Legion Posted January 12, 2011 Share Posted January 12, 2011 Okay, okay, so some of you might want to give me a hard time about citing Rosen's work too frequently. But I can't help it. What if you had lived during the time of Newton? Further, what if you had stumbled onto his work before many others had? What if you didn't completely grasp all the implications of his work but had an intuition that it was an addition to human understanding? Well, I think Rosen is the Newton of biology. However here I don't want to speak of organisms; rather let us speak of machines. This is a small sample of what's in chapter 9 of Life Itself which is entitled "Relational Theory of Machines". I present this here in the hopes that some of you may think about it, see if it agrees or disagrees with your reason, and perhaps share your thoughts. I may have condensed this too much and been overly brief. This will be a rapid abstraction. If the reader feels that I have skipped vital steps then I invite them to tell me where clarification is required. We are aiming for a formal representation of machines. This will capture the organizational aspect of machines and make the entailments (implications) associated with them apparent. Rosen begins with a definition of machines. A machine is a natural system such that 1) all models are simulable and 2) there is a model which is a mathematical machine. It is meaningful to speak of all the states of a machine or a mechanism. We utilize the fact that a machine has a model which is a mathematical machine. We can thereby segregate all of its states into hardware states and software states. Hardware is a processor of software. It therefore has an inherent polarity; it has input and output. This polarity manifests the flow of time in the system dynamic; the flow is from input to output. With respect to hardware, input and output are both software. However there can be plenty of software which neither input nor output. Hardware induces a flow on software from input to output via software states which are neither input nor output. This can all be rather tersely represented as... hardware ==> (input --> output) ... where ==> is hardware induction and --> is name of flow on software from inputs to outputs induced by hardware. Written with a bit more abstraction... f => (a --> f(a)) And in categorical terms it is written... f: A --> B This is a mapping f with domain A and codomain B. Given this mapping we can ask, "Why f(a)?" and there are two answers: 1)* because a {- A, and 2) because f. That's it. How does this sit with you so far? * the symbol "{-" indicates membership Link to comment Share on other sites More sharing options...
Legion Posted January 13, 2011 Author Share Posted January 13, 2011 No one wants to play with me? Link to comment Share on other sites More sharing options...
Ouroboros Posted January 13, 2011 Share Posted January 13, 2011 I'm not sure what to contribute. How do we play the game? Link to comment Share on other sites More sharing options...
Legion Posted January 13, 2011 Author Share Posted January 13, 2011 It's pretty much a stepwise progression, I think Hans. You could start at the definition of machines here... A machine is a natural system such that 1) all models are simulable and 2) there is a model which is a mathematical machine. ... and voice your thoughts about it. Agree? Disagree? Why? Don't know what some of the terms might mean? Ask me and I'll try to clarify. Link to comment Share on other sites More sharing options...
Ouroboros Posted January 13, 2011 Share Posted January 13, 2011 Okay. Well, I both agree and disagree with the definition. I think it will be really difficult to explain exactly what I mean. Perhaps it is easier to stay with his definition and accept it as that, and also maybe we need to clarify it or rephrase it to make sure we know what we're talking about. Let me paraphrase the definition and let me know if it's proper: A machine is 1) a natural system 2) it simulates or emulates something else 3) it follows a mathematically predictive behavior (algorithmic). While you're responding to my definition, would a quantum computer still be qualified within this definition, and if so, how? Link to comment Share on other sites More sharing options...
Legion Posted January 13, 2011 Author Share Posted January 13, 2011 I think it will be really difficult to explain exactly what I mean. I'll bear with you if you bear with me. I am sorting through thoughts as we go. Let me paraphrase the definition and let me know if it's proper: A machine is 1) a natural system Yes. e.g. a toaster, a car, a laptop computer 2) it simulates or emulates something else No, all of it's models can be simulated. Like they do with satellite orbits and try to do with weather. 3) it follows a mathematically predictive behavior (algorithmic). I pretty sure that math is not limited to algorithmic things. Math is richer than algorithm alone. Edit: The natural system's behavior can be predicted using a model which is a mathematical machine (e.g. Turing machine, von Neuman machine, cellular atomata). While you're responding to my definition, would a quantum computer still be qualified within this definition, and if so, how? Can a quantum computer be partitioned into hardware states and software states? Link to comment Share on other sites More sharing options...
