On a Networked Automata Spacetime and its Application to the Dynamics of States in Uniform Translatory Motion

Transcription

1 On a Networked Automata Saetime and its Aliation to the Dynamis of States in Uniform Translatory Motion Othon Mihail Deartment of Comuter Siene, University of Liverool, Liverool, UK Othon.Mihail@liverool.a.uk Reent advanes in a number of disilines indiate that a variety of hysial and biologial systems are governed by information-based laws and an be formalized and aurately analyzed through omutational aroahes. However, these develoments have been rather ad ho and a general onnetion between omuting and the hysial world has not been drawn, one of the reasons being that there is urrently no generally agreed unifying model of all forms of omuting when omuting entities are merged with a ossibly hysial environment. We here exlore suh a model based on networked automata and by adoting the viewoint that information might be a building blok of hysial systems, we aly this model to the dynamis of states (reresenting light and bodies) in uniform translatory motion. This allows us to obtain formal desritions and roofs for a number of hysial settings and henomena, inluding deiding simultaneity and determining absolute seeds. Moreover, we highlight that in this formal setting the rinile of relativity (and the resulting Lorentz transformation) orresonds to a seial method of measuring, whih gives a non-otimal worst-ase aroximation of the absolute quantities involved. Among other onsequenes, these findings leave oen the ossibility of relativity being a roerty of measurements erformed under inomlete knowledge. The landmark work of Turing 1, among other ontributions, equied us with a model of omutation so simle yet so owerful that algorithms are now defined as exatly those dynami roedures that an be exeuted by a Turing mahine. Sine the invention of the Turing mahine and u to now, the main, if not the only, method that one an emloy to haraterize the omutational aabilities of any system, is to develo a formal model of that system and rove its equivalene to a Turing mahine or a seial ase of it (e.g., a bounded-sae one). This is usually done by establishing a formal simulation of some Turing mahine in the model under onsideration. Our resent knowledge allows us to quite safely onlude that any disrete dynami system an be simulated by a Turing mahine. On the other hand, desite the general agreement on the universality of the Turing mahine, a notable number of researh areas of Comuter Siene have dearted from it to various degrees either as an essential ste to suessfully analyze the systems under onsideration or aiming at exloring if there is anything beyond it. For examle, in Online Comutation the inut is not known in advane and is instead revealed to the mahine during its ourse. Parallel Comuting mainly onerns itself with how more than one suh mahines an be interonneted and rogrammed to seed-u omutation. Distributed Comuting studies how a number of omuting entities (Turing mahines or weaker) an solve tasks that require oordination, suh as leader-eletion, onsensus, or omutation of funtions over the distributed inut as a whole, when those entities are deloyed in some environment, having to fae a number of (ossibly adversarial ) fators whih are not under the ontrol of the system itself (e.g., loal knowledge, failures, dynami ommuniation toology). In Artifiial Intelligene and Mahine Learning one or more suh omuting entities are usually studied not against a stati inut any more but a hallenging and ossibly omliated environment in whih they should be able to arry out some desired tasks by interating with the environment, learning it, and beoming aable of rediting its future behavior. These develoments have been largely driven by the growing realization that omuting is not just a mere artifiial roessing on abstrat (mathematial) inuts and that omuting entities an atually be deloyed, merged, and interat in meaningful ways with the natural world. One ould justifiably argue that eah of the aforementioned settings an be enoded by some omutation of a single Turing mahine, simulating both the environment and all the omuting entities involved. Still, this is almost never done in ratie for two main reasons: 1. A single-mahine simulation of a olletion of interating systems introdues unneessary omliations and hides the atual dynamis, 2. Due to a general belief that (hysial) environments are fundamentally different from omutation and annot be fully atured by it. Instead, the tyial aroah is to distinguish two involved systems in any setting under onsideration: A omuting system (whih ould be a olletion of omuting entities) and the environment in whih that system oerates. This has some imortant inonvenient, if not misleading, onsequenes. One is that omuting is still, and desite the ever inreasing indiations to the ontrary, regarded as an artifiial human-engineered onstrut that under some onditions an be emloyed and oerate in the real world. Another is that this has effetively led to the develoment of an unlimited set of different omuting models (artly resulting from the different tyes of environments that a omuting system an be oerating in), whih are usually inomarable to eah other as they are rarely redued to a ommon basis (given that no suh generally aeted ommon basis exists). Atually, this multiliity of models together with the fat that slight modifiations of a model may result in totally different formal roerties have led to a long-standing oen roblem in Comuter Siene, and Distributed Comuting in artiular: How an we unify these models and study them on a ommon basis?

