Lin Gao Qiming

Chapter 1956 True or False

"Compound propositions are synthesized from one or more simple propositions, and the way they are synthesized, we call them 'connectors'. For example, 'This card is not a slave', 'This card is a man over 16 years old',' This card is a person originally from Fujian or Hainan', these are three compound propositions."

"The first proposition is a negation of the simple proposition 'this card is a slave', and the synthesis method is 'no'; the second proposition is a combination of 'this card is a person over 16 years old' and ' This card is composed of two simple propositions for men, and the synthesis method is "and", that is, when the two simple propositions are "true" at the same time, the compound proposition is "true"; and the third proposition is composed of "this card Is a person originally from Fujian" and "This card is a person originally from Hainan" are composed of two simple propositions, and the synthesis method is "or", that is, when any one of the two simple propositions is "true", the compound proposition is " real'."

"So, we have three means of connecting multiple propositions to make them into larger propositions, and, or, and not. In fact, there are other two, but they are not related to the design of the classifier for the time being, so I will skip them here for now."

"We use symbols to represent propositions and connective words, so any query can be expressed as an expression. Obviously, the card that makes the expression 'true' is the card we are looking for. The function of the sorting machine is, It is to judge whether this expression is 'true' for all cards."

"Therefore, any expression that our classifier can judge 'true/false' is a problem that we can solve, and any expression that our classifier cannot judge true or false is a problem that we cannot solve."

"That's our initial abstraction of the problem."

Fonno wrote down a few strange symbols ∨ (or), ∧ (and), ┐ (not) on the blackboard, which looked like greater than and less than signs rotated 90 degrees, as well as reversed Latin letters L.

"Okay, now you can write

The expression of the proposition 'people originally from Fujian or Hainan' is 100 in Hainan and 122 in Fujian, so we make
Proposition A: 'the first digit of the area code is 1',
Proposition B: 'the second digit of the area code is 2',
Proposition C: 'the third digit of the area code is 3',
Proposition D: 'the second digit of the area code is 2',
Proposition E: 'the third digit of the area code is 3',
Then, the expression of compound proposition is: '(A∧B∧C)∨(A∧D∧E)'. "

"How does our sorting machine judge whether it is true or false? It is by checking whether the punched card is perforated, that is to say, each card reading unit of the sorting machine can judge the true or false of a simple proposition in the compound proposition. At the same time, through With a control relay, we can allow each card reading unit to judge the truth or falsehood of a compound proposition with only one 'not' connective word, that is, a simple proposition and a non-proposition."

"If we had only one card reader, that's all. But now we have 1 card readers, so things are more complicated. But it is still analyzable. Please note that the side of each card reader Card Pockets, Cards Included Features:
The cards in the k card bag are the 'and' and 'and' k propositions of the 'not' propositions of the 1~k-1 propositions.

The remaining cards passing through card reading unit k are the 'and' of the 'not' propositions that satisfy the propositions judged by numbers 1~k.

The cards in the card pockets numbered 1~k are taken together to be an 'or' that satisfies the propositions judged by numbers 1~k.

Assume that the simple propositions (or non-propositions of simple propositions) judged by our card reading unit are p1, p2, ., p10.

Then the propositional expression we can judge is:
Card pocket No. 1: p1
Card pocket 2: ┐p1∧p2
Card pocket 3: ┐p1∧┐p2∧p3
4号卡袋：┐p1∧┐p2∧┐p3∧p4
10号卡袋：┐p1∧┐p2∧.∧┐p9∧p10
Final remaining cards: ┐p1∧┐p2∧.∧┐p10
Finally, since these cards are separated from each other, we can finally freely choose any number of cards in the card pocket to combine together, which is the 'or' between the above expressions; the most important of which is the continuous number from 1 to k The cards in the k card pockets are combined, and the result is: p1∨.∨pk, that is, the continuous 'or' operation starting with p1;
The remaining cards on the machine after passing through card reader unit k can be expressed as ┐p1∧.∧┐pk, which is the continuous 'AND' operation starting with ┐p1. "

"Therefore, all propositions that can be transformed into the above-mentioned formal expressions can be found by the classifier, otherwise, they cannot be found by the classifier."

"The question I asked Kana, to find out the cards in the Sanya region except slaves, can be decomposed into the following simple propositions or non-propositions of simple propositions:
Proposition A: 'the first digit of the area code is not 1',
Proposition B: 'the second digit of the area code is not 2',
Proposition C: 'the third digit of the area code is not 3',
Proposition D: 'the fourth digit of the area code is not 4',
Proposition E: 'the third digit of the area code is 5',
Proposition F: 'The fifth digit of the area code is not 5'

Proposition G: 'The sixth digit of the area code is not 6'

Proposition H: 'The seventh digit of the area code is not 7'

┐A∧┐B∧┐C∧┐D∧E, this is 10011, Sanya Yulin, it conforms to the expression of No. 5 card pocket, so these cards are located in No. 5 card pocket, which can be recorded as p5.

┐A∧┐B∧┐C∧┐D∧┐E∧┐F∧G, this is 100120~100128, Sanya Tiandu 11~89 commune, it conforms to the expression of No. 7 card bag, so these cards are located at No. 7 In the card pocket, it can be recorded as p7.

┐A∧┐B∧┐C∧┐D∧┐E∧┐F∧┐G∧H, this is 1001290~1001298, Sanya Tiandu 90~98 commune, it conforms to the expression of No. 8 card bag, so these cards Located in the No. 8 card pocket, it can be recorded as p8.

The combination of the latter two, that is, p7∨p8, is Sanya Tiandu, but does not include slaves.The combination of all three, that is, p5∨p7∨p8, is the result we want.Because this expression fits our form above, the classifier can solve it. "

"And '(A∧B∧C)∨(A∧D∧E)', no matter how we transform it, cannot be transformed into the above expression, so it cannot be solved by the current classification machine."

"Okay, here comes the question, how to change the expression?" At this moment, he looked at Feng Shan.

"This is the Boolean algebra of 0 and 1." Feng Shan replied with a look of fascination in her eyes.

Feng Nuo nodded. Qian Yuzhi and Li Jianai were completely clueless before, but they came to their senses when they heard Boolean algebra.

Vonno only taught them the simplest Boolean algebra, so that they thought that Boolean algebra was the Boolean algebra of 0 and 1.

"And then?" Feng Nuo continued to guide.

"Boolean algebra has a complementary distributive lattice! The intersection operation is 'and', the union operation is 'or', and the complement operation is 'not', which satisfies the law of commutation, associativity, and absorption, and 'and' and 'or' satisfy each other. Law! 0-1 Boolean algebra also satisfies the law of idempotence!"

This is the theoretical part of Boolean algebra, and Qian Yuzhi and Li Jianai are confused again.

"Very good." Feng Nuo praised.

"However," he added, "the basic operation laws of lattices are only between the operations of 'and' and 'or', including commutative law, associative law, absorption law, idempotent law, distributive law, etc. In In propositional logic, the nature of 'not' must also be considered. Here I will only say two points: first, the law of double negation. Obviously, the non-proposition of a proposition is itself. The form of its expression is— —”

Feng Nuo wrote on the blackboard:

┐┐A = A;

"Secondly, virtue... Well, let's call it the 'and-or conversion law'. The conjunction of two propositions is the disjunction of the negation of two propositions; the disjunctive negation of two propositions is the disjunction of two propositions. The conjunction of the negation of a proposition. Its expression is of the form—"

He wrote again:

┐(A∧B)= ┐A∨┐B,

┐(A∨B)=┐A∧┐B.

"Let me give you two examples to understand, 'not a man over the age of 16' means 'a person under the age of 16' or a 'woman'; 'not a person originally from Hainan or Fujian' means "People who are not originally from Hainan" and "people who are not originally from Fujian".

Then he continued, "According to these laws of operation, the expressions of logical propositions can be transformed into various forms, but generally we will transform into the 'or' of continuous 'and', or the 'and' of continuous 'or', These are called disjunctive normal form and conjunctive normal form."

"Okay, with theoretical tools, we can find that there are limitations in the design of the current classifier. If the classifier can handle general disjunctive paradigms or conjunction paradigms, there will be no problems that cannot be solved by design. ——such as 'Find people who are originally from Fujian or Hainan'."

"This requires each of our card-reading units to not only be able to judge the truth or falsehood of a simple proposition, but to be able to judge the truth or falsehood of the conjunctive or disjunctive items formed by multiple simple propositions. Reflected in the design of the sorting machine , is to transform the simple circuit of the card reading unit, which currently only includes 1 working relay and 1 control relay, into a switching circuit including multiple relays."

"Yuzhi, you have been very familiar with circuits during this period of time. You come to assemble a circuit with two switches and a light bulb, and ask 'only when both switches are closed can the light bulb turn on'."

Feng Nuo pointed to the workbench beside him.There was a mass of wires, relays, light bulbs, and switches on the workbench, under which lay two heavy clock batteries, and a multimeter and several other instruments were tucked away in the corner of the workbench.

Qian Yuzhi skillfully came to the workbench and started to work. He first drew the wires from the positive and negative terminals of the battery, and then connected the light bulb into the circuit, and the light bulb turned on.Then, he connected the two switches with wires, and connected them with the light bulb and the battery.

Feng Nuo asked all three students to try to see if the light bulb would light up only when two switches were closed, and if any one switch was off, the light bulb would turn off.

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