The Markov property is described in Wikipedia as "the memoryless property of a stochastic process". Ok, so there are (at least) two concepts here that need further description: memoryless(ness), and stochasticity. A Markov process is one whose future states don't depend on its past states, roughly speaking. A common analogy is the drunken walk, or a drunk man looking for his lost keys, staggering from one street light to another who also"forgets" where he's been. Then a future location for the drunk man depends on where he is at present. Hence, roughly speaking some more, time is defined only in a present tense; the past has no influence and there are many possible futures. Another analogous idea is that of a switchboard where "operators" wait for a request to be connected to somewhere; this waiting period is random for each operator (a single input). Each operator forgets previous requests and acts when they have a "present" request. The stochastic part is, again roughly, the emergence of patterns from the randomness (of wait times for each input of the switching network). Does randomness mean we need to review what we mean by "the past", or by "deterministic"? Or is randomness just a kind of "mathematical trick", since memory and remembering the past are obviously not some kind of trick? Heh.
There are plenty of examples of memoryless "systems", such as board games where players throw a pair of dice to determine their next move. Obviously this move depends on the dice and the player's present position, and not on which squares (or what have you) a player has been on in the past. So, a notion of independence exists -- the past and the future in terms of "events", don't depend on each other. Found a good link that discusses Markov chains and something called Monte Carlo simulation here: http://stats.stackexchange.com/questions/165/how-would-you-explain-markov-chain-monte-carlo-mcmc-to-a-layperson . But, this term "Markov property" is used in what seems to be a completely different sense (or context perhaps), in Kauffman's book Knots and Physics: "[Vaughan Jones] defined a trace \( tr: M_n \rightarrow \mathbb C \) (complex numbers), such that it satisfied the Markov property: \( tr( \omega e_i) = \gamma\, tr( \omega) \) for \( \omega\) in the algebra generated by \( M_0, e_1, ..., e_{i-1} \)." Hmm.
In a board game, does not your next move, depend on your current position, which depends on past moves? In that sense the past defines what possible future moves exist.
Quite so, but think of this: given a position on the board, how many different past moves could have gotten you there? And you're at some position now, so if you throw the dice, how do any past moves determine where you will land next since this is determined by the dice you just threw?
Once you have advanced to a specific location, only the precise past moves that brought you there are of any consequence. And while it is true that the roll of dice alone may determine your next move, that next move is first limited by the location your past moves have left you with.
However, any previous roll of the dice has no influence on the next roll, correct? Therefore, the next roll of the dice and your present position are all that determines the next move, not any previous roll or any previous position. It's not complicated. What about operators on a switchboard? Does any previous wait time influence the time the next request will arrive?
The board game is a far easier situation. What I was pointing out is that even though the next roll is random, the choices it represents is dependent on past moves. The past is fixed. If in the last move you rolled a seven your possible next moves are different than if it had been a three. Where you are right now does have an affect on where you can move to next, even while your next roll is random... And where you are now is no longer a variable, nor are any of the past rolls or decissions that led to where you are. The switchboard situation is far more complex, but at its simplest, when the next request arrives should have nothing to do with the operator, other than whether the operator is available to respond. All of the past and present decisions are independent of the operator's actions. Operator availability is the only, operator dependent factor.
I'll go over your points again I believe that must be a misconception. If you are at a specific location on the board (where else could you be?), then there is a set of past moves that could have gotten you there; remembering the actual throws for your past moves has no relevance to the "game", does it? That's true, likewise the drunk man doesn't have to stagger very far to the next street light. I think you're thinking that because the positions on the board are numbered, say. Past moves got you to where you are and now you have the same randomly generated "choice", which obviously isn't a choice at all. The drunk man "chooses" to stagger in some direction and eventually finds another street light, forgetting not just where he was, but how he got there and which direction he staggered from. No, you have only one choice after rolling the dice; whether you are on the seventh square or the third square makes no difference to this fact. Yes, the effect is that the randomly generated "next" move won't be the start position, unless it's your first throw. It's about what you can say is "true" about the system; what are the facts? You made a mistake, your position is always a variable and obviously generated randomly (with constraints). Consider a Rubik's cube. If you wrote down all the moves made when you "scrambled" it, then you can just reverse them. In general, people don't remember how they scrambled it, and yet, the cube is some "distance" from the solution. Do you try a random approach; do you try some kind of algorithm and is it deterministic or does it employ some randomness? etc
Well, it would seem that the Markov "property" depends on the context after all. A stochastic process can have this property as discussed; the other context is that of braid groups. So, at first glance, the stochastic process property and the braid property are unrelated. Heh
If i've understood yourself and arfa correctly. I think what Arfa is say is that if you're on the 6th square, then where you land next is a function of the next dice role and the fact that you're on the 6th square. It doesn' matter whethere you got there in one roll or six, and it doesn't matter if you got there with a 5 and a 1 or three consecutive twos.