2 The revailing dihotomy highlighted above artly exlains why omutational roesses of nature are still regarded as an exetion and inversely why the ability of omutation to exlain natural henomena has been so far treated with surrise or susiion. But this is in ontrast to a large number of reent and not so reent disoveries indiating that omutation might be something more than an artifiial human onstrut. Comuters and sueromuters are now tyially arrying out effiient and aurate simulations of any natural system, from whole ells 2 to the Universe itself 3. Chemial reations are essentially omuting and formal models of omutation have been roved equivalent to hemial reations 4;5;6. The Plank length seems to suggest that by restriting attention to a disretization of sae no information about the underlying dynamis of the Universe would be lost. The Standard Model of artile hysis 7 suorts, in agreement with exerimental evidene, that every hysial roess is the (higher-level) outome of the oexistene and interation of artiles from a relatively small disrete set of elementary artiles (as muh as Chemistry is the result of different arrangements and interations of around 118 elements). The mass-energy equivalene 8 indiates that different hysial quantities and henomena may just be different manifestations of a single underlying quantity. Biologial organisms rely on information maintenane and interretation (stored in their genome) 9;10, while biologial roesses like ell division and embryoni growth have started to be regarded as being essentially algorithmi 11. Relativisti effets learly show u in Distributed Comuting systems, when the omuting entities have only a loal limited ersetive and inomlete knowledge (Lamort s ausality 12 ). Artifiial Intelligene aroahes suessfully reliate various asets of the skills 13, behavior, and intelligene 14 of organisms outerforming them in many ases 15. Motivated in art by these findings, we here exlore an alternative viewoint that onsiders information and omutation as the underlying building bloks of nature. To this end, we emloy a formal model of omutation based on networked automata and use it to define a reasonable saetime and to reresent in it at the same time both the omutational roedures (e.g., observers making measurements) and the various hysial arameters. An imortant benefit of adoting a formal model like this is that it allows to rigorously argue about hysial roesses (e.g., rove imossibility results). To demonstrate this, after defining the model we aly it to formally analyze the dynamis of states in uniform translatory motion and to rove a number of statements about feasibility and imossibility of observations and measurements in this setting. Our findings (being at least formally valid in this model) inlude: absolute quantities (sae, time, seeds) are erfetly onsistent with relativisti effets, relativity turns out to be a roerty of measuring under inomlete knowledge, methods of measuring absolute quantities an be develoed and formally roved orret, rigorous imossibility results an be roved for a number of other inferene roblems, and how an observer ereives the various events deends only on how those events ausally influene the loal state of the observer and on the measuring aaity of the observer. The Model We begin with a definition of a Universe. A Universe U onsists of a set V of state laeholders (or nodes or ositions) and a network G = (V, E), where E V 2. We here restrit ourselves to finite V and onneted undireted simle grahs G. The Universe also has a finite set of states Q assoiated with it, where ε Q denotes the emty state, and a transition funtion δ seifying how states are to be udated. We only fous on deterministi transition funtions in the resent study. In artiular, the transition funtion is defined as δ : Q (Q { }) (G) Q, reresenting the fat that the state of a node x is udated (deterministially) based on its resent state and the resent states of its neighbors, (G) denoting the maximum degree of a node in G and the absene of a neighbor. As we shall here exlusively fous on k-regular grahs, we may simlify the definition of the transition funtion as δ : Q k+1 Q. For examle, a 2-dimensional universe ould onsist of a square grid urved around a shere, so that every node is on a irle of latitude having one east and one west neighbor and a irle of longitude having one north and one south neighbor, in whih ase δ would have been defined as δ : Q 5 Q, where in δ(q 0, q 1, q 2, q 3, q 4), q 0 would denote the state of the node under udate and q 1, q 2, q 3, q 4 the states of its north, east, south, and west neighbors, resetively. The global state of the Universe at any given time is omletely desribed by the states of all nodes. Formally, a onfiguration of the Universe is a maing C : V Q seifying the state of eah node in V. The Universes onsidered here are dynami systems, starting from an initial onfiguration and rogressing in disrete synhronous stes. In every ste