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Reductions

Reductions. 邓小铁 上海交通大学. 计算机科学的 Reduction. （ von Neuman 的电脑 ENIAC ）. 计算机科学. 计算机应用. 逻辑电子元件. 逻辑电子线路. 用 AND-OR-NOT 元件组合的线路 Universal ：可以计算任何（变量有限）的逻辑函数 应用领域 自动机 计算机，空调，电梯，洗衣机，电视，手机， GPS 导航，电子游戏，以及许多我们今天已经难以离开的现代电子科技产品。. 可计算性理论. 可计算性理论. 图灵机 有限态控制器：状态集合，字母集合，转移规则 输入／输出纸带 等价计算体系 递归函数

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Reductions

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  1. Reductions • 邓小铁 • 上海交通大学

  2. 计算机科学的Reduction

  3. （von Neuman的电脑ENIAC）

  4. 计算机科学

  5. 计算机应用

  6. 逻辑电子元件

  7. 逻辑电子线路 • 用AND-OR-NOT元件组合的线路 • Universal：可以计算任何（变量有限）的逻辑函数 • 应用领域 • 自动机 • 计算机，空调，电梯，洗衣机，电视，手机，GPS导航，电子游戏，以及许多我们今天已经难以离开的现代电子科技产品。

  8. 可计算性理论

  9. 可计算性理论 • 图灵机 • 有限态控制器：状态集合，字母集合，转移规则 • 输入／输出纸带 • 等价计算体系 • 递归函数 • λ演算

  10. Hilbert（第二）问题 • 数学是完备的吗？ • 面对那些正确的数学陈述，我们是否总能找出一个证明？数学真理是否总能被证明？ • Godel:No • 数学是一致的吗？ • 数学是否前后一致，不会得出某个数学陈述又对又不对的结论？数学是否没有内部矛盾？ • Godel:No • 数学是可判定的吗？ • 能够找到一种方法，仅仅通过机械化的计算，就能判定某个数学陈述是对是错？数学证明能否机械化？ • HaltingProblem

  11. 2.3 停机问题 • 是否有无理数？ • 对角化证明：将所有在［0，1］间的有理数写成二进制数，排为一序列：r1,r2,…,rn… • 构造 r：r的第i位和ri的第i位不同。 • 结论：r不可以是有理数。 • 能否确定一个图灵机会停机？ • 将图灵机排序（按有限态控制器）：T1,T2,…,Tn,… • 构造T：如果机器Ti在输入Ti时停机，T在输入Ti时不停机，反之停机。 • T停机与否成为不可判定。

  12. Apply Reduction • A不可判定 • A可reduce到B • B也不可判定

  13. 其他计算工具

  14. Other computational reductions • Mechanic: Analytical Engine by Babbage • Quantum: D-Wave on sale • DNA

  15. A two player game implementing “+” • Player one: Six Strategies • Input: Nodes (a,0), (a,1), (b,0) and (b,1) • Output: nodes (c,0), (c,1) • Player two: intermediate nodes (d,0), (d,1) • Nontrivial pure strategy pairs: • s1=<(a, 1), (d, 1)>; s2=<(b, 1), (d, 1)>; s3=<(c, 1), (d, 0)>;s4=<(c, 1), (d, 1)>;s5=<(c, 0), (d, 0)>; • Payoffs • For Player one: one for s4, s5; zero otherwise • For Player two: one for s1, s2, s3; zero otherwise.

  16. a c b • Encoding by Equilibrium Probability • X: the probability of playing (a,1) • Y: the probability of playing (b,1) • Z: the probability of playing (c,1) • For Player 2: • Utility playing (d,0): Z • Utility of playing (d,1): X+Y • At equilibrium player two has the same utility choosing its pure strategies: (d,0) or (d,1) • Z=X+Y d

  17. Other operations • Can be implemented by a two player game in a similar manner. • Arithmetic operations (can be done approximately). • +, -, equal_to, assign_C, multiply_by_C • Comparator: < (must be done approximately). • Logic operations (exactly done). • ∨ • ∧ • Negation: ⌐