Ouroboros Posted January 13, 2011 Share Posted January 13, 2011 I'll bear with you if you bear with me. I am sorting through thoughts as we go. Fair enough. Yes. e.g. a toaster, a car, a laptop computer Even biological life would be a natural system, right? No, all of it's models can be simulated. Like they do with satellite orbits and try to do with weather. I'm not sure I understand. With "models" do you mean the same thing as "functions"? If so, are you saying that it is a machine just because it's functions (models) can be simulated? And if they can't be simulated, then it's not a machine? I pretty sure that math is not limited to algorithmic things. Math is richer than algorithm alone. Edit: The natural system's behavior can be predicted using a model which is a mathematical machine (e.g. Turing machine, von Neuman machine, cellular atomata). I'm not sure exactly what you mean that it's a "mathematical" machine. We're about to define what a machine is, so to say that a machine is a mathematical machine becomes tautological. Or are you saying that a machine is a device that can be simulated by a mathematical machine? (Similar to previous point) Can a quantum computer be partitioned into hardware states and software states? I'm not sure. That's why I'm asking. Are you suggesting that the hardware and software states also are part of the definition of a machine? Link to comment Share on other sites More sharing options...
Legion Posted January 13, 2011 Author Share Posted January 13, 2011 Even biological life would be a natural system, right? Yes, organisms are natural systems. But I was hoping we could concentrate on machines. With "models" do you mean the same thing as "functions"? Models are mathematical mirrors. If we have a model of a natural system then we can accurately predict its behavior and we can look at the implications within the math for explanations. If a natural system has a model which cannot be simulated then it is not a machine. I'm not sure exactly what you mean that it's a "mathematical" machine. I was assuming that you knew what a Turing machine is. It is an example of a mathematical machine. It exists in the abstract. It's a mathematical construct. Are you suggesting that the hardware and software states also are part of the definition of a machine? All of a machine's states can be partitioned into hardware states and software states. If a quantum computer has a set states that can be partioned into hardware and software then it is a machine. I guess the question to ask is... Can a quantum computer be given software? Link to comment Share on other sites More sharing options...
Ouroboros Posted January 13, 2011 Share Posted January 13, 2011 Yes, organisms are natural systems. But I was hoping we could concentrate on machines. Right now, we're defining what constitutes a machine, which means that we have to go from higher order systems to lower order systems. We can't concentrate on machines unless we define what a machine really is, and if we define a machine as a machine or a machine as something that is not an organism, we haven't really defined anything. We have to say what it is, and what makes it different from the other natural systems. So the first point includes organisms, and the subsequent points limits the scope step by step. Models are mathematical mirrors. If we have a model of a natural system then we can accurately predict its behavior and we can look at the implications within the math for explanations. Ok. If a natural system has a model which cannot be simulated then it is not a machine. Then quantum computers would most likely fall outside of the definition, and perhaps even evolutionary computing. I was assuming that you knew what a Turing machine is. It is an example of a mathematical machine. It exists in the abstract. It's a mathematical construct. But it's physical too. It's based on some basic mathematical principles, but what is the difference between a Turing or Von Neuman mathematical machine and a computer? I know what Turing machine is and Von Neuman's computer model (which should be attributed to other people, he stole the ideas). But to say a machine is a machine, or a machine is something that other machines can replicate through simulation, starts to become tautological. Its definition goes in circles. All of a machine's states can be partitioned into hardware states and software states. If a quantum computer has a set states that can be partioned into hardware and software then it is a machine. I guess the question to ask is... Can a quantum computer be given software? I guess it can, in some way, but it can maintain an exponential amount of states, so a 300 qubit quantum computer would not be possible to simulate with another computer. The only way to "simulate" a quantum computer is to build another quantum computer doing exactly the same thing as the first one (a clone, not a simulation). Link to comment Share on other sites More sharing options...