i, the transition funtion δ is alied onurrently to every node u V, taking as inut (C i(u), C i(u 1), C i(u 2),..., C i(u k )), where u j denotes the jth neighbor of u, and rodues a new state C i+1(u) for u. In this manner, a new onfiguration C i+1 is rodued from the resent onfiguration C i and we write C i C i+1 to denote that onfiguration C i rodues (or goes to) C i+1. It should be noted that δ being deterministi imlies that for any onfiguration C, C C an only be satisfied by exatly one onfiguration C. An exeution of a Universe is a finite or infinite sequene of onfigurations C 0, C 1, C 2,..., where C 0 is an initial onfiguration and C i C i+1, for all i 0. The model that we have just defined an be viewed as a generalization of Ullam s and von Neumann s Cellular Automata 16;17 to arbitrary networks. Our modeling assumtions so far, indiate that suh Universes have an inherent definition of sae and time. From the most general ersetive, sae is defined by the grah G and time is defined by the number of stes elased sine the exeution started, whih we shall often all true (or absolute) sae and time, resetively. We should emhasize at this oint that we make no assumtions about how the grah is drawn (e.g., lengths of edges onneting the nodes) neither about how muh time elases between two onseutive stes; all of our onstrutions and results shall be indeendent of suh undefined arameters (or in other words, working equally well for all ossible suh hoies). In what follows, (true) time an be interhanged by number of stes (or global tiks) in the exeution. We say that a time-node (u, t) otentially influenes (u, t ) if there is in G a simle ath uu of length at most t t. Potential influene reresents the fat that, in rinile, a state starting from (u, t) an reah (u, t ) through a (shortest) uu ath. To define atual influene we must take into aount that (u, t ) is atually influened by (u, t) only through a sequene of state-hanges (or events). We say that an event e ours at time-node (u, t) if C t(u) C t 1(u), that is, if a state-hange takes lae at node u at time t. We an now define (atual) ausal influene as a hain of events originating at one node and ending at some other node.

3 Prinile 1. A system S an observe the t-state of a system S only if the latter influenes a t -state of S, for some t > t. In other words, any system S observes any other system S in the Universe only through loal state hanges haening at S and being aused by S. These loal state hanges form what S ereives about S. If the loal exeution (or loal history) at S is the same in two distint sub-exeutions of the Universe then we may say that the two ases are indistinguishable for S. We distinguish a veloity omonent v(u, t) in the states of nodes satisfying the following roerties: 1. The emty state ε has no veloity omonent. 2. The veloity omonent of every non-emty state breaks down to a delay omonent l(u, t) and a diretion omonent r(u, t). 3. The delay omonent l(u, t) is a ositive integer in {MIN l, MIN l + 1,..., MAX l, }, for some redetermined finite integers MIN l and MAX l. The diretion omonent, defined for a k-regular grah, is an integer in {1, 2,..., k}. 4. A state with delay l(u, t) = l and diretion r(u, t) = r moves every l true stes of the Universe to the neighbor z with index 1 r. For examle, if in the 2-dimensional ase we define r(u, t) {,,, }, then a state with delay 100 and diretion moves every 100 global tiks one osition to the right. 5. Given the delay l(u, t), we may define the seed of a state as 1/l(u, t). A state with seed 0 (orresonding to l(u, t) = ) is alled stationary (or at absolute rest) and does not move at all. Then seed = 1/MIN l is the maximum and seed 1/MAX l the minimum ossible (nonzero) seeds in the Universe. 6. In the systems in uniform translatory motion that we shall onsider (inertia systems), the veloity omonent of a state, one set, is never altered unless it is exliitly said so. We leave the latter ossibility to allow for exeriments inluding, for examle, states being observed and then simly disarded or states being erfetly refleted (in whih ase their diretion omonent is reversed ). Remark 1. In general, it is not neessary that the minimum ossible delay oinides with the global tik of the Universe, therefore we assume here that the minimum ossible delay is a MIN l 1 and we allow in rinile for in between (faster) events to be only internal events of the states of the nodes, that is, the state of a node hanging without affeting the states of its neighbors. In aordane with Physis, we shall all = 1/MIN l the seed of light. We imagine that states move aording the following welldefined roedure: A state with delay l(u, t) = l and diretion r(u, t) = r, imlements an l-ounter ount starting from ount = l. With every global tik, if ount = 1 then the state moves to the neighbor of u in diretion r and ount l, otherwise the ounter dereases by 1, i.e., ount ount 1. We are now ready to state the fundamental equation of motion that