sábado, 4 de outubro de 2014

A interpretação de Copenhague

A interpretação de Copenhague

Autor: 
Joshua Clark

A teoria dos Muitos Mundos da mecânica quântica (em inglês) supõe que para cada resultado possível de qualquer ação, o universo se divide para acomodar cada um deles. Esta teoria tira o observador da equação. Não somos mais capazes de influenciar o resultado de um evento simplesmente por observá-lo, como afirma o Princípio da Incerteza de Heisenberg.
O gato de Schrodinger
Mas a teoria dos Muitos Mundos vira de cabeça para baixo uma teoria muito aceita da mecânica quântica. E no imprevisível universo quântico, isso quer dizer muita coisa.
Por uma boa parte do último século, a explicação mais aceita para uma mesma partícula quântica se comportar de maneiras diferentes foi a interpretação de Copenhague. Apesar de desafiar a interpretação dos Muitos Mundos, muitos físicos (em inglês) quânticos ainda acreditam que a interpretação de Copenhague esteja correta. A interpretação de Copenhague foi proposta pela primeira vez pelo físico Neils Bohr (em inglês), em 1920. Ela diz que uma partícula quântica não existe em um estado ou outro, mas em todos os seus possíveis estados de uma vez só. É somente quando observamos seu estado que a partícula quântica é forçada a escolher uma probabilidade, e este é o estado que observamos. Como ela pode ser forçada a se apresentar em um estado observável diferente cada vez, isto explica porque as partículas quânticas têm um comportamento irregular.
Este estado de existir em todos os estados possíveis de uma vez é chamado de superposição coerente de um objeto. O total de estados possíveis em que um objeto pode existir - por exemplo, em forma de onda ou partícula para os fótons (em inglês) que se movimentam em duas direções ao mesmo tempo - forma a sua função onda. Quando observamos um objeto, a superposição cai e o objeto é forçado a assumir um dos estados da sua função onda.
A interpretação de Copenhague de Bohr da mecânica quântica foi teoricamente provada, pelo que se tornou um experimento mental famoso envolvendo um gato e uma caixa. É chamado de 'gato de Schrödinger', e foi apresentado pelo físico vienenseErwin Schrödinger (em inglês), em 1935.
Em seu experimento teórico, Schrödinger colocou seu gato em uma caixa, junto com um pouco de material radioativo e umcontador Geiger (em inglês) - dispositivo para detectar radiação. O contador Geiger foi montado de maneira que quando percebesse o decaimento do material radioativo, acionaria um martelo posicionado para quebrar um frasco contendo ácido cianídrico que, quando liberado, mataria o gato.
Para eliminar qualquer incerteza sobre o destino do gato, o experimento deveria acontecer dentro de uma hora, tempo longo o suficiente para que o material radioativo pudesse decair um pouco, mas também curto para que também fosse possível que nada acontecesse.
No experimento de Schrödinger, o gato estava fechado dentro de uma caixa. Durante o período em que estivesse ali dentro, o gato passaria a existir em um estado desconhecido. Como não poderia ser observado, não seria possível dizer se estava vivo ou morto. Ao invés disso, existia no estado de vida e morte. Em outras palavras, o gato está vivo e morto enquanto não se observa. É mais ou menos a resposta da física quântica para a velha pergunta zen (em inglês): se uma árvore cair no meio da mata, e ninguém estiver perto para escutar, faz barulho?
Uma vez que a interpretação de Copenhague diz que, quando observado, um objeto é forçado a assumir um estado ou outro, o experimento do suicídio quântico não funciona - de acordo com esta teoria. Como o sentido do quark medido pelo gatilho pode ser observado, no final das contas o quark será forçado a assumir o sentido horário que vai disparar a arma e matar o homem.
Mas isso tudo não é só uma bobagem? Estes experimentos mentais e interpretações quânticas nos ensinam alguma coisa de verdade? Na próxima seção falaremos sobre as possíveis implicações dessas idéias.

