Randomized algorithms are completely different to random program code
The distinction between data and code is mostly useful, but in some situations we intentionally cross the boundary. If you have a text editor, then the separation is valid—the program is code, and the text documents are data. But if you have a compiler, you have a compiler program, which is code, and the compiled program, which is data, but on a different level it is also code. If you have a dynamically loaded library, while you load it, it is data, but then you start treating it as code by running it. If you have an interpreter, which is a code, you emulate the program, which on one level is data and simultaneously on another level is code. If you have a decompiler or debugger, which is code, it takes another code and treats it like data. A bootloader loads data and then starts them as a code. Etc.
This means that treating data as code, or code as data, is not necessarily a mistake or confusion of levels, and sometimes it is unavoidable. How else could you for example start a program, which originally exists only as a sequence of bytes in a file on a disk? You treat it as data when loading the file to a memory, and then you treat it as a code.
A unit test cannot ever tell you that your program is free of errors.
Some maths here: let X be a problem we want to solve, and let’s suppose that the problem is algorithmically solvable. Further, let P be a set of all programs, PX a set of programs that correctly solve problem X, U is a unit test and PU is a set of programs that passes the unit test.
We will call U a correct unit test for problem X if each program from PX belongs to PU (a correct program always passes the unit test) and some programs in P-PX don’t belong to PU (some incorrect programs don’t pass the unit test). In other words, PX is a subset of PU, which is a strict subset of P.
A random code generator will provide a random distribution R of programs. Now the question is whether R ∩ PX is non-empty, in other words whether the generator is capable of generating a correct solution assuming it is incredibly lucky. This depends on the generator and the problem. For example if generator only creates random programs up to 100 lines of code, but the shortest possible code that solves the problem has 150 lines, then the generator cannot create a correct program. But if the generator is capable to create a program of any length, or more precisely capable of creating any program, then if the program is algorithmically solvable, the generator will generate the correct program with a non-zero probability. Just imagine that you wrote the correct program—well, with some non-zero probability, the output of the random generator will be exactly the same text. The only problem is that p(x in PX | x in R) is very small. Sometimes so small that even if we could reliably automatically test whether a program belongs to PX, we just wouldn’t have enough time to wait until first such program is generated. But let’s assume that it is a simple enough program, so if we generate a few millions of programs, we can expect that some of them belong to PX.
Yeah, we have a couple assumptions here. Again, they are: (1) a correct program exists, and (2) some correct program has Kolmogorov complexity small enough (i.e. has a short code, if written optimally) that we can realistically wait until a random generator generates it. Actually, also (3) some correct program with small Kolmogorov complexity runs fast enough that it will pass the unit test in given time limit.
Assuming this all, we have a set of generated random programs R, some of them belong to PX, and we are trying to find them. If we replace the set R with R ∩ PU, we have a smaller set, and R ∩ PX is a subset of R ∩ PU, so we did not lose any solution. In probabilistic terms, p(x in R ∩ PX | x in R ∩ PU) is greater than p(x in PX | x in R), in human speech: if we choose a random program that passes the unit test, we have better chance than if we choose any random program.
Question is, how much does the test U reduce the set of all programs. If only 1% of programs passes the unit test, then p(x in R ∩ PX | x in R ∩ PU) = 100×p(x in R ∩ PX | x in R), so by doing the unit test we increased our luck 100 times… though the result may be still pretty close to zero. We can filter the results by another test, but now we need that only a few of programs that have passed the previous test can pass the next test. Which may be difficult; and also pretty difficult to estimate in advance. We can only measure it afterwards, how many programs that passed the first test were filtered by the second one. The idea is, if a generated program has chance 1:10^10 to be correct, if we can make a sequence of 5 unit tests where each one filters out 99% of programs that have passed the previous tests, we have a decent chance at the end.
Of course to make all this work, we need a decent generator, which will give programs reasonable prior probabilities, therefore it would be better to generate parse trees than source texts, etc. We could even invent a new language that could allow us to write shorter programs. For example it should have an instruction “forAllItemsInSet(SET) { X | CODE }” made of 4 symbols, instead of “for (X = 1; X < SET.size; X++) { CODE }” with 11 symbols, etc.
The distinction between data and code is mostly useful, but in some situations we intentionally cross the boundary. If you have a text editor, then the separation is valid—the program is code, and the text documents are data. But if you have a compiler, you have a compiler program, which is code, and the compiled program, which is data, but on a different level it is also code. If you have a dynamically loaded library, while you load it, it is data, but then you start treating it as code by running it. If you have an interpreter, which is a code, you emulate the program, which on one level is data and simultaneously on another level is code. If you have a decompiler or debugger, which is code, it takes another code and treats it like data. A bootloader loads data and then starts them as a code. Etc.
This means that treating data as code, or code as data, is not necessarily a mistake or confusion of levels, and sometimes it is unavoidable. How else could you for example start a program, which originally exists only as a sequence of bytes in a file on a disk? You treat it as data when loading the file to a memory, and then you treat it as a code.
Some maths here: let X be a problem we want to solve, and let’s suppose that the problem is algorithmically solvable. Further, let P be a set of all programs, PX a set of programs that correctly solve problem X, U is a unit test and PU is a set of programs that passes the unit test.
We will call U a correct unit test for problem X if each program from PX belongs to PU (a correct program always passes the unit test) and some programs in P-PX don’t belong to PU (some incorrect programs don’t pass the unit test). In other words, PX is a subset of PU, which is a strict subset of P.
A random code generator will provide a random distribution R of programs. Now the question is whether R ∩ PX is non-empty, in other words whether the generator is capable of generating a correct solution assuming it is incredibly lucky. This depends on the generator and the problem. For example if generator only creates random programs up to 100 lines of code, but the shortest possible code that solves the problem has 150 lines, then the generator cannot create a correct program. But if the generator is capable to create a program of any length, or more precisely capable of creating any program, then if the program is algorithmically solvable, the generator will generate the correct program with a non-zero probability. Just imagine that you wrote the correct program—well, with some non-zero probability, the output of the random generator will be exactly the same text. The only problem is that p(x in PX | x in R) is very small. Sometimes so small that even if we could reliably automatically test whether a program belongs to PX, we just wouldn’t have enough time to wait until first such program is generated. But let’s assume that it is a simple enough program, so if we generate a few millions of programs, we can expect that some of them belong to PX.
Yeah, we have a couple assumptions here. Again, they are: (1) a correct program exists, and (2) some correct program has Kolmogorov complexity small enough (i.e. has a short code, if written optimally) that we can realistically wait until a random generator generates it. Actually, also (3) some correct program with small Kolmogorov complexity runs fast enough that it will pass the unit test in given time limit.
Assuming this all, we have a set of generated random programs R, some of them belong to PX, and we are trying to find them. If we replace the set R with R ∩ PU, we have a smaller set, and R ∩ PX is a subset of R ∩ PU, so we did not lose any solution. In probabilistic terms, p(x in R ∩ PX | x in R ∩ PU) is greater than p(x in PX | x in R), in human speech: if we choose a random program that passes the unit test, we have better chance than if we choose any random program.
Question is, how much does the test U reduce the set of all programs. If only 1% of programs passes the unit test, then p(x in R ∩ PX | x in R ∩ PU) = 100×p(x in R ∩ PX | x in R), so by doing the unit test we increased our luck 100 times… though the result may be still pretty close to zero. We can filter the results by another test, but now we need that only a few of programs that have passed the previous test can pass the next test. Which may be difficult; and also pretty difficult to estimate in advance. We can only measure it afterwards, how many programs that passed the first test were filtered by the second one. The idea is, if a generated program has chance 1:10^10 to be correct, if we can make a sequence of 5 unit tests where each one filters out 99% of programs that have passed the previous tests, we have a decent chance at the end.
Of course to make all this work, we need a decent generator, which will give programs reasonable prior probabilities, therefore it would be better to generate parse trees than source texts, etc. We could even invent a new language that could allow us to write shorter programs. For example it should have an instruction “forAllItemsInSet(SET) { X | CODE }” made of 4 symbols, instead of “for (X = 1; X < SET.size; X++) { CODE }” with 11 symbols, etc.