relates sae, time, and seed: 1 In some ases, we shall allow this movement regardless of whether z is in the emty state ε and regardless of whether more than one neighbors of z qualify to transfer states to z at the same time (in whih ase, it will be imliitly assumed that a node an store a number of veloity omonents that deends on the size k of its neighborhood). In other ases, we shall exloit the non-emty state of a node in order to imlement henomena, suh as erfet refletion. Proosition 1. Let a state with delay v 1 and diretion r be at a (true) osition x 0 at (true) time t 0. Let its ount value at t 0 be ount(t 0) and define t ahead = v 1 ount(t 0). Then its osition x f in the r diretion at any time t f t 0, is given by x f = x 0 + v(t f t 0 + t ahead ) or = v(dt + t ahead ). (1) When t ahead = 0 and v dt is integer, this simlifies to = v dt. (2) Throughout, we make the assumtion that any system S with veloity v an emit a light beam in any diretion and the beam will travel at seed in that diretion, indeendently of v, and that any observer knows the seed of light. Also, sae that is not exliitly stated to be ouied by seifi states shall always be interreted as vauum, i.e., emty states. Simultaneity In the model under onsideration there is an unambiguous and straightforward definition of simultaneity: Two events e and e are simultaneous if they our at the same global tik. Moreover, any set of airwise non-simultaneous events (u i, t i) an be totally ordered based on the ordering of their t is. Still, e and e being simultaneous does not imly that any given observer will ereive or be able to infer this fat. Consider a 1-dimensional line of the Universe grah. Assume first that a stationary observer is laed at some osition u o and that two events e and e are to take lae at some unknown ositions u and u and at unknown times t 0 and t 0, resetively. Let also x = dist(u o, u) and x = dist(u o, u ) be the sae-distanes of the events from the observer and define = x x. Assume that eah of these events is simly the emission of light towards the observer starting from the orresonding osition of the event. Based on these two emissions, the observer ereives two times T 1 and T 2, where T 1 denotes the arrival of one of the hotons and T 2 of the other, where T 2 T 1, and we define T 2 T 1. Note that without additional information the observer annot tell for any of these loal events the original event to whih it orresonds. Theorem 1. If the observer an base his/her deision only on dt, x, and x, then deiding whether e and e are simultaneous is imossible. In light of this imossibility result, we assume that the observer an additionally distinguish whih of the arriving hotons orresonds to the losest and whih to the furthest event (e.g., the losest always emits blue light and the furthest always red light); see Figure 1 for an illustration. In this ase, the observer an deide simultaneity as follows: If the order has not been reserved, then deide non-simultaneous. If the order has been reserved, then if / deide simultaneous, otherwise nonsimultaneous. Moreover, now the observer an also determine the exat time-differene dt = t 0 t 0 between the two events as follows: If the hoton e that arrived first is the losest, imlying x x, then x x 0 = x x and the observer omutes dt = dt / in this ase. If, on the other hand, e is the furthest then the observer omutes dt = dt +/. Theorem 2. If the observer an additionally distinguish whih loal event orresonds to the losest and whih to the

4 furthest of e and e, then he/she an deide simultaneity of e and e and an determine their exat time-differene. time x+ / time-differene ereived by observer x/ t 0 = t 0 e e = 0 time x + u o u u x / 0 x x (a) / time-differene ereived by observer e t 0 t 0 = 0 e time x + x / u o u u 0 x x (b) / time-differene ereived by observer e t 0 t 0 = 0 e u o u u 0 x x () simultaneous events sae non-simultaneous events sae non-simultaneous events sae Figure 1: An observer (e.g., human or mahine) lies at u o and two events e and e are to haen at (u, t 0 ) and (u, t 0 ) emitting hotons and towards the observer, resetively. (a) The two events are simultaneous and the observer measures a loal time-differene dt. (b) The two events are now non-simultaneous but the observer will measure again dt and, as he annot distinguish their order, ases (a) and (b) are indistinguishable for the observer. () Now the observer an distinguish the order (blue losest and red furthest) and, as the order has not been reserved, he an determine that the two events annot have been simultaneous. In knowledge of dt above, we have assumed that the observer ossesses a erfet lok, whih an be seen as a ounter that inreases with every global tik of the Universe. Having aess to suh a ounter imlies having aess to the true number of time-stes searating two loal events. Similarly, a erfet rod is one that rovides knowledge of the exat number of ositions forming a ath between two given ositions (as we assumed in knowledge of x and x above). In ontrast, we shall define an imerfet lok as any lok that gives a time τ whih is a funtion of the true time t. In artiular, we fous on imerfet loks that are linear funtions of true time, that is, of the form τ = (t/a) + b, for integer onstants a, b, usually assuming b = 0. Similarly, an imerfet rod gives a χ = (x/a ) + b or simly χ = x/a, where x orresonds to true distane. A way to imlement a universal imerfet lok is to have any system S transmit a state moving at a known seed w relative to it and onsider as a tik the time it takes for that state to travel a known imerfet distane χ = x/a (measured equally by all observers through a universal imerfet rod). In what follows, loks and rods are usually assumed to be erfet, but our results hold also for imerfet loks and rods, tyially being aroximations of the true quantities within fators a 1 and (a ) 1 (the lok and rod sales, resetively). Inferring Absolute Seed In this setion, we exlore methods by whih an observer on a moving system S (i.e., at rest relative to S) an infer the true seed v of S and/or its diretion and some inherent limitations in doing so. We start by restriting attention to methods based only on loal erfet loks to generate light signals and to measure the arrival times of inoming signals. Our goal is to answer the following question: Can one or more observers on S, by emitting any sequene of light signals and measuring the way they are ereived, determine the true seed and/or diretion of system S? Method (Refleted Light Pair). The observer emits a light air ( 1, 2) at a redetermined loal time-ga t between the two beams, emitting both beams in the same diretion. We also assume that the observer an dynamially insert (or determine all these in advane) any number of refletion oints/walls at rest relative to S (both to his left and to his right) and at some oint deide to use any osition on S for him to absorb the light air and measure the ereived time-ga dt (that is, the air is not neessarily absorbed at the observer s original osition on S). Theorem 3 (No Internal Doler Effet). In the Refleted Light Pair method it will always hold that t indeendently of the hosen sequene (and number) of refletions, of the starting diretion of the light air, of the starting and ending osition of the air, and of the veloity v of system S (for v < ). The No Internal Doler Effet of Theorem 3 has the following diret onsequene: Theorem 4. By only measuring time-gas between onseutive light emissions it is imossible for an observer to determine his own absolute seed or his diretion of motion. In artiular, the ereived time-gas are always indeendent of both these quantities imlying that all seed-diretion ombinations are indistinguishable from one another based on this seifi way of measuring. In light of the imossibility of Theorem 4 for measurements based on light time-gas, we roeed with an alternative aroah in whih the observer measures the total travel-time of a single beam that he emits. The observer is again allowed to hoose any sequene of refletions but now stays ut, meaning that at the end he will absorb the beam at the same osition on S from whih it was emitted. Proosition 2. For any suh single-beam travel initiated by the observer at some time t 0 and then absorbed bak at some later time t f at the original osition on S from whih it was emitted, the total travel time t total = t f t 0 is indeendent of the diretion of v (but not of its magnitude) and is given by t total = 1 ( 1 2 v + 1 ) x total. (3) + v

5 Proosition 2 gives a method for the observer to omute the absolute seed v of S but not its diretion of motion. Theorem 5. If an observer knows the total distane x total traveled by the beam relative to S, then he an omute the seed v of S (but not its diretion) by v = 1 x total. (4) t total Due to the imossibilities of Theorems 4 and 5, showing that single-observer time-measurements of light events annot rovide omlete information of seed and diretion, we now onsider the ase in whih there are two observers at different ositions of the system S, who an oordinate in order to infer their absolute diretion and seed. Method (Coordinating Observers). The observers have two erfet and omletely synhronized loks. Eah of them emits a light beam towards the other at a redetermined time t 0 and let without loss of generality t 0 = 0. Given that their loks are synhronized they an emit simultaneously, but in general this is not neessary: in ratie it is suffiient for eah observer to know the emission time of the other aording to the former s loal lok. Then eah observer measures the absolute time until the other s beam is reeived, yielding two time measurements t 1 and t 2 for Observer 1 and Observer 2, resetively. What we an safely say without further assumtions about their relative ositions, is that one of them will measure x 0/( + v) and the other will measure x 0/( v). Assuming erfet oordination they an both know