There is no dispute about the random nature of any future roll of the dice and future moves. The past moves and rolls on the other hand, while they may have been random.., once made the random nature of past events is lost. They become completely determined.... There are many possible futures but only one past... And that one fixed past put you where you stand to start the next move. The role of the dice is random but there was a single fixed path to the square that role is made from. Even for the drunk who does not remember his path, there was only one path that lead him to where he is. You can imagine an almost infinite number of paths that might have lead to where you are today, but there is only one that actually brought you to where you are. In his first post he said the past has no influence, which is not entirely true. The past does not affect the next roll of the dice, but it does determine which square that role moves forward from. Thus the past does have some influence even on random next moves.
The randomness is "lost" in respect of a position in the present, because that position is a definite one. In the board game scenario, you trace out a definite path which is randomly generated, the drunk man does this too. I would agree that the past does have an influence, but only on the present position. Past events (as in which street lights the drunk man has visited before) don't have any influence on future events, which is what I meant to say. This is true th0ugh only for a process with completely random inputs.
I was nitpicking the use of real world analogy, in describing the influence that past actions/paths have on future moves. And to be honest it may have been, at least in part, the way I read the OP. Your first sentence above clarifies the issue. A board game like Monopoly is a good real world example, because no matter what series of past moves leaves you on a particular square, the dice and your current position solely determine the next move. When you introduce even a drunk man walking or switchboard operator, there are too many additional variables, most of which even if they are subconscious (the drunk walk) are still the product of past moves/experience. You concluded the OP with, The concept of time is one that has been long debated and has yet to reach any general consensus. Statistically time is just a measurement of change.., and other than conceptually, change is something that occurs between the present and the future. The past is, as you mentioned above fixed. Then.., conceptually, as a mater of experience, time becomes far more complex, and involves more than just statistical changes, because we conceptually associate change with the sequence of past events/moves, even where there can be no real variation. A drunk man may not know why he moves from one particular pole to the next, but there is a significant probability that his past plays a role... I don't think that any statistical system can be successfully applied to an individual, even where patterns do develope in large groups/communities. For an individual accurate knowledge of past actions is needed, even to begin to predict future actions/decisions/moves. Thus I don't believe that the Markov process can be applied to individuals. It would take a great deal more thought to say one way or the other, in the case of large groups/communities, where past experience still plays a significant role in future decisions, but I am skeptical that it could be applied to any situation where there is a memory of past, whether that is a conscious or not. An example... There are drugs that are manufactured through different processes, different sequences of chemical reactions.., and yet test in vitro as identical molecules. They could then be thought of as starting out on the same square so to speak. The problem is that sometimes, they do not metabolize the same in vivo. In a sense you could think of each as having and undetectable in vitro memory that affects their in vivo effect. ... A Markov process is more ridgely defined than it seems the question in your OP suggests. In other words not all random actions fit within the confines of a Markov process. The concepts of time, as in past, present and future, and determinism, seem to me, far to complex.
for some reason I can't paste anything into the editor at the moment. No matter, the definition of a Markov process is: a stochastic process that satisfies the Markov property. If the future state of a process can be predicted given its present state, and if this prediction is the same ("as good") as that based on its complete history, then it satisfies the Markov property. So in the board game scenario what you can say is the next throw of two dice will be a number from 2 to 12 inclusive, and that the history is irrelevant since you're already at some position (determined by that history). In the switchboard scenario, each operator is either busy connecting a request for a channel or waiting for a request, again, the history is irrelevant to what is happening "now" at the switchboard. What is a reasonable way to describe, mathematically, what happens over time at the switchboard? Suppose each operator in wait mode has an input stream of zeros, this changes to a 1 when a request arrives. So the length of the string of 0s corresponds to a wait time, it's an interval timer for each operator. So for n operators (these are abstractions, they could be mechanisms of some kind) there are n inputs. If these are completely random there is no way to predict how any of them will change over time, singly or together. The best we can do is look for patterns. If however, the inputs are not completely random or are constrained by something else (for instance there are 12 independent faces on two dice, but you "throw" a number from the set: {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}), then things are, well, different. Perhaps more complicated. So if you suppose there is a predictable pattern in time, over n inputs to a switchboard, then you should be able to expect certain things: such as when it will be busy or idle overall. In an actual old-style phone network, the operators were human and so were the requests, so it tended to be busy during the day and idle at night; the constraint here is human behaviour over a 24-hour cycle. For example . . .