Legion Posted January 13, 2011 Author Share Posted January 13, 2011 Right now, we're defining what constitutes a machine,... I thought we were discussing a definition which had been offered. I am mainly interested to know if you agree or disagree with it and why. I was assuming that you knew what a Turing machine is. It is an example of a mathematical machine. It exists in the abstract. It's a mathematical construct. But it's physical too. It's based on some basic mathematical principles, but what is the difference between a Turing or Von Neuman mathematical machine and a computer? A computer, like the natural system sitting on my desk and which I am using at the moment, can realize a Turing machine, but a Turing machine itself is a thing which exists in formal languages alone. Basically there is the idea of a computer (an idea), and the realization of a computer (a physical thing). Link to comment Share on other sites More sharing options...
Ouroboros Posted January 13, 2011 Share Posted January 13, 2011 I thought we were discussing a definition which had been offered. I am mainly interested to know if you agree or disagree with it and why. I'm trying to understand the definition, and I think it's too vague. A computer, like the natural system sitting on my desk and which I am using at the moment, can realize a Turing machine, but a Turing machine itself is a thing which exists in formal languages alone. Basically there is the idea of a computer (an idea), and the realization of a computer (a physical thing). But not so with the von Neumann machine. Am I supposed to assume an mathematical machine in principle, but not in practice? In other words, are you suggesting that this mathematical machine, which against we would baseline what machines are, is just a hypothetical machine, but not a machine that could be made? Or put it this way, a machine is a thing from which the models of the same machine could be simulated in a hypothetical mathematical machine. Is that what you're saying? If that's what you're saying, then I guess a quantum computer should be considered a machine, since it's not practically possible to create a computer that would simulate the quantum states, it is however possible to thing of a theoretical machine that would be able to simulate it. Correct? Link to comment Share on other sites More sharing options...
Legion Posted January 13, 2011 Author Share Posted January 13, 2011 Let's take a different approach Hans and see if we can get somewhere. Let's take a specific example. Would you agree with me that a toaster is a machine? According to the definition all of it's models must be simulable and at least one of those models must be a mathematical machine. Do you think a toaster's behavior can be completely simulated? And do you think a Turing machine could model a toaster? Link to comment Share on other sites More sharing options...
RankStranger Posted January 14, 2011 Share Posted January 14, 2011 I know quite a bit about machines- and math. And I really have no idea what you're talking about. Link to comment Share on other sites More sharing options...
Ouroboros Posted January 14, 2011 Share Posted January 14, 2011 Let's take a different approach Hans and see if we can get somewhere. Let's take a specific example. Would you agree with me that a toaster is a machine? According to the definition all of it's models must be simulable and at least one of those models must be a mathematical machine. Do you think a toaster's behavior can be completely simulated? And do you think a Turing machine could model a toaster? Yes. I think so. A counter question: can a Turing machine model a Turing machine? Sorry. That's a stupid question. Let's go with it. A Turing machine can model other machines. So the definition of a machine is that it can be modeled by a Turing machine. Where do we go from here? Link to comment Share on other sites More sharing options...
Legion Posted January 14, 2011 Author Share Posted January 14, 2011 A counter question: can a Turing machine model a Turing machine? Yes, I'm fairly certain it can. Even realized computers can emulate one another, aside from issues of processing power and memory. Let's go with it. A Turing machine can model other machines. So the definition of a machine is that it can be modeled by a Turing machine. Where do we go from here? Yes, this is one of two requirements for a natural system to be a machine and the one will will now utilize. Seeing that a model of the machine is a Turing machine, then among all of its states, some are hardware states and the remainder are software states. Would you agree or disagree and why? Link to comment Share on other sites More sharing options...
Ouroboros Posted January 14, 2011 Share Posted January 14, 2011 A counter question: can a Turing machine model a Turing machine? Yes, I'm fairly certain it can. Even realized computers can emulate one another, aside from issues of processing power and memory. And we assume that this is only true in principle, not in practice. An emulator/simulator will always be slower than the original object. Let's go with it. A Turing machine can model other machines. So the definition of a machine is that it can be modeled by a Turing machine. Where do we go from here? Yes, this is one of two requirements for a natural system to be a machine and the one will will now utilize. Seeing that a model of the machine is a Turing machine, then among all of its states, some are hardware states and the remainder are software states. Would you agree or disagree and why? What do you consider hardware states and software states? Can you give examples. Link to comment Share on other sites More sharing options...
par4dcourse Posted January 14, 2011 Share Posted January 14, 2011 I'm immersed in Vlatko Vedral's "Decoding Reality" and he touches on many of these arguments. He expands Shannon Information Theory into a link between quantum and classical physics. Interesting discussion guys. Link to comment Share on other sites More sharing options...