quinta-feira, 2 de outubro de 2014

Logical Spreadsheets and Websheets

Logical spreadsheets: Similar to a traditional spreadsheet where the cells can be related using formulas, cells in a logical spreadsheet are related / constrained using logical expressions. Consider the following example of a logical spreadsheet which has three cells: start, end, and duration and two logical constraints (a) start time < end time and (b) start time + duration = end time.
StartEndDuration
Play around with the above logical spreadsheet. For instance, try entering 9:00 am as the start time and then choose 1:00 pm as the end time. The duration will be automatically filled in as 4 hours. The duration is italicized to indicate that it is a computed value rather than one entered directly. Now, remove your choice of 1:00 pm for the end time by choosing the empty value (the top choice) from the menu, and choose a value for the duration. Note that the end time is automatically filled in. Traditional spreadsheets for e.g., MS Excel do not support bi-directional propagation of values.
In addition, logical spreadsheets can sometimes resolve conflicts automatically. For example, if you choose 9:00 am as the start time and then choose 8:00 am as the end time, the start time will automatically be erased because it is inconsistent with your choice of end time. Alternatively, a spreadsheet can keep the inconsistent values but alert the user to the problem. For example, try entering a start time of 9:00 am, an end time of 1:00 pm, and a duration of 1 hour. These three choices are incompatible, so the spreadsheet indicates the inconsitency by coloring the three choices red.
Websheets: A logical spreadsheet need not look like a traditional spreadsheet (e.g. MS Excel) at all i.e. the cells of a logical spreadsheet need not be necessarily laid-out in a grid. The idea behind websheets is that any DOM element (for e.g. checkbox, dropdown, radio, input field, or button) of a HTML page can be turned into a cell of a logical spreadsheet. One can then characterize the behavior of the webpage using logical rules that constrain the value of these cells. One practical application of websheets is to design, validate and manage smart HTML-forms in a declarative fashion.
The programsheets used by CS Masters students at Stanford is a real-world example of websheets in action. Click on the following URL to view a Stanford MSCS programsheet. Note that the programsheet has been altered for demonstration purposes.
Stanford MSCS programsheet
Every form element in the programsheet (for e.g., the checkboxes  or , or the SPAN element that captures the Breadth requirement "At least 3 of the following:") can be seen as cells in a logical spreadsheet. Interacting with these form elements changes the truth values of the associated cells. For example, clicking on the checkbox labeled CS 103 sets the truth value of the cell / proposition cs103 to true.
In addition, you'll observe that after certain interactions for e.g., when three breadth courseshave been selected, the color of the SPAN element corresponding to the Breadth Requirement turns from red to black. This visual feedback is used to let students (who are filling in the programsheet) know whether or not they have satisfied the appropriate program requirements. We note that the logical constraints that capture the program requirements are not explicitly rendered in the above programsheet (only the English equivalent is rendered). Examples of (modified) logical constraints that used in the programsheet are as follows.
cs103 ∨ logicApprovallogicReqSatisfied
cs103¬logicApproval
logicApproval¬cs103
The above logical sentences constrain truth value of the propositon logicReqSatisfied to be true(causing the color of SPAN element "Logic, Automata, and Complexity" to turn from red toblack) if one of the following actions is performed.
  • The checkbox  is checked (which causes the proposition cs103 to be true), or
  • One of the non-empty options under the dropdown Approval is selected (causing logicApproval to be true).
In addition, these two actions are mutually exclusive i.e. selecting CS 103 (and thereby setting the truth value of cs103 to true) causes the value of the dropdown menu Approval to be empty (thus changing the truth value of logicApproval to false).
Thought-exercise: Can you think of another simple application of websheets in action that you have encountered in this course?
Note: In this application there is uni-directional propagation of values.
Hint: Feedback is the key!

