"A compound proposition is composed of one or more simple propositions, and the way it is synthesized is called a 'connective word'. 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', which 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 composed of 'This card is a person over 16 years old' and'
This card is composed of two simple propositions, and the synthesis method is 'and', that is, when two simple propositions are 'true' at the same time, the compound proposition is 'true'; and the third proposition is composed of 'this card
It is composed of two simple propositions: "I am a person originally from Fujian" and "This card is a person originally from Hainan". 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. There are actually two other methods, but they are not related to the design of the classification machine for the time being, so we will skip them here."
"We use symbols to represent propositions and connectives, then any query can be expressed as an expression. Obviously, the card whose expression is 'true' is the card we are looking for. And the role of the classification machine is,
It is to determine whether this expression is 'true' for all cards."
"Therefore, any expression that our classifier can judge as true/false is a problem that we can solve. Any expression that our classifier cannot judge as true or false is a problem that we cannot solve."
"This is our initial abstraction of the problem."
Vonn wrote several strange symbols on the blackboard: ∨ (or), ∧ (and), ┐ (not), which looked like a greater than sign and a less than sign that were rotated 90 degrees, as well as inverted Latin letters.
l.
"Okay, now you can write
The expression of the proposition ‘people originally from Fujian or Hainan’, Hainan is 100 and Fujian is 122, so we make
Proposition a: ‘The first digit of the area code is 1’,
Proposition b: ‘The second digit of the area code is 0’,
Proposition c: ‘The third digit of the area code is 0’,
Proposition d: ‘The second digit of the area code is 2’,
Proposition e: ‘The third digit of the area code is 2’,
Then, the expression of the compound proposition is: ‘(a∧b∧c)∨(a∧d∧e)’.”
"How does our sorting machine determine the truth or falsehood? It is by checking whether the punched card is perforated. In other words, each card reading unit of the sorting machine can determine the truth or falsehood of a simple proposition in a compound proposition. At the same time, by
With a control relay, we can allow each card reading unit to determine the true or false of a compound proposition with only one 'not' connective, that is, a non-proposition of a simple proposition."
"If we only have 1 card reading unit, then that's it. But now we have 10 card reading units, so things are a little more complicated. But it can still be analyzed. Please pay attention to the
Card pocket, features of loaded cards:
The card in the card bag numbered k is the 'and' of the 'not' proposition of propositions numbered 1~k-1, and then 'and' the proposition numbered k.
The remaining cards that have passed through the card reading unit No. k are the "AND" of the "NOT" propositions that satisfy the propositions judged by No. 1~K.
The cards in the card bag numbered 1~k, together they are the 'or' that satisfies the proposition judged by numbered 1~k.
Suppose 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 No. 2:┐p1∧p2
Card pocket No. 3:┐p1∧┐p2∧p3
Card pocket No. 4:┐p1∧┐p2∧┐p3∧p4
...
Card bag No. 10:┐p1∧┐p2∧...∧┐p9∧p10
The final remaining cards:┐p1∧┐p2∧...∧┐p10
Finally, since these cards are separated from each other, we can finally freely choose the cards from any number of card pockets to be combined, which is the 'or' between the above expressions; the most important of which is the continuous number from 1 to k
When the cards in k card pockets are put together, the result is: p1∨...∨pk, which is a continuous 'OR' operation starting with p1;
The remaining cards on the machine after passing through the card reading unit k can be expressed as ┐p1∧...∧┐pk, which is a continuous ‘AND’ operation starting with ┐p1.”
"So, any proposition that can be transformed into the above formal expression can be searched by the classification machine, otherwise, it cannot be searched by the classification machine."
"The question I asked Kanai to find the cards in the Sanya region except slaves can be broken down 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 0’,
Proposition c: ‘The third digit of the area code is not 0’,
Proposition d: ‘The fourth digit of the area code is not 1’,
Proposition e: ‘The fifth digit of the area code is 1’,
Proposition f: ‘The fifth digit of the area code is not 2’
Proposition g: ‘The 6th digit of the area code is not 9’
Proposition h: ‘The seventh digit of the area code is not 9’
┐a∧┐b∧┐c∧┐d∧e, this is 10011, Sanya Yulin, which conforms to the expression of card bag No. 5, so these cards are located in card bag No. 5 and 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 card bag No. 7, 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 card bag No. 8, so these cards
It is located in card pocket No. 8 and can be recorded as p8.
The latter two combined, 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 conforms to our above form,
So 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 transform the expression?" At this time, 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 Janai were completely confused before, but after hearing Boolean algebra, they somewhat came to their senses.
Vonn only taught them the simplest Boolean algebra, so that they thought Boolean algebra was Boolean algebra of 0 and 1.
"Then what?" Feng Nuo continued to guide.
"Boolean algebra is a complemented lattice! The intersection operation is 'and', the union operation is 'or', the complement operation is 'not', and it satisfies the commutation law, associative law, absorption law, 'and' and 'or' each other
It satisfies the distributive law! 0-1 Boolean algebra also satisfies the idempotent law!”
This is the theoretical part of Boolean algebra, and Qian Yuzhi and Li Ganai were confused again.
"Very good." Feng Nuo praised.
"However," he added, "the basic operational laws of the lattice are only between the two operations of 'and' and 'or', including commutative law, associative law, absorption law, idempotent law, distributive law, etc. In
In propositional logic, we also need to consider the nature of 'not'. Here I will only talk about two points for now: First, the law of double negation. Obviously, the non-proposition of a proposition is itself. The form of its expression is -
—”
Vonn wrote on the blackboard:
┐┐a=a;
"Second, virtue...well, let's call it the 'and or transformation law'. The non of the conjunction of two propositions is the disjunction of the non of two propositions; the non of the disjunction of two propositions is the non of two propositions.
The conjunction of the negations of propositions. The form of its expression is——"
He further wrote:
┐(a∧b)=┐a∨┐b,
┐(a∨b)=┐a∧┐b.
"I will give you two examples and you will understand. 'Not a man over 16 years old' means 'a person under 16 years old' or a 'woman'; 'Not a person originally from Hainan or Fujian' means a 'person who is not originally from Hainan or Fujian'."
"Not from Hainan" and "Not from Fujian".
Then he continued, "According to these operational laws, the expressions of logical propositions can be transformed into various forms. However, generally we will transform into 'OR' of continuous 'AND', or 'AND' of continuous 'OR',
They are called disjunctive normal form and conjunctive normal form.”
"Okay, with theoretical tools, we can find that there are limitations in the design of current classification machines. If the classification machine can handle general disjunctive paradigms or conjunctive paradigms, there will be no problems that cannot be solved by design.
——For example, 'find people who are originally from Fujian or Hainan'."
"This requires that each of our card reading units can not only judge the truth or falsehood of a simple proposition, but also be able to judge the truth or falsehood of the conjunction or disjunction composed of multiple simple propositions. This is reflected in the design of the classification machine
, is to transform the simple circuit of the card reading unit, which currently only includes one working relay and one control relay, into a switching circuit containing multiple relays."
"Yu Zhi, you have become very familiar with circuits during this time. Come and assemble a circuit with two switches and a light bulb. The requirement is that 'the light bulb will light up only when both switches are closed'."
Von Nuo pointed to the workbench beside him. There were a lot of wires, relays, light bulbs and switches on the workbench. Two bulky bell batteries were placed under the workbench. A multimeter and several other instruments were thrown in the corner of the workbench.
.
Qian Yuzhi came to the workbench skillfully and got busy. He first led out the wires from the positive and negative terminals of the battery, and then connected the light bulb to the circuit. The light bulb lit up. Then, he connected the two switches with wires, and connected them with
The light bulb and battery are connected together.
Feng Nuo asked three students to try to see if the light bulb only lights up when two switches are closed. If any one switch is open, the light bulb goes out.
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Next update: Volume 7 - Guangdong and Guangxi Strategy Chapter 61