不完全性定理入門 from「コンピュータは数学者になれるのか」
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コンピュータサイエンスの文脈で不完全性定理を考えます。「コンピュータは数学者になれるのか」という本の2章をまとめています。こちらの本を読む際に、まとめとして本スライドを活用してもらえれば幸いです。
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1:42-History of C language
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✅C++ Career Prospects:
With just C++ programming expertise, you will have excellent job opportunities, salaries, and career prospects. However, for a career based on programming languages such as Java and Python (which are in more demand than C++) or for careers based on front-end, back-end, and full-stack
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This presentation was designed to provide strategic recommendations for a brand in decline. The deck also incorporates a situational assessment, including a brand identity, positioning, architecture, and portfolio strategy for the Brand.
Presentation originally created for NYU Stern's Brand Strategy course. Design by Erica Santiago & Chris Alexander.
Good Stuff Happens in 1:1 Meetings: Why you need them and how to do them well
According to the latest State of the American Manager report from Gallup, employees who have regular meetings with their managers are almost three times as likely to be engaged as those who don’t. These regular check-ins keep managers and employees in sync and aligned. Want to see better manager/employee relationships in your organisation? Then make an all-in commitment to 1:1 meetings. Not sure how? You’ve come to the right place.
In this webinar with Jamie Resker, Founder and Practice Leader for Employee Performance Solutions (EPS), and Teala Wilson, Talent Management Consultant at Saba Software, you’ll get the inside track on how to hold effective 1:1 meetings, including tips for getting managers on board.
• Go beyond discussing the status of everyday work to higher level topics, including recognition, performance, development, and career aspirations
• Learn how to decide meeting frequency, what to cover, as well as roles and responsibilities of the manager and employee
• Understand how managers can build trust and make it comfortable for employees to provide upward feedback
• Unite your organisation with a unified approach to 1:1 meetings
Join us for this 1-hour webinar to get practical tips for building better manager-employee relationships with intention and purpose.
About the Speakers
Jamie Resker - Founder and Practice Leader for Employee Performance Solutions (EPS)
Jamie Resker, Practice Leader and Founder of Employee Performance Solutions, is a recognized innovator in performance management. She is the originator of the-the Performance Continuum Feedback Method® and Conversations to Optimize Employee Performance training program; tools and training that reshape communications between managers and employees to drive and align performance. Jamie is on the faculty for the Northeast Human Resources Association, is a contributor to Halogen Software's Talent Space Blog, and is an editorial advisory board member for HR Examiner.
Teala Wilson - Senior Consultant, Strategic Services, Saba Software
Teala is a Talent Management Consultant at Halogen Software, now a part of Saba Software. She has worked with teams on a national and global level supporting human resources in areas such as performance management, recruitment, employee benefit programs, training and talent development, workforce planning and internal communications. Teala also has a personal passion for visual arts and design.
Want to learn more? Join us for an upcoming Product Tour!
http://bit.ly/2yitfqu
Introduction to C Programming Language
This video by Simplilearn will explain to you Introduction to C Programming Language. Introduction to C Programming Language Tutorial For Beginners will explain to you the C language's history, C's importance, its features, real-world applications, and some of its advantages and disadvantages.
00:00 Introduction to C
1:42-History of C language
Dennis Ritchie, a computer scientist, could identify the gaps and tap out the best features from both B and BCPL languages to invent a new hybrid.
Hence, C was born in 1972 at Bell Laboratories. A remarkably simple and highly readable programming language resulted in groundbreaking advancements in the IT industry.
2:48-Importance and unraveling the powerful capabilities of C,
The widespread use of C started to take over the IT industry. Unraveling the potential of C, the designers began to discover new possibilities that led them to focus on the big picture.
3:56-C's cutting-edge features
The designers at Bell Laboratories ensured that their programming language solved the issues with B and BCPL and the ones they had foreseen.
6:35-The popular real-world applications of C
-UNIX operating system
-google file system
-Mozilla
-Graphical user interface
8:30-The advantages and disadvantages of C
10:34-The popular IT companies and their domains that employ C
· MasterCard
· IBM
· Flipkart
· Dell
· Twitter
· GitHub and twitch
11:09-First c program.
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#IntroductiontoCProgrammingLanguage #CLanguage #CProgramming #CProgram #CProgrammingLanguage #LearnCProgramming #HowToCodeInCForBeginners #CTutorialForBeginners #LearnCProgramming #Simplilearn
Dennis Ritchie, a computer scientist, was able to identify the gaps and tap out the best features from both B and BCPL languages to invent a new hybrid.
Hence, C was born in 1972 at Bell Laboratories. A remarkably simple and highly readable programming language resulted in groundbreaking advancements in the IT industry.
✅What is C++ Programming?