the measured times t 1 and t 2 (and whih belongs to whom), and based on this knowledge they an deide as follows: If t 1 = t 2, then v = 0, that is, S is stationary. If t 1 > t 2, then the diretion of v is towards Observer 1. If t 2 > t 1, then the diretion of v is towards Observer 2. One having determined the diretion of v as above, one observer knows that for his measured t it holds that t = x 0/( + v) and the other knows that for his t it holds that t = x 0/( v). Consider the observer knowing his time t and that t = x 0/( + v) (1) holds. From our oordination assumtions he also knows the other observer s time t and that t = x 0/( v) (2). Therefore he an solve the system (1), (2) and determine v by v = t t. (5) t + t Finally, note that the other observer an aly exatly the same roedure. Theorem 6. Two oordinating observers on a system S an omute both the diretion and magnitude of the veloity v of S rovided that they lie at a non-zero distane x 0 from eah other. Both quantities an be determined without knowledge of the value of x 0. Observe that any degree of oordination would not hel to determine the seed and/or diretion in ase two or more observers deide solely by omaring original time-gas with ereived time-gas of airs of light beams (no matter who transmits and who absorbs), due to the No Doler Effet that they would exeriene (by Theorem 3). One may also arrive at inferene methods for a single observer, if the observer s aabilities are strengthened by measuring methods inluding measurement of sae-distane between beams, synhronized loks, and exloiting an additional dimension. We now resent one of these methods. Method (Meeting Beams). We start with the assumtion that the observer on system S knows the axis along whih S is moving, but not neessarily its diretion of movement. At the end we show how this assumtion an be droed. Two light soures s 1 and s 2 are laed on system S (i.e., at rest relative to it) at distane x 0 from eah other along the axis of S s motion. We further assume that the observer knows the exat osition on S of the middle oint x 0/2 (as, e.g., measured from light soure s 1) and that he an set u two erfetly synhronized loks on S, one at s 1 and the other at s 2, with the only requirement being that the two loks will trigger the simultaneous emission of light from s 1 and s 2. The sae between the two soures is assumed to be emty aart from a solid line-ruler (at rest relative to S and the light soures) that we imagine to be arallel to the line joining the two light soures, to be used for distane measurements. Now, given that the two light beams are emitted simultaneously, they will always meet eah other at distane x 0/2 from the original osition of s 1 (i.e., setting that original osition as x = 0) and in the seial ase in whih v = 0 this will oinide with the middle of the ruler. Moreover, meeting will take lae after absolute time t meet = x 0/2. But this is enough time for S to move an absolute distane vt meet (towards s 2 in ase v is ositive and towards s 1 in ase v is negative). In both ases, there will be a deviation of the meeting oint from the middle of the ruler of x diff = (vx 0)/(2). Assuming that the interferene of the two beams rodues a mark on the observer s ruler that he an then use to measure the deviation x diff from the middle of his ruler, the observer an solve the above equation for v to determine the absolute seed v of S as v = 2 x diff x 0, (6) where x 0, x diff, and are all assumed to be known by the observer. Moreover, by observing whether the interferene was in the middle of his ruler, in [s 1, middle), or in (middle, s 2] (observe that the meeting oint an never be outside the ruler s boundaries s 1 and s 2 and these extreme ases an only result if v = or v =, resetively) the observer an determine whether v = 0, v direted from s 1 to s 2, or from s 2 to s 1, resetively. To dro the assumtion of knowing the axis of S s motion it is suffiient to observe that the maximum deviation will always result from the exeriment erformed in arallel to the diretion of v. A method is then for the observer to reeat the exeriment in all diretions (either in arallel, e.g., inside a sherial ball with its surfae filled with light soures ointing towards the enter or sequentially). Then the axis of motion will be the one yielding the maximum deviation max φ {x diff (φ)}, where φ is the angle between the veloities of light and S, and the diretion of v an be determined by the orientation of the deviation as above and its magnitude an be alulated by using max φ {x diff (φ)} in lae of x diff in Eq. 6. Measuring Light We now onern ourselves with measuring methods that internal observers an aly in order to infer the true values involved in a sequene of events. By sequene of events we shall mainly mean here the motion of light or any other system of states for a fixed amount of time. By a slight abuse of terminology we shall refer to suh sequenes of events as an event or an exeriment.

6 We start this disussion by onsidering the following simle sequene of events. Event (Moving Light). The events take lae on a horizontal line of ositions. A beam of light starts at some initial true osition u s and moves to the right towards some final true osition u f (through a vauum), therefore the beam travels a true distane x = dist(u s, u f ) and the true travel time is given by t = x/. For an external full-knowledge observer there is no ambiguity involved in the above event and suh an observer an verify that the event is omletely desribed by the model s fundamental equation of motion x = t (Equation 2 of Proosition 1). The same is under reasonable assumtions true for an internal observer who an aly one of the methods resented in the revious setion, based on whih he an measure his own true veloity v o or who an deloy loks in the viinity of the event and get absolute time measurements. In what follows, we restrit attention to observers that do not ossess suh means of diretly omuting the true values and have to infer them indiretly through relative distane and seed measurements and loal time measurements. The Measuring Method of Relativity We start from the method that results if one onstrains the observers to satisfy the rinile of relativity 18;19;20;21. To this end, let us return to our light event and onsider observers trying to make deisions about it while satisfying the rinile of relativity. Assume that an observer (Observer 1) lies somewhere to the left of the light event and is moving to the right at seed v 1. Let another observer (Observer 2) be traveling at seed v 2 v 1 in the same diretion as the light beam. Moreover, assume that Observer s 2 system onsists of two vertial walls at true horizontal distane x 2 from eah other. The whole system, inluding Observer 2 and the two walls is traveling right at v 2. It is assumed that both observers know their relative seed v = v 2 v 1 but that they do not know and annot infer their absolute seeds. The light beam is to be measured by both observers for the journey from the left wall to the right wall. This orresonds to a true distane traveled x = t. Observer 1, by having aess to two marks left by the beam on his line (horizontal axis of his oordinate system) and by using rods that are at rest with reset to him will measure a distane x 1 traveled by the beam. We assume that observers do not measure time diretly, but rather they infer time through the equations of motion. Now, if the observers have to satisfy the rinile of relativity, then Observer 1 must aly t 1 = x 1/ in order to omute the time needed by the beam to travel distane x 1 (whih is the one that Observer 1 an measure). But if, additionally, Observer 1 knows the length x 2, given that he observes the right wall moving away from him at seed v (and light moving away from him at seed, again by the rinile of relativity), it must also hold for him that his time for the duration of the event is t 1 = x 2/( v). Therefore, by insisting on the rinile of relativity, Observer 1 must omute t 1 = x2 v = x1. (7) Similarly, for Observer 2 the distane traveled by the beam is x 2, therefore to satisfy the rinile of relativity he shall omute t 2 = x 2/. Moreover, if Observer 2 knows the length x 1, given that he observes the meeting oint on Observer s 1 line aroahing him at seed v, it must also hold for him that t 2 = x 1/( + v). Therefore, by insisting on the rinile of relativity, Observer 2 must omute t 2 = From Eq. 7 and 8 we get t 2 = x1 + v = x2. (8) 1 (1 v/)t1. (9) 1 v2 /2 Therefore, by insisting on satisfying the rinile of relativity and alying to any observers the measuring method imlied by it, we arrived at the Lorentz transformation 19. By straightforward substitutions of Eq. 7 and 8 in Eq. 9 we an get the general form of the transformations. A useful interretation of the above derivation is that the transformations of seial relativity result in our model by following a seial ase of measuring, namely the one that insists on satisfying the relativity rinile. We next argue on how good of an aroximation of the true events this artiular way of measuring is. Aroximating the True Events Consider Observer 1, who is traveling at seed v 1. Reall that there is an underlying true event (desribed in the beginning of the resent setion), that is, light traveling a distane x in time t away from and in the same diretion as the observer s motion. Based on his rods, at rest relative to him, the observer measures an x 1 = x v 1t = (1 v 1/)x. Definition 1. We all a sae measuring method an f- aroximation method, f 1, if for all true distanes x, the x measured omuted by the method satisfies 1 f x measured x true f. (10) If x measured /x true is ever 0 then we an say that the method does not aroximate x in the worst ase, or that it is an -aroximation. Similarly for time and seