constructed an apparently hard problem and posed it to his students, including Gauß, who demonstrated a quick method to solve it, but. . .

Impagliazzo’s Personal The five worlds View of Average-Case Complexity H.H.Helgason Abstract Questions regarding the average case complexity of Introduction problems in NP have five possible answers Five possible worlds Algorithmica Heuristica Pessiland Benefits for AI, VLSI, Cryptography, Comp Sec, Minicrypt Cryptomania Derandomization of algorithms Definitional issues Carl Friedrich Gauß, and his teacher ◮ B¨ uttner constructed an apparently hard problem and posed it to his students, including Gauß, who demonstrated a quick method to solve it, but. . . ◮ . . . what if B¨ uttner had been a complexity expert, and the main questions abour average-case complexity had been resolved? How would he fare in the five worlds?

Impagliazzo’s Personal P = NP View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Algorithmica Heuristica Pessiland Minicrypt Cryptomania Definitional issues

Impagliazzo’s Personal P = NP View of Average-Case Complexity H.H.Helgason B¨ uttner less likely to succeed: Abstract Introduction Five possible worlds Algorithmica Heuristica Pessiland Minicrypt Cryptomania Definitional issues

Impagliazzo’s Personal P = NP View of Average-Case Complexity H.H.Helgason B¨ uttner less likely to succeed: To produce easily demonstratable Abstract solutions in class, B¨ uttner is confined to NP, now P-solvable Introduction Five possible worlds Algorithmica Heuristica Pessiland Minicrypt Cryptomania Definitional issues

Impagliazzo’s Personal P = NP View of Average-Case Complexity H.H.Helgason B¨ uttner less likely to succeed: To produce easily demonstratable Abstract solutions in class, B¨ uttner is confined to NP, now P-solvable Introduction Five possible worlds Revolution in comp sci Algorithmica Heuristica Pessiland Minicrypt Cryptomania Definitional issues

Impagliazzo’s Personal P = NP View of Average-Case Complexity H.H.Helgason B¨ uttner less likely to succeed: To produce easily demonstratable Abstract solutions in class, B¨ uttner is confined to NP, now P-solvable Introduction Five possible worlds Revolution in comp sci Algorithmica Heuristica ◮ VLSI would be optimized: Heuristics no more Pessiland Minicrypt Cryptomania Definitional issues

Impagliazzo’s Personal P = NP View of Average-Case Complexity H.H.Helgason B¨ uttner less likely to succeed: To produce easily demonstratable Abstract solutions in class, B¨ uttner is confined to NP, now P-solvable Introduction Five possible worlds Revolution in comp sci Algorithmica Heuristica ◮ VLSI would be optimized: Heuristics no more Pessiland Minicrypt ◮ AI: Inductive learning instead of expert systems: Cryptomania Definitional issues

Impagliazzo’s Personal P = NP View of Average-Case Complexity H.H.Helgason B¨ uttner less likely to succeed: To produce easily demonstratable Abstract solutions in class, B¨ uttner is confined to NP, now P-solvable Introduction Five possible worlds Revolution in comp sci Algorithmica Heuristica ◮ VLSI would be optimized: Heuristics no more Pessiland Minicrypt ◮ AI: Inductive learning instead of expert systems: Feed an Cryptomania Definitional issues algorithm a training set and it produces the simplest algorithm that produced the same results as an expert system

Impagliazzo’s Personal P = NP View of Average-Case Complexity H.H.Helgason B¨ uttner less likely to succeed: To produce easily demonstratable Abstract solutions in class, B¨ uttner is confined to NP, now P-solvable Introduction Five possible worlds Revolution in comp sci Algorithmica Heuristica ◮ VLSI would be optimized: Heuristics no more Pessiland Minicrypt ◮ AI: Inductive learning instead of expert systems: Feed an Cryptomania Definitional issues algorithm a training set and it produces the simplest algorithm that produced the same results as an expert system Loss of informational-based distinction ◮ No way of telling people, computers apart with information ◮ Any code could be easily broken