This random input stream as a timer between requests is just a first stab at constructing a reasonable model. If this stream is separate from those used by the requesters, as in, an operator doesn't know or need to know about what's being connected, then it's also information about the states of each operator, they are either waiting or busy. You might assume that the operator has enough time between requests, to process a request (by connecting an input to an output); perhaps there should be a string of 1s long enough to mark this time-to-process a request, between the random strings of 0s. Or you could enable a requester to see (hear) that an operator is busy--try again later, and so on. As with the drunken walk, there are two states: the drunk man is either at some street light, or moving; in the board game you are at some location on the board, or moving to the next location. The busy/waiting "function" looks the same in each case.
Time is not random, but it may indeed be stochastic, or at least partly so. As long as real EM propagates in a single direction, time has a definitive arrow, if not a definitive rate, in all reference frames. That is not random. Its future state is predictable with any clock. Hence, not Markov. Inertia is essentially dynamic memory of motion, but it only remembers the present state; not any previous states or how a particle or bound energy attained its current FoR.
Well, if time is an input, it can be random; that's as an input to a process. If it's an input, it's an input not a process . . . (?) I was only wanting to discuss what relevance this idea of a process with random inputs might have to time itself. Randomness does seem to affect how we can describe (usefully or otherwise) a process; is that like a "law" of nature, say? What is a law of nature, or physics (ye cannae change the laws of physics, Jim)? Apparently, it's an old-fashioned way of describing a "principle" of physics pertaining to motion, energy, conservation (of mass or momentum), and so on. What about information? Is there a set of "laws" such as: "the total amount of information is conserved"? Are the laws or principles different (in which case, how are they different) between classical and quantum domains in which "information" exists and can be "read or written" (i.e. measured or prepared)? See it? BTW, I'm fairly sure I'm not the only person asking questions like this.
See: http://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory and also, http://en.wikipedia.org/wiki/Jarzynski_equality that one actually has some interesting temporal flow properties related to thermodynamics These appear to be relevant to your query., but also a bit out of my depth and area of interest. It's just too easy to partition a thermodynamic system wrong and wind up with a design for a perpetual motion machine, which would quickly be excluded from discussion in a forum such as this.
Randomness is a well-defined property of processes, but what about numbers or sets? Suppose you randomly sprinkle some pointlike objects on a surface. Suppose further that the set of points is distributed in a group, there is a region occupied by the points, and a region in which no points appear--the set has a boundary, or you can draw a closed curve around the whole set. These seem to be reasonable assumptions (!) So you have n distinct points which can be supposed to be identical; the "distinctness" is entirely a property of their different positions. That is, exchanging any two points leaves the space unchanged (a definition of an identity transformation!). If you label the points though, say with natural numbers, they are all distinguished from each other since each has a unique number attached. There are n! ways to number n points; any random labeling (or random walk using successive numbers to label each vertex) over the set of points such that the n points are mapped one-to-one to the first n natural numbers, leaves the n points distinguished from each other. If you connect points together with edges, any edge is described by its endpoints. There is a much larger set of possible edges over the set of points, hence a much larger space for randomly generated graphs. The first kind of randomness, sprinkling points on a surface, still assumes some kind of locality; you can't sprinkle n points on a surface and cover an infinite region, the finiteness of n implies a finite region, but that's geometry. The second kind of randomness in labeling n points with n numbers (starting with 1) is constrained by n. Once you have n labeled points, edges between them also means a direction can be defined, for random or other kinds of graph traversal. A set of edges over some graph defines a kind of fixed structure, however (it won't change under topological deformations: bending, stretching or shrinking). Randomness, or a way to measure it, seems to require that something be fixed, and not random at all.
