Years ago, I have heard that someone was about to demonstrate that every computer program could be solved with just three instructions:
Assignment
Conditional
Loop
Please I would like to hear your opinion. I mean representing any algorithm as a computer program. Do you agree with this?
No need. The minimal theoretical computer needs just one instruction. They are called One Instruction Set Computers (OISC for short, kinda like the ultimate RISC).
There are two types. The first is a theoretically "pure" one instruction machine in which the instruction really works like a regular instruction in normal CPUs. The instruction is usually:
subtract and branch if less than zero
or variations thereof. The wikipedia article have examples of how this single instruction can be used to write code that emulates other instructions.
The second type is not theoretically pure. It is the transfer triggered architecture (wikipedia again, sorry). This family of architectures are also known as move machines and I have designed and built some myself.
Some consider move machines cheating since the machine actually have all the regular instructions only that they are memory mapped instead of being part of the opcode. But move machines are not merely theoretical, they are practical (like I said, I've built some myself). There is even a commercially available family of CPUs built by Maxim: the MAXQ. If you look at the MAXQ instruction set (they call it transfer set since there is really only one instruction, I usually call it register set) you will see that MAXQ assembly looks rather like a standard accumulator based architecture.
This is a consequence of Turing Completeness, which is something that was established many decades ago.
Alan Turing, the famous computer scientist, proved that any computable function could be computed using a Turing Machine. A Turing machine is a very simple theoretical device which can do only a few things. It can read and write to a tape (i.e. memory), maintain an internal state which is altered by the contents read from memory, and use the internal state and the last read memory cell to determine which direction to move the tape before reading the next memory cell.
The operations of assignment, conditional, and loop are sufficient to simulate a Turing Machine. Reading and writing memory and maintaining state requires assignment. Changing the direction of the tape based on state and memory contents require conditionals and loops. "Loops" in fact are a bit more high-level than what is actually required. All that is really required is that program flow can jump backwards somehow. This implies that you can create loops if you want to, but the language does not need to have an explicit loop construct.
Since these three operations allow simulation of a Turing Machine, and a Turing Machine has been proven to be able to compute any computable function, it follows that any language which provides these operations is also able to compute any computable function.
Edit: And, as other answerers pointed out, these operations do not need to be discrete. You can craft a single instruction which does all three of these things (assign, compare, and branch) in such a way that it can simulate a Turing machine all by itself.
The minimal set is a single command, but you have to choose a fitting one, for example - One instruction set computer
When I studied, we used such a "computer" to calculate factorial, using just a single instruction:
SBN - Subtract and Branch if Negative:
SBN A, B, C
Meaning:
if((Memory[A] -= Memory[B]) < 0) goto C
// (Wikipedia has a slightly different definition)
Notable one instruction set computer (OSIC) implementations
This answer will focus on interesting implementations of single instruction set CPUs, compilers and assemblers.
movfuscator
https://github.com/xoreaxeaxeax/movfuscator
Compiles C code using only mov x86 instructions, showing in a very concrete way that a single instruction suffices.
The Turing completeness seems to have been proven in a paper: https://www.cl.cam.ac.uk/~sd601/papers/mov.pdf
subleq
https://esolangs.org/wiki/Subleq:
https://github.com/hasithvm/subleq-verilog Verilog, Xilinx ISE.
https://github.com/purisc-group/purisc Verilog and VHDL, Altera. Maybe that project has a clang backend, but I can't use it: https://github.com/purisc-group/purisc/issues/5
http://mazonka.com/subleq/sqasm.cpp | http://mazonka.com/subleq/sqrun.cpp C++-based assembler and emulator.
See also
What is the minimum instruction set required for any Assembly language to be considered useful?
https://softwareengineering.stackexchange.com/questions/230538/what-is-the-absolute-minimum-set-of-instructions-required-to-build-a-turing-comp/325501
In 1964, Bohm and Jacopini published a paper in which they demonstrated that all programs could be written in terms of only three control structures:
the sequence structure,
the selection structure
and the repetition structure.
Programmers using Haskell might argue that you only need the Contional and Loop because assignments, and mutable state, don't exist in Haskell.
Related
I wanted to know how computers calculate logarithms?
I don't mean the related functions. For example, Python uses math.log() function. But I want to know what exactly does this function do? And can it be simulated again and more accurately?
Is there a formula for it? Or an algorithm? (I don't think the computer has a log table!)
Thanks
The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that logarithms are calculated directly from the hardware.
