四色定理及其计算机证明

为了黑这个：“OpenAI发文表示，他们已经为Lean创建了一个神经定理证明器，用于解决各种具有挑战性的高中奥林匹克问题，包括两个改编自IMO的问题和来自AMC12、AIME竞赛的若干问题。该证明器使用一个语言模型来寻找形式化命题(formal statement)的证明。”

The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken after many false proofs and counterexamples (unlike the five color theorem, proved in the 1800s, which states that five colors are enough to color a map)...

The Appel and Haken proof attracted a fair amount of criticism. Part of it concerned the proof style: the statement of the Four Colour Theorem is simple and elegant so many mathematicians expected a simple and elegant proof that would explain, at least informally, why the theorem was true - not opaque IBM 370 assembly language programs.

用Bing搜(ibm 370 1976), https://www.ibm.com/ibm/history/exhibits/mainframe/mainframe_PP3148.html

System/370 Model 148

The new model also offers increased system throughput -- the amount of time it takes to perform a given amount of work -- compared to the Models 135 and 145. The Model 148 is available with 1,048,576 or 2,097,152 characters of memory. 高达1MB或2MB内存。同期Apple I默认4KB内存，CPU主频是1MHz. The Model 148, under the 48-month contract, can be leased for $17,280 a month with one million characters of memory and for $22,650 a month with two million characters. Monthly rental prices are $19,000 and $24,900. Purchase prices are $689,000 and $859,000. 2021年黄金价格约为1976年黄金价格的14倍。1.05⁵⁰ = 11.48

https://www.ibm.com/ibm/history/exhibits/mainframe/mainframe_FS370B.html

CPU是个啥？Cycle Time (nanoseconds), Model 115 480ns, Model 148 180ns。按200ns算，there are a billion nanoseconds in a second, 主频5MHz。显然是单核，相当显然乘法需要不止一个时钟周期。

• 现在台式机CPU主频接近5G, 10核20线程 (Intel i9)，服务器CPU主频低些，核数更多。5G是5M的1000倍。
• OpenAI有可能用了上千台服务器，每服务器可能22核。每服务器内存按64GB算，1000 * 64GB / 2MB = ?
• DDR4内存带宽是DDR内存的多少倍？System/370有CPU片内cache吗？Intel 80486

OpenAI的卖点是“使用一个语言模型”。比如说鸡兔同笼问题，不用晦涩地输入5, 16两个数，而是“OpenAI啊，鸡和兔共5个，腿共16条，鸡兔各几只？”

老笑话“小明购书两本花了八角，三角三角，几何几何？”，OpenAI目前可能还解决不了。用机器翻译给翻成了英文"Xiao Ming bought two books at the cost of octagons, trigonometry, geometry?" How thoughtful am I. Aren't I sweet? “我想得多周到啊，我不是很可爱吗？” [这个翻译真心佩服 :-)] 

几何定理机器证明_360百科 (so.com) OpenAI能做数学题是厉害的，但包了个花哨的界面，再经媒体一渲染，就神乎其神好卖股票了(没产品可卖:-))。

76年证四色定理花了1200小时，90年代算法改进了，计算和存储能力提高了，只花了24小时。现在CPU单核的性能不是继续不断翻倍，而是快到顶了。有些问题已被证明计算量就是非常大。所以窃以为在革命性的技术比如通用量子计算机出现之前，AI又要进入冬天了。现在的n万亿参数，4096张显卡，千张人脸等有点像“上帝要让它灭亡，先让它疯狂。” 就算64核的CPU降到640甚至64元，电费呢？

In the history of artificial intelligence, an AI winter is a period of reduced funding and interest in artificial intelligence research. The term was coined by analogy to the idea of a nuclear winter. The field has experienced several hype cycles, followed by disappointment and criticism, followed by funding cuts, followed by renewed interest years or decades later. 有"核冬天"这么严重吗？有些人为了拿经费也是拼了。:-)

牛津词典: Her artificial gaiety disguised an inner sadness. 她以矫揉造作的快乐掩饰着内心的悲痛。[affected; insincere; not genuine，名字都没起好]

There were two major winters in 1974–1980 and 1987–1993[6] and several smaller episodes, including the following:

• 1966: failure of machine translation
• 1970: abandonment of connectionism

Period of overlapping trends:

• 1971-75: DARPA's frustration with the Speech Understanding Research program at Carnegie Mellon University
• 1973: large decrease in AI research in the United Kingdom in response to the Lighthill report
• 1973-74: DARPA's cutbacks to academic AI research in general
• 1987: collapse of the LISP machine market
• 1988: cancellation of new spending on AI by the Strategic Computing Initiative
• 1993: resistance to new expert systems deployment and maintenance
• 1990s: end of the Fifth Generation computer project's original goals

In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet. It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. Since then the proof has gained wide acceptance, although some doubters remain.

  

As far as is known, the conjecture was first proposed on October 23, 1852, when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed. At the time, Guthrie's brother, Frederick, was a student of Augustus De Morgan (the former advisor of Francis) at University College London. Francis inquired with Frederick regarding it, who then took it to De Morgan (Francis Guthrie graduated later in 1852, and later became a professor of mathematics in South Africa). According to De Morgan:

"A student of mine [Guthrie] asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be any how divided and the compartments differently colored so that figures with any portion of common boundary line are differently colored - four colors may be wanted but not more - the following is his case in which four colors are wanted. Query cannot a necessity for five or more be invented…" (Wilson 2014, p. 18)

"F.G.", perhaps one of the two Guthries, published the question in The Athenaeum in 1854, and De Morgan posed the question again in the same magazine in 1860. Another early published reference by Arthur Cayley (1879) in turn credits the conjecture to De Morgan.

六级/考研单词: lean, elegant, mathematics, opaque, assemble, exhibit, lease, million, billion, adjacent, curve, segment, mere, compute, notify, inquiry, graduate, professor, accord, compartment, portion, query, necessity, pose