Legion Posted January 16, 2011 Author Share Posted January 16, 2011 What do you consider hardware states and software states? Can you give examples. Hans in terms of cellular automata (a mathematical machine) I think hardware would correspond with the "cells" and their transition rules, whereas software would be the states (e.g. {on, off}) of the cells. According to Rosen the hardware of a Turing machine is its reading head and software is the state of the tape. In von Neumann machines I guess hardware is CPU and software states are memory. The easiest example for me here are the cellular automata. Edit: Here's a cellular automata called the Game of Life processing a pattern called the Canada goose. Link to comment Share on other sites More sharing options...
Legion Posted January 16, 2011 Author Share Posted January 16, 2011 http://www.youtube.com/watch?v=XcuBvj0pw-E&feature=related Link to comment Share on other sites More sharing options...
Ouroboros Posted January 16, 2011 Share Posted January 16, 2011 Hans in terms of cellular automata (a mathematical machine) I think hardware would correspond with the "cells" and their transition rules, whereas software would be the states (e.g. {on, off}) of the cells. According to Rosen the hardware of a Turing machine is its reading head and software is the state of the tape. In von Neumann machines I guess hardware is CPU and software states are memory. The easiest example for me here are the cellular automata. Sounds good to me Edit: Here's a cellular automata called the Game of Life processing a pattern called the Canada goose. Ok. I've actually seen it before. Something cool that you might know about, the evolutionary process of mutation, selection, drift, flow, etc, is used in computer science for finding optimal solutions to non-linear problems. And my understanding is that it's used in research for artificial intelligence. Link to comment Share on other sites More sharing options...
Legion Posted January 17, 2011 Author Share Posted January 17, 2011 Okay Hans, I appreciate your participation here. We're moving right along. You seem to agree that we can partition a machine's states into hardware states and software states. Do you agree or disagree with the following?... Hardware is a processor of software. Software can be partitioned into input states, output states, and all other software states. Hardware induces a flow on software from input to output via software states which are neither input or output. Link to comment Share on other sites More sharing options...
Ouroboros Posted January 17, 2011 Share Posted January 17, 2011 Hardware is a processor of software. Sounds very restrictive. Hardware is also storage, interfaces, communication channels, and more. Without those parts, a processor can't do anything. The processor is only the translator of software code into physical events. Software can be partitioned into input states, output states, and all other software states. Most of the instructions are internal, like mathematical and data manipulation. Some of them are also there to change the state of the actual hardware, like interrupt mode, error handling, clock speed, etc. Hardware induces a flow on software from input to output via software states which are neither input or output. Ok. That relates to what I said above. Link to comment Share on other sites More sharing options...
andyjj Posted January 17, 2011 Share Posted January 17, 2011 Just to confuse matters (I am an electronics engineer) there isn't necessarily a great distinction between hardware and software - I work on things that are called Field Programmable Gate Arrays that are integrated circuits that you can program to do anything that you want...... more sort of 'firmware' if you like. One of the functions I need to implement frequently are "Finite State Machines' Link to comment Share on other sites More sharing options...
Legion Posted January 19, 2011 Author Share Posted January 19, 2011 Hans, Andy, I have not abandoned this thread. I just have other things going on over here. Link to comment Share on other sites More sharing options...
Legion Posted January 24, 2011 Author Share Posted January 24, 2011 Just to confuse matters (I am an electronics engineer) there isn't necessarily a great distinction between hardware and software - I work on things that are called Field Programmable Gate Arrays that are integrated circuits that you can program to do anything that you want...... more sort of 'firmware' if you like. One of the functions I need to implement frequently are "Finite State Machines' Okay Andy, let's do this. Here's my question for you. Could these FPGAs be modelled by a Turing machine? Link to comment Share on other sites More sharing options...
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