Hence one should ask: what algorithm is used for calculating logarithms by computers?
Depends on the CPU, For intel IA64, they use the Taylor series combined with a table.
More info can be found here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5177
and here: http://www.computer.org/csdl/proceedings/arith/1999/0116/00/01160004.pdf
This is a hugely open, broad and "depends-on".
For every programming language, every different core library, every different system or so forth, different algorithms/mechanisms and machine-code instructions may be existing for performing mathematical (and any other type of) calculations.
Furthermore, even if all the programming languages, in this world, would have been using same algorithm "X", it still does not mean, that Computer calculates logarithm in way X, because, computer still will (most likely) be doing its machine-level job differently, in different circumstances, disregarding the point, that algorithms are the same (which is, very less likely).
Bear in mind, that Computer Architectures differ, Operating Systems differ, and Assembler Instructions can be very different from CPU to CPU.
I really think, that you should come up with more specific and concrete questions on this website.
Any algorithm you can implement in a HLL you can implement in assembly. On the other hand, there are many algorithms you can implement in assembly which you cannot implement in a HLL. - Randall Hyde
I found this statement in the forward to a book on assembly. The book is here: https://courses.engr.illinois.edu/ece390/books/artofasm/fwd/fwd.html#109
Does anyone know an example of this type of algorithm?
It's plain wrong.
You can implement any algorithm (in the CS sense of the word) in any turing complete programming language.
On the other hand, if he would have said something a like: "Some algorithms can be implemented very efficiently, and with ease in assembly, much more so than what is possible in most high level programming languages", then his statement would have made sense...
Interesting text though....
There is a sense in which it is trivially false: in the worst case, you could write an emulator in the HLL and then run the algorithm in there. But that's cheating a bit because now the HLL does not directly implement the algorithm.
A concrete example of what many HLL's can't do (or maybe they can in practice, but it is not guaranteed that they can do it), is directly implementing a XOR linked list. In many languages you just cannot XOR pointers, and/or it wouldn't make sense even if you could (consider garbage collection). Of course you can refer to every node by an integer ID and XOR those, but that's a workaround, not a direct implementation.
HLL's often have trouble implementing unstructured control flow, though many (particularly older) languages offer a goto. That means you may have to jump through hoops to implement a state machine (using a switch in a loop or whatever), instead of letting the state be implied by the program counter.
There are also many algorithms and data structures that rely on operations that don't exist in typical HLL's, for example popcnt or lzcnt, which can again be emulated, but then so can everything.
In case you have strict limitations in terms of memory and/or execution time, you might be forced to use assembly language.
High level languages typically require a run-time library which might be too big to fit into your program memory.
Think of a time-critical driver routine. An interrupt service routine for example. If there are only a few nanoseconds available for the routine, assembly language might be the only viable option.
How about this? You need to write some assembly code in order to access system registers and tables. But onces the setup is done, no CPU instructions are executed (everything's done by the complex CPU exception handling mechanisms) and yet the thing is Turing-complete and can "run" programs.
Let's say we have two integers a and b. which way is faster for swapping their values?
c=a;
a=b;
b=c;//(edited typo)
or
a=a+b;
b=a-b;
a=a-b;
or bitwise xor
a=a^b;
b=a^b;
a=a^b;
I'll test its performance differences when I'll be able but I'd like to know it now. Is it bitwise?
Firstly, you cannot quantify the speed of an algorithm independent of the program language, the compiler and the platform on which it is run. An algorithm is a mathematical abstraction.
Having said that:
for a typical programming language,
and a typical compiler, and
a typical execution platform,
the first version will typically be faster because it will typically compile to fewer native instructions that take less clock cycles to execute. The first version only requires load and save operations. The other two versions have (at least) the same number of loads and saves, and some additional arithmetic or bit manipulation instructions.
However, even that is not cut-and-dry.
The 2nd and 3rd examples are performing the swap without using a temporary variable. This is something you might do if using an extra temporary variable was expensive. This might happen on a machine which didn't provide enough general purpose registers, and the relative cost of loading / saving to memory was large. In some circumstances, the native code equivalents could be optimal.
However ... and this is the real point ... the best strategy is to leave this kind of decision to the compiler. Unless you are prepared to put a huge amount of effort into micro-optimizing, the compiler is likely to be able to a better job than you can. Indeed, writing code in "cunning ways" is liable to make it harder for the compiler to optimize. (In the 3rd case for example, the compiler would need to figure out that that sequence is actually swapping 2 variables before it can substitute the optimal instruction sequence. Chances are that the optimizer won't be able to do that.)
I have recently begun working with logic level design as an amateur hobbyist but have now found myself running up against software, where I am much less competent. I have completed designing a custom 4 bit CPU in Logisim loosely based on the paper "A Very Simple Microprocessor" by Etienne Sicard. Now that it does the very limited functions that I've built into it (addition, logical AND, OR, and XOR) without any more detectable bugs (crossing fingers) I am running into the problem of writing programs for it. Logisim has the functionality of importing a script of Hex numbers into a RAM or ROM module so I can write programs for it using my own microinstruction code, but where do I start? I'm quite literally at the most basic possible level of software design and don't really know where to go from here. Any good suggestions on resources for learning about this low level of programming or suggestions on what I should try from here? Thanks much in advance, I know this probably isn't the most directly applicable question ever asked on this forum.
I'm not aware of the paper you mention. But if you have designed your own custom CPU, then if you want to write software for it, you have two choices: a) write it in machine code, or b) write your own assembler.
Obviously I'd go with b. This will require that you shift gear a bit and do some high-level programming. What you are aiming to write is an assembler program that runs on a PC, and converts some simple assembly language into your custom machine code. The assembler itself will be a high-level program and as such, I would recommend writing it in a high-level programming language that is good at both string manipulation and binary manipulation. I would recommend Python.
You basically want your assembler to be able to read in a text file like this:
mov a, 7
foo:
mov b, 20
add a, b
cmp a, b
jg foo
(I just made this program up; it's nonsense.)
And convert each line of code into the binary pattern for that instruction, outputting a binary file (or perhaps a hex file, since you said your microcontroller can read in hex values). From there, you will be able to load the program up onto the CPU.
So, I suggest you:
Come up with (on paper) an assembly language that is a simple written representation for each of the opcodes your machine supports (you may have already done this),
Learn simple Python,
Write a Python script that reads one line at a time (sys.stdin.readline()), figures out which opcode it is and what values it takes, and outputs the corresponding machine code to stdout.
Write some assembly code in your assembly language that will run on your CPU.
Sounds like a fun project.
I have done something similar that you might find interesting. I also have created from scratch my own CPU design. It is an 8-bit multi-cycle RISC CPU based on Harvard architecture with variable length instructions.
I started in Logisim, then coded everything in Verilog, and I have synthesized it in an FPGA.
To answer your question, I have done a simple and rudimentary assembler that translates a program (instructions, ie. mnemonics + data) to the corresponding machine language that can then be loaded into the PROG memory. I've written it in shell script and I use awk, which is what I was confortable with.
I basically do two passes: first translate mnemonics to their corresponding opcode and translate data (operands) into hex, here I keep track of all the labels addresses. second pass will replace all labels with their corresponding address.
(labels and addresses are for jumps)
You can see all the project, including the assembler, documented here: https://github.com/adumont/hrm-cpu
Because your instruction set is so small, and based on the thread from the mguica answer, I would say the next step is to continue and/or fully test your instruction set. do you have flags? do you have branch instructions. For now just hand generate the machine code. Flags are tricky, in particular the overflow (V) bit. You have to examine carry in and carry out on the msbit adder to get it right. Because the instruction set is small enough you can try the various combinations of back to back instructions and followed by or and followed by xor and followed by add, or followed by and or followed by xor, etc. And mix in the branches. back to flags, if the xor and or for example do not touch carry and overflow then make sure you see carry and overflow being a zero and not touched by logical instructions and carry and overflow being one and not touched, and also independently show carry and overflow are separate, one on one off, not touched by logical, etc. make sure all the conditional branches only operate on that one condition, lead into the various conditional branches with flag bits that are ignored in both states insuring that the conditional branch ignores them. Also verify that if the conditional branch is not supposed to modify them that it doesnt. likewise if the condition doesnt cause a branch that the conditional flags are not touched...