  18. Power of 2 Player Nash • It can solve the fixed point problem of a polynomial time computable function.

  19. 计算复杂性

  20. 复杂性：P vs. NP 问题 • P：多项式时间可以计算的问题 • NP：多项式时间可以验证“正确解”的问题

  21. P：多项式时间可以计算的问题 • 匹配：给定二部图，G＝(V1,V2;E), 如何找到边集合的子集M使得V1和V2中的每个点与E相交一条边。 • 如学生申请大学

  22. Matching Algorithm • Reduction to Network Flow problem: the latter has a polynomial time solution.

  23. NP：多项式时间可以验证“正确解”的问题 • 匹配：给定一图，G＝(V,E), 如何找到边集合E的子集C形成一个圈图， • Hamiltonian 圈：C经过G中每一点恰好一次。 • 见右图 • 欧拉圈：C经过G中每一边恰好一次。 • 如左图国王堡七桥问题

  24. NP-Hard？ • 可以在多项式时间 将3SAT问题reduce到Hamiltonain圈 3SAT is NP-hard So is Hamiltonain圈 • 欧拉圈多项式时间可解。

  25. 应用领域

  26. 这是AI吗？ • "It is my conviction that intentional phenomenology has for the first time made spirit as spirit the field of systematic scientific experience, thus effecting a total transformation of the task of knowledge.” • Edmund Husserl, Crisis of European Humanity, Pt. II, 1935

  27. 图灵测试

  28. 反向图灵测试

  29. Machine Translation

  30. Eidetic reduction • By which the philosopher moves from • the consciousness of individual and concrete objects • to • the transempirical realm of pure essences • and thus achieves an intuition of the eidos (Greek: “shape”) of a thing— • i.e., of what it is in its invariable and essential structure, apart from all that is contingent or accidental to it. • From Encyclopeadia Britannica

  31. The Art of Computer Programming • Fundamental Algorithms • SeminumericalAlgorithms • Sorting and Searching • Combinatorial Algorithms • By D. Kunth

  32. Deep Learning

  33. What does big data bring us? • 逻辑关系？ • Wisdom of the crowd？ • Network analysis?

  34. Life as it could be • The invention of the computer has revolutionized science. With respect to finding the essential structures of life, for example, it has enabled scientists not only to investigate empirical examples, but also to create and study novel hypothetical variations by means of simulation: ‘life as it could be’. • Tom Froese & Shaun Gallagher (2010). Phenomenology and Artificial Life: Toward a Technological Supplementation of Phenomenological Methodology.Husserl Studies 26 (2):83-106.

  35. Money as it could be

  36. Stone money • In Yap island, called ‘fei’ • Made of particular variety and quality of limestone • Transferable. • Ownership Public Recognized Source: http://www.visit-fsm.org

  37. Role of E-Money • Support activities over the Internet.

  38. Design of a Internet Financial Market: Bitcoin • Aim to recreate the concept of cash-based shopping over the Internet. • Use cryptographic techniques and protocols • New thinking.

  39. Outline Bitcoin • Origin of funds: Out of thin air (无中生有) • Bitcoin Mining: Anyone can join the effort to mine new bitcoins. • The protocol is preset and agreed by all who participate, enforced by computational complexity. • Internet commercial activities are easily supported.

  40. The Bitcoin Design

  41. Record all transactions and moves of money • 每一笔交易被纪录下来。 • 最初的交易由创始人发行bitcoin促成。 • Bitcoin由交易转换它的拥有者。 • 每个人拥有的Bitcoin是社区的common knowledge。 • 对比 Fei：almost the same.

  42. 新发Bitcoin • 纪录每一笔交易的人获得交易费用。 • 纪录每一笔交易的人获得新发行的bitcoin。 • 纪录交易有计算难度（用hash function） • 新发行的bitcoin按指数递减，直到发行完。

  43. Bitcoin的优点 • 用历史纪录保证安全性：难以伪造。 • 公钥体系保证匿名性。 • 总数有限，不会通货膨胀。 • 公钥保证不能重复使用。 • 货币体系安全由全体矿工保证。

  44. Bitcoin可能的弱点 • 依赖历史纪录。 • 对普通人可能产生其他安全漏洞。

  45. Bitcoin的创新性 • Money的交易特性而非其价值特性通过bitcoin体现出来。 • 没有准备金没有银行的货币体系。 • 用电子货币的特征，新建incentive机制。

  46. Summary

  47. Computer Science at Internet Age： • 有大量创新（甚至“创世纪”）空间： • 放弃表象，重新审视世界和社会基本元素： reduction

  48. 谢谢大家！

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