Nota a

Introduction to Logic

Alice in Wonderland
Lewis Carroll included a number of instances of specious logical reasoning in Alice in Wonderland. The following is a famous example.
'[Y]ou should say what you mean,' the March Hare went on.
'I do,' Alice hastily replied; 'at least - at least I mean what I say - that's the same thing, you know.'
'Not the same thing a bit!' said the Hatter. 'You might just as well say that "I see what I eat" is the same thing as "I eat what I see"!'
'You might just as well say,' added the March Hare, 'that "I like what I get" is the same thing as "I get what I like"!'
'You might just as well say,' added the Dormouse, who seemed to be talking in his sleep, 'that "I breathe when I sleep" is the same thing as "I sleep when I breathe"!'
Alice seems to believe that the sentence p ⇒ q is the same as q ⇒ p; and this is a logical error, as made clear by the comments of the Mad Hatter, the March Hare, and the Dormouse. Clearly, Alice needs to sign up for Introduction to Logic.

Curso de lógica

Announcements

Week 1

Okay. We are on our way! First week of class begins now.
This week, your goal is to master the material in lesson 1. This should not be too hard. The lesson is mostly overview. That said, you should not shortchange the material. This lesson talks about the main ideas of Logic and how they relate to each other, and it provides a framework for organizing the rest of the material in the course.
This week, you should also master the art of doing online problems. You should check out the Notes and the Extras and the Puzzles, and you should figure out how to use Discussion Forum.
A word about certificates. There are approximately 70 problems in this course, and your grades on these problems will determine your overall grade for the course. We will be issuing certificates of accomplishment to all students who complete the course with a cumulative grade of 70% of the possible maximum score, and we will offer Statements of Distinction to all students who complete the course with a cumulative grade of 90% or more.
Finally, a few words about the online problems. First of all, you can submit your answers to any problem as often as you like, and the system will take your highest grade. Second, all problems provide immediate feedback. When you check a checkbox or make a selection from a menu or compete a proof, the system will tell you immediately whether the answer is correct, even BEFORE you submit your answers for grading. The upshot is that, for some problems, there is no reason not to get a perfect score.
Yes, we realize that it is possible to "game" the system by dumbly trying all answers until you get the right one and then submitting that answer. However, this is already possible. There is nothing stopping you from signing up twice, getting the right answers from one account and using them in your other account. Besides, for many problems, mostly proofs, finding a correct answer is a challenge, and you will have to work hard to get that coveted green checkmark. And we do not reveal proofs until after the hard deadline.
The main reason for this seemingly lax approach to grading is our belief that there is great pedagogical value in immediate feedback rather than waiting for hours or days or even weeks to see the results. This is the first time doing things this way. When the course is over, let us know what you think.
mrg and abhijeet and daniel
Mon 29 Sep 2014 12:00 PM PDT

Welcome!

We are happy to have you join us for this introductory course on Logic. This is the fifth time we are offering the course in this online format. We believe the online format has the potential for improving Logic education and for bringing the material to a wide audience.
The course consists of ten lessons on different aspects of Logic. Each lesson consists of several sections, each with its own video and notes and exercises. Our intent is to proceed through one or two lessons each week, which means you need to view a few hours of video and do a handful of exercises each week.
In addition to these lessons, the course contains some additional material, puzzles, and ancillary readings for those of you who want to explore beyond the primary material of the course. This additional material is not required. However, you are encouraged to look at these materials as they reinforce and extend the course in interesting ways.
The course begins officially on September 29. You should study the first lesson during the first week and do the corresponding problems by the end of the week. If you miss the nominal deadline, you have an additional week to do the problems.
Importantly, you should take advantage of the discussion forum to communicate with your fellow students and with the instructors. Use the forum to ask questions, answer questions, and add additional material to the course.
We hope the course will be a rewarding, educational, and entertaining experience for you. We are sure that it will be a learning experience for us; and we are looking forward to working with you.
mrg and abhijeet and daniel