| aminer68@gmail.com: Mar 31 12:01PM -0700 Hello, About Markov chains in mathematics and more.. In mathematics, many Markov chains automatically find their own way to an equilibrium distribution as the chain wanders through time. This happens for many Markov chains, but not all. I will talk about the conditions required for the chain to find its way to an equilibrium distribution. If in mathematics we give a Markov chain on a finite state space and asks if it converges to an equilibrium distribution as t goes to infinity. An equilibrium distribution will always exist for a finite state space. But you need to check whether the chain is irreducible and aperiodic. If so, it will converge to equilibrium. If the chain is irreducible but periodic, it cannot converge to an equilibrium distribution that is independent of start state. If the chain is reducible, it may or may not converge. So i will give an example: Suppose that for the course you are currently taking there are two volumes on the market and represent them by A and B. Suppose further that the probability that a teacher using volume A keeps the same volume next year is 0.4 and the probability that it will change for volume B is 0.6. Furthermore the probability that a professor using B this year changes to next year for A is 0.2 and the probability that it again uses volume B is 0.8. We notice that the matrix of transition is: 0.4 0.6 0.2 0.8 The interesting question for any businessman is whether his market share will stabilize over time. In other words, does it exist a probability vector (t1, t2) such that: (t1, t2) * (transition matrix above) = (t1, t2) [1] So notice that the transition matrix above is irreducible and aperiodic, so it will converge to an equilibrum distribution that is (t1, t2) that i will mathematically find, so the system of equations of [1] above is: 0.4 * t1 + 0.2 * t2 = t1 0.6 * t1 + 0.8 * t2 = t2 this gives: -0.6 * t1 + 0.2 * t2 = 0 0.6 * t1 - 0.2 * t2 = 0 But we know that (t1, t2) is a vector of probability, so we have: t1 + t2 = 1 So we have to solve the following system of equations: t1 + t2 = 1 0.6 * t1 - 0.2 * t2 = 0 So i have just solved it with R, and this gives the vector: (0.25,0.75) Which means that in the long term, volume A will grab 25% of the market while volume B will grab 75% of the market unless the advertising campaign does change the probabilities of transition. So notice that in my previous post i have talked about invariants of the system, here is my previous post read it carefully: About Turing completeness and parallel programming.. You have to know that a Turing-complete system can be proven mathematically to be capable of performing any possible calculation or computer program. So now you are understanding what is the power of "expressiveness" that is Turing-complete. For example i am working with the Tool that is called "Tina"(read about it below), it is a powerful tool that permits to work on Petri nets and be able to know about the boundedness and liveness of Petri nets, for example Tina supports Timed Petri nets that are Turing-complete , so the power of there expressiveness is Turing-complete, but i think this level of expressiveness is good for parallel programming and such, but it is not an efficient high level expressiveness. But still Petri nets are good for parallel programming. Read the rest of my previous thoughts to know more: About deadlocks and race conditions in parallel programming.. I have just read the following paper: Deadlock Avoidance in Parallel Programs with Futures https://cogumbreiro.github.io/assets/cogumbreiro-gorn.pdf So as you are noticing you can have deadlocks in parallel programming by introducing circular dependencies among tasks waiting on future values or you can have deadlocks by introducing circular dependencies among tasks waiting on windows event objects or such synchronisation objects, so you have to have a general tool that detects deadlocks, but if you are noticing that the tool called Valgrind for C++ can detect deadlocks only happening from Pthread locks , read the following to notice it: http://valgrind.org/docs/manual/hg-manual.html#hg-manual.lock-orders So this is not good, so you have to have a general way that permits to detect deadlocks on locks , mutexes, and deadlocks from introducing circular dependencies among tasks waiting on future values or deadlocks you may have deadlocks by introducing circular dependencies among tasks waiting on windows event objects or such synchronisation objects etc. this is why i have talked before about this general way that detects deadlocks, and here it is, read about it in my following thoughts: Yet more precision about the invariants of a system.. I was just thinking about Petri nets , and i have studied more Petri nets, they are useful for parallel programming, and what i have noticed by studying them, is that there is two methods to prove that there is no deadlock in the system, there is the structural analysis with place invariants that you have to mathematically find, or you can use the reachability tree, but we have to notice that the structural analysis of Petri nets learns you more, because it permits you to prove that there is no deadlock in the system, and the place invariants are mathematically calculated by the following system of the given Petri net: Transpose(vector) * Incidence matrix = 0 So you apply the Gaussian Elimination or the Farkas algorithm to the incidence matrix to find the Place invariants, and as you will notice those place invariants calculations of the Petri nets look like Markov chains in mathematics, with there vector of probabilities and there transition matrix of probabilities, and you can, using Markov chains mathematically calculate where the vector of probabilities will "stabilize", and it gives you a very important information, and you can do it by solving the following mathematical system: Unknown vector1 of probabilities * transition matrix of probabilities = Unknown vector1 of probabilities. Solving this system of equations is very important in economics and other fields, and you can notice that it is like calculating the invariants , because the invariant in the system above is the vector1 of probabilities that is obtained, and this invariant, like in the invariants of the structural analysis of Petri nets, gives you a very important information about the system, like where market shares will stabilize that is calculated this way in economics. About reachability analysis of a Petri net.. As you have noticed in my Petri nets tutorial example (read below), i am analysing the liveness of the Petri net, because there is a rule that says: If a Petri net is live, that means that it is deadlock-free. Because reachability analysis of a Petri net with Tina gives you the necessary information about boundedness and liveness of the Petri net. So if it gives you that the Petri net is "live" , so there is no deadlock in it. Tina and Partial order reduction techniques.. With the advancement of computer technology, highly concurrent systems are being developed. The verification of such systems is a challenging task, as their state space grows exponentially with the number of processes. Partial order reduction is an effective technique to address this problem. It relies on the observation that the effect of executing transitions concurrently is often independent of their ordering. Tina is using "partial-order" reduction techniques aimed at preventing combinatorial explosion, read more here to notice it: http://projects.laas.fr/tina/papers/qest06.pdf About modelizations and detection of race conditions and deadlocks in parallel programming.. I have just taken further a look at the following project in Delphi called DelphiConcurrent by an engineer called Moualek Adlene from France: https://github.com/moualek-adlene/DelphiConcurrent/blob/master/DelphiConcurrent.pas And i have just taken a look at the following webpage of Dr Dobb's journal: Detecting Deadlocks in C++ Using a Locks Monitor https://www.drdobbs.com/detecting-deadlocks-in-c-using-a-locks-m/184416644 And i think that both of them are using technics that are not as good as analysing deadlocks with Petri Nets in parallel applications , for example the above two methods are only addressing locks or mutexes or reader-writer locks , but they are not addressing semaphores or event objects and such other synchronization objects, so they are not good, this is why i have written a tutorial that shows my methodology of analysing and detecting deadlocks in parallel applications with Petri Nets, my methodology is more sophisticated because it is a generalization and it modelizes with Petri Nets the broader range of synchronization objects, and in my tutorial i will add soon other synchronization objects, you have to look at it, here it is: https://sites.google.com/site/scalable68/how-to-analyse-parallel-applications-with-petri-nets You have to get the powerful Tina software to run my Petri Net examples inside my tutorial, here is the powerful Tina software: http://projects.laas.fr/tina/ Also to detect race conditions in parallel programming you have to take a look at the following new tutorial that uses the powerful Spin tool: https://mirrors.edge.kernel.org/pub/linux/kernel/people/paulmck/perfbook/perfbook.html Here is how to install Spin Model Checker on windows: I invite you to look at this video to learn how to install Spin model checker and iSpin: https://www.youtube.com/watch?v=MGzmtWi4Oq0 I have installed them and configured them correctly and i am working with them in parallel programming to detect race conditions etc. This is how you will get much more professional at detecting deadlocks and race conditions in parallel programming. Thank you, Amine Moulay Ramdane. |
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