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Why Undecidable Problems Must Exist
Why Undecidable Problems Must Exist
While it is tricky to prove that a specific problem, such as the “hello- world problem” discussed here, must be undecidable, it is quite easy to see why almost all problems must be undecidable by any system that involves programming. Recall that a “problem” is really membership of a string in a language. The number of different languages over any alphabet of more than one symbol is not countable. That is, there is no way to assign integers to the languages such that every language has an integer, and every integer is assigned to one language.
On the other hand programs, being finite strings over a finite alphabet (typically a subset of the ASCII alphabet), are countable. That is, we can order them by length, and for programs of the same length, order them lexicographically. Thus, we can speak of the first program, the second program, and in general, the ¿th program for any integer i.
As a result, we know there are infinitely fewer programs than there are problems. If we picked a language at random, almost certainly it would be an undecidable problem. The only reason that most problems appear to be decidable is that we rarely are interested in random problems. Rather, we tend to look at fairly simple, well-structured problems, and indeed these are often decidable. However, even among the problems we are interested in and can state clearly and succinctly, we find many that are undecidable; the hello-world problem is a case in point.
about a single, 22-line program, then the general problem of telling whether a given program, on a given input, prints hello, world must be hard indeed. In fact, any of the problems that mathematicians have not yet been able to resolve can be turned into a question of the form “does this program, with this input, print hello, world? Thus, it would be remarkable indeed if we could write a program that could examine any program P and input I for P, and tell whether P, run with I as its input, would print hello, world. We shall prove that no such program exists.
While it is tricky to prove that a specific problem, such as the “hello- world problem” discussed here, must be undecidable, it is quite easy to see why almost all problems must be undecidable by any system that involves programming. Recall that a “problem” is really membership of a string in a language. The number of different languages over any alphabet of more than one symbol is not countable. That is, there is no way to assign integers to the languages such that every language has an integer, and every integer is assigned to one language.
On the other hand programs, being finite strings over a finite alphabet (typically a subset of the ASCII alphabet), are countable. That is, we can order them by length, and for programs of the same length, order them lexicographically. Thus, we can speak of the first program, the second program, and in general, the ¿th program for any integer i.
As a result, we know there are infinitely fewer programs than there are problems. If we picked a language at random, almost certainly it would be an undecidable problem. The only reason that most problems appear to be decidable is that we rarely are interested in random problems. Rather, we tend to look at fairly simple, well-structured problems, and indeed these are often decidable. However, even among the problems we are interested in and can state clearly and succinctly, we find many that are undecidable; the hello-world problem is a case in point.
about a single, 22-line program, then the general problem of telling whether a given program, on a given input, prints hello, world must be hard indeed. In fact, any of the problems that mathematicians have not yet been able to resolve can be turned into a question of the form “does this program, with this input, print hello, world? Thus, it would be remarkable indeed if we could write a program that could examine any program P and input I for P, and tell whether P, run with I as its input, would print hello, world. We shall prove that no such program exists.
0/5000
Tại sao vấn đề Undecidable phải tồn tạiTrong khi nó là khó khăn để chứng minh rằng một vấn đề cụ thể, chẳng hạn như thế giới Xin chào "vấn đề" thảo luận ở đây, phải là undecidable, nó là khá dễ dàng để xem tại sao hầu hết các vấn đề phải được undecidable bởi bất kỳ hệ thống nào liên quan đến việc lập trình. Nhớ lại rằng một "vấn đề" thực sự là thành viên của một chuỗi trong một ngôn ngữ. Số lượng các ngôn ngữ khác nhau hơn bất kỳ bảng chữ cái biểu tượng nhiều hơn một là không đếm được. Có nghĩa là, không có cách nào để chỉ định số nguyên cho các ngôn ngữ như vậy mà mỗi ngôn ngữ có một số nguyên, và mỗi số nguyên được gán cho một ngôn ngữ.Mặt khác, chương trình, là hữu hạn dây trên một bảng chữ cái hữu hạn (thường là một nhóm bảng chữ cái ASCII), được đếm được. Đó là, chúng tôi có thể đặt chúng theo chiều dài, và cho các chương trình của chiều dài tương tự, để chúng lexicographically. Vì vậy, chúng tôi có thể nói chuyện của chương trình đầu tiên, chương trình thứ hai, và nói chung, chương trình ¿th cho bất kỳ số nguyên tôi.Kết quả là, chúng tôi biết có rất vô hạn chương trình ít hơn không có vấn đề. Nếu chúng tôi đã chọn một ngôn ngữ ngẫu nhiên, gần như chắc chắn nó sẽ là một vấn đề undecidable. Lý do duy nhất mà hầu hết vấn đề xuất hiện để được decidable là rằng chúng tôi hiếm khi có quan tâm đến vấn đề ngẫu nhiên. Thay vào đó, chúng tôi có xu hướng để xem xét vấn đề khá đơn giản, cơ cấu tốt, và thực sự đây là thường xuyên decidable. Tuy nhiên, ngay cả trong số những vấn đề chúng tôi quan tâm và có thể nhà nước rõ ràng và ngắn gọn, chúng tôi tìm thấy nhiều mà được undecidable; vấn đề thế giới Xin chào là một trường hợp tại điểm.về một chương trình duy nhất, 22-dòng, sau đó vấn đề chung kể cho dù một chương trình nào đó, trên một đầu vào cho trước, in hello, thế giới phải được thực sự khó khăn. Trong thực tế, bất kỳ vấn đề mà nhà toán học được nêu ra đã không thể giải quyết có thể được biến thành một câu hỏi về các hình thức "có chương trình này, với này xin chào đầu vào, in, thế giới? Do đó, nó sẽ là đáng kể thực sự nếu chúng tôi có thể viết một chương trình mà có thể kiểm tra bất kỳ chương trình P và đầu vào tôi cho P, và cho biết cho dù P, chạy với tôi như là đầu vào của nó, sẽ in chào, thế giới. Chúng tôi sẽ chứng minh rằng không có chương trình như vậy tồn tại.
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Why Undecidable Problems Must Exist
While it is tricky to prove that a specific problem, such as the “hello- world problem” discussed here, must be undecidable, it is quite easy to see why almost all problems must be undecidable by any system that involves programming. Recall that a “problem” is really membership of a string in a language. The number of different languages over any alphabet of more than one symbol is not countable. That is, there is no way to assign integers to the languages such that every language has an integer, and every integer is assigned to one language.
On the other hand programs, being finite strings over a finite alphabet (typically a subset of the ASCII alphabet), are countable. That is, we can order them by length, and for programs of the same length, order them lexicographically. Thus, we can speak of the first program, the second program, and in general, the ¿th program for any integer i.
As a result, we know there are infinitely fewer programs than there are problems. If we picked a language at random, almost certainly it would be an undecidable problem. The only reason that most problems appear to be decidable is that we rarely are interested in random problems. Rather, we tend to look at fairly simple, well-structured problems, and indeed these are often decidable. However, even among the problems we are interested in and can state clearly and succinctly, we find many that are undecidable; the hello-world problem is a case in point.
about a single, 22-line program, then the general problem of telling whether a given program, on a given input, prints hello, world must be hard indeed. In fact, any of the problems that mathematicians have not yet been able to resolve can be turned into a question of the form “does this program, with this input, print hello, world? Thus, it would be remarkable indeed if we could write a program that could examine any program P and input I for P, and tell whether P, run with I as its input, would print hello, world. We shall prove that no such program exists.
While it is tricky to prove that a specific problem, such as the “hello- world problem” discussed here, must be undecidable, it is quite easy to see why almost all problems must be undecidable by any system that involves programming. Recall that a “problem” is really membership of a string in a language. The number of different languages over any alphabet of more than one symbol is not countable. That is, there is no way to assign integers to the languages such that every language has an integer, and every integer is assigned to one language.
On the other hand programs, being finite strings over a finite alphabet (typically a subset of the ASCII alphabet), are countable. That is, we can order them by length, and for programs of the same length, order them lexicographically. Thus, we can speak of the first program, the second program, and in general, the ¿th program for any integer i.
As a result, we know there are infinitely fewer programs than there are problems. If we picked a language at random, almost certainly it would be an undecidable problem. The only reason that most problems appear to be decidable is that we rarely are interested in random problems. Rather, we tend to look at fairly simple, well-structured problems, and indeed these are often decidable. However, even among the problems we are interested in and can state clearly and succinctly, we find many that are undecidable; the hello-world problem is a case in point.
about a single, 22-line program, then the general problem of telling whether a given program, on a given input, prints hello, world must be hard indeed. In fact, any of the problems that mathematicians have not yet been able to resolve can be turned into a question of the form “does this program, with this input, print hello, world? Thus, it would be remarkable indeed if we could write a program that could examine any program P and input I for P, and tell whether P, run with I as its input, would print hello, world. We shall prove that no such program exists.
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