I like to use randomization but it may be more work than you are after. I like to independently develop a software simulator of the instruction set, which I find easier to use that the logic also sometimes easier to use in batch testing. you can then randomize some short list of instructions, varying the instruction and the registers, naturally test the tester by hand computing some of the results, both state of registers after test complete and state of flag bits. Then make that randomized list longer, at some point you can take a long instruction list and run it on the logic simulator and see if the logic comes up with the same register results and flag bits as the instruction set simulator, if they vary figure out why. If the do not try another random sequence, and another. Filling registers with prime numbers before starting the test is a very good idea.
back to individual instruction testing and flags go through all the corner cases 0xFFFF + 0x0000 0xFFFF+1, things like that places just to the either side of and right on operands and results that are one count away from where a flag changes at the point where the flag changes and just the other side of that. for the logicals for example if they use the zero flag, then have various data patterns that test results that are on either side of and at zero 0x0000, 0xFFFF 0xFFFE 0x0001 0x0002, etc. Probably a walking ones result as well 0x0001 result 0x0002, ox0004, etc.
hopefully I understood your question and have not pointed out the obvious or what you have already done thus far.
Isn't every language compiled into low-level computer language?
If so, shouldn't all languages have the same performance?
Just wondering...
As pointed out by others, not every language is translated into machine language; some are translated into some form (bytecode, reverse Polish, AST) that is interpreted.
But even among languages that are translated to machine code,
Some translators are better than others
Some language features are easier to translate to high-performance code than others
An example of a translator that is better than some others is the GCC C compiler. It has had many years' work invested in producing good code, and its translations outperform those of the simpler compilers lcc and tcc, for example.
An example of a feature that is hard to translate to high-performance code is C's ability to do pointer arithmetic and to dereference pointers: when a program stores through a pointer, it is very difficult for the compiler to know what memory locations are affected. Similarly, when an unknown function is called, the compiler must make very pessimistic assumptions about what might happen to the contents of objects allocated on the heap. In a language like Java, the compiler can do a better job translating because the type system enforces greater separation between pointers of different types. In a language like ML or Haskell, the compiler can do better still, because in these languages, most data allocated in memory cannot be changed by a function call. But of course object-oriented languages and functional languages present their own translation challenges.
Finally, translation of a Turing-complete language is itself a hard problem: in general, finding the best translation of a program is an NP-hard problem, which means that the only solutions known potentially take time exponential in the size of the program. This would be unacceptable in a compiler (can't wait forever to compile a mere few thousand lines), and so compilers use heuristics. There is always room for improvement in these heuristics.
It is easier and more efficient to map some languages into machine language than others. There is no easy analogy that I can think of for this. The closest I can come to is translating Italian to Spanish vs. translating a Khoisan language into Hawaiian.
Another analogy is saying "Well, the laws of physics are what govern how every animal moves, so why do some animals move so much faster than others? Shouldn't they all just move at the same speed?".
No, some languages are simply interpreted. They never actually get turned into machine code. So those languages will generally run slower than low-level languages like C.
Even for the languages which are compiled into machine code, sometimes what comes out of the compiler is not the most efficient possible way to write that given program. So it's often possible to write programs in, say, assembly language that run faster than their C equivalents, and C programs that run faster than their JIT-compiled Java equivalents, etc. (Modern compilers are pretty good, though, so that's not so much of an issue these days)
Yes, all programs get eventually translated into machine code. BUT:
Some programs get translated during compilation, while others are translated on-the-fly by an interpreter (e.g. Perl) or a virtual machine (e.g. original Java)
Obviously, the latter is MUCH slower as you spend time on translation during running.
Different languages can be translated into DIFFERENT machine code. Even when the same programming task is done. So that machine code might be faster or slower depending on the language.
You should understand the difference between compiling (which is translating) and interpreting (which is simulating). You should also understand the concept of a universal basis for computation.
A language or instruction set is universal if it can be used to write an interpreter (or simulator) for any other language or instruction set. Most computers are electronic, but they can be made in many other ways, such as by fluidics, or mechanical parts, or even by people following directions. A good teaching exercise is to write a small program in BASIC and then have a classroom of students "execute" the program by following its steps. Since BASIC is universal (to a first approximation) you can use it to write a program that simulates the instruction set for any other computer.
So you could take a program in your favorite language, compile (translate) it into machine language for your favorite machine, have an interpreter for that machine written in BASIC, and then (in principle) have a class full of students "execute" it. In this way, it is first being reduced to an instruction set for a "fast" machine, and then being executed by a very very very slow "computer". It will still get the same answer, only about a trillion times slower.
Point being, the concept of universality makes all computers equivalent to each other, even though some are very fast and others are very slow.
No, some languages are run by a 'software interpreter' as byte code.
Also, it depends on what the language does in the background as well, so 2 identically functioning programs in different languages may have different mechanics behind the scenes and hence be actually running different instructions resulting in differing performance.