Computer assisted proof and their effect on mathematicsStudent name ?Alan Yang Student ID? 1689664 IntroductionAt the beginning of this article demonstrates what exactly is mathematical proofs, by giving a series of examples, furthermore explaining how computer program solves mathematical problems and emphasise the advantages of using computer program to check proofs. What are proofs?Basically proofs are the arguments which convince people that something are credible and believable, Here is a simple example of a mathematical proof: Assume X = 0.9999…….10X would be ? 10 * 0.9999….. = 9.9999…….Then 10X -X=9X, 9.9999….. – X =9 Hence, X = 9/9=1Therefore X=0.9999…=1 Another example : As the diagram shown above, assume that the first circle has a point on its boundary, second one with two and so on. When each boundary point links together, they will divide the circle into different regions, then we can count the number of regions for each circle. Boundary point :1 Boundary point:2 Boundary point : 3 Boundary point : 4 Boundary point : 5 Region : 1 Region : 2 Region : 4 Region : 8 Region : 16 We can recognise that as the number of boundary point increase, the regions of the circles double each time, whence we can deduce the equation of the regions for nth boundary point : 2 ^n-1However, all the examples shown above do not have a conclusive evidence. In maths, it is different form physics or other fields, it is not dependent on partially known physical principles or uncertain behaviors from human. In our real world, we set the rules such as addition and multiplication, then show what we get from a+b and b+a are the same. For those routine calculation, we require more persuasive proofs. Computers not only help human to solve the mathematics problems, but also play a key role in discovering the proof of maths theorems. One of the famous example is the four-colour theorem.First stated in 1852,but proved with the aid of computer until 1976. 4-colour 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. (Wikipedia) (To colour this blank square, we only use two colours because two section share the common edge can not have the same colour) Here is the Europe map, we use 4 colours. However ,for any pattern or map the maximum colours we can use are 4. Understanding what is automated theorem provingAccording to the Wikipedia, automated proof checking is the process of using software for checking the correctness of the proofs. We focus the one more related to computer science which is automated theorem proving. ATP is a process of dealing with proving mathematical theorems by computer programs that some statement is a logical consequence of a set of axioms and hypotheses.Here are some examples of ATP in real application : l The human cost of bug: computer are used in safety-critical systems where a failure could cause loss of life; e.g. Aircraft, Car engine management systems, heart pacemakers. l Complexity of designs: Market pressures are leading to more and more complex designs where bugs are more likely, such as a 4-fold increase in bugs in Intel processor designs per generation. l Limits of testing: Bugs are usually detected by extensive testing, like pre-silicon simulation. computer logic techniques used in ATPThe following example using the propositional logic ( a declarative statement cannot be true and also false) to solve the real world problem. Firstly, we need to understand the meaning of each logic symbols, here is the table. SymbolDefinition~Not^AndvOr If … then… If and only if For example, if we say ” if today is sunny and warm, then I will not go out.” ,first we write down the proposition of this sentence by using logic symbol. A= today is going to rain B=toady is warm C= go out ( A B and C are also called atomic position) A ^ B ~C we can solve something more complicated , e.g.If Alan studies music, then Bob studies it too. If either Alan or Bob studies music,then Alan definitely does. Therefore, both Alan and Bob study music. We want to check the validity of this argument, write down the atomic position first?Atomic proposition : A= Alan studies music B=Bob studies music A?BThen as the argument shown above, we separate the argument into different pieces of comprise. Argument comprise: Comprise1 : A ? BComprise2: A?B ?AConclusion: A?B We put all the atomic positions and comprises in a truth table, a truth table is mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus. Comprise 1 Comprise 2 Conclusion A B A?BA ? BA?B ?AA?BTTTTTTTFTFTFFTTTFFFFFTTF To get into the point, primarily we consider all the possibilities for A and B , there are only 4 : A and B both True ; A is true B is False; A is false B is true ; A and B are both false. That’s why the table shown has 4 rows. Then we judge A ? B and A ? B , For A ? B , if one of the element is true, then it is true. For A ? B, it is true when both of them are true. For A ? B ( A implies B ), it is special because if the latter is false, then it is false, otherwise it is true. When we finish the truth table, we check the truth of all comprises, if one of them doesn’t conform the conclusion, then this argument is invalid. Return to the table, the bottom line indicates both comprises are true but the conclusion is false, therefore this argument is invalid. Conclusion Through the whole article, we know that with the assistance of computer logic, more detailed and logical proof can be shown a. Additionally, most systems allow you to provide feedback on question result, for mistakes or correct answers. Computer assisted facilitates a detailed analysis of test results with minimal effort.