C++ is an enhanced and extended version of C programming language, developed by Bjarne Stroustrup in 1979 as part of his Ph.D. project. Bjarne developed what he called ‘C with Classes’ (later renamed C++) because he felt limited by the existing programming languages that were not ideal for large scale projects. He used C to build what he wanted because C was already a general-purpose language that was efficient and fast in its operations.
✅C++ Career Prospects:
With just C++ programming expertise, you will have excellent job opportunities, salaries, and career prospects. However, for a career based on programming languages such as Java and Python (which are in more demand than C++) or for careers based on front-end, back-end, and full-stack
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不完全性定理入門 from「コンピュータは数学者になれるのか」
- 6. ゲーデル符号化のおかげで論理式 について述べることができる 論理式についての性質をいろいろと記述できる ℕ ⊨ formula(n) ⟺ nは論理式(のゲーデル数)である ℕ ⊨ sentence(n) ⟺ nは文（のゲーデル数）である これらは「文法」についての性質 →「意味的な性質」について初等数論の論理式を用いて 記述できるか
- 7. 定理2.3 不動点 どんな1変数論理式A(x)に対しても、文DAが存在して次のこ とが成り立つ ℕ ⊨ DA ⟺ A(文DAのゲーデル数) 数学の不動点とは、関数を作用させても動かない点のこと 初等数論の言語においても成り立つ
- 8. 定理2.4 真理定義不能性 次の性質を満たす1変数論理式 true(x)は存在しない。任意の 文Aについて ℕ ⊨ A ⟺ ℕ ⊨ true(Aのゲーデル数) 初等数論の言語の表現力に一定の限界がある ある言語における真理概念は同じ言語を用いたのでは定義で きない
- 10. 計算課題の決定可能性 次のようなプログラムPが存在するとき、Xは決定可能 𝑛∈X ⇒ P(𝑛) = yes 𝑛∉X ⇒ P(𝑛) = no 次のようなプログラムQが存在するとき、Xは半決定可能 𝑛∈X ⟺ Q(𝑛) = yes Pは必ず停止するが、Qは何千兆年経っても止まらないかもし れない
- 11. コンピュータの限界（チューリ ングの決定不能性） 停止問題 入力 プログラムPと自然数n 質問 P(n)は停止するか？ プログラムを用いてプログラムの停止を判定しなければならない 自己参照的だ… 定理 2.10 停止問題は半決定可能ではあるが、決定可能ではない
- 12. 𝜮₁論理式と𝜫₁論理式 ・∃x₁ … ∃x𝑛 A を𝜮₁論理式 真なときには、検証を続けていけばいつかは真だとわかる ・∀x₁ … ∀x𝑛 A を𝜫₁論理式 偽なときには、検証を続けていけばいつかは偽とわかる → 正しさを確かめるのは𝜮₁文のほうが圧倒的にかんたん 𝜮₁と𝜫₁はコインの裏表の関係 「すべてのxについてA(x)が成り立つ」の否定は、「あるxが存在し てA(x)が成り立たない」（ドモルガンの法則）
- 15. 定理2.17 形式系TはQのRE拡大であるとする。このときTのゲーデ ル文Gtについて次が成り立つ Tは無矛盾 ⟺ TはGtを証明できない ⟺ Gtは真 Tが無矛盾である限り、Tには証明できない 真理が存在する
- 17. 自分の証明能力について認識できてい る 自分自身についての事柄をも証明する能力を得てしまった Q ⊦ A ならば Q ⊦ provQ(Aのゲーデル数) もしもQ氏がAを証明できるならば、彼女は「自分はAを証明 できる」ことも証明できてしまう 自己言及できるならば、対角線論法の餌食になってしまう
- 18. 第二不完全性定理が出てくるわけ 第一不完全性定理の証明の手続きは、すべて初等数論でかける。人が 証明してきた第一不完全性定理を、今度は人工数学者に証明させる。 すると T ⊦ Cont ⟷ Gt （Contは「Tは無矛盾である」を表す論理式） 第一不完全性定理(1)より、Tが無矛盾ならばTはGtを証 明できない
- 19. 第二不完全性定理 T ⊦ Cont ⟷ Gt だから 第一不完全性定理(1)より、Tが無矛盾ならばTは「T ⊦ Cont」 を証明できない 無矛盾なTは、自分が無矛盾であることを証明できない 自己完結的な無矛盾性の可能性を否定。 自己正当化しつつ数学を土台から構築していく可能性はついえた
- 20. 補足 不完全性定理（結城浩 数学ガール ゲーデルの不完全性定理）より 第一不完全性定理 ある条件を満たす形式系には、以下の両方が成り立つ文Aが存在 する ・その形式系には、Aの形式的証明は存在しない ・その形式系には、¬Aの形式的証明は存在しない 第二不完全性定理 ある条件を満たす形式系には、自己の無矛盾性を表現する文の形 式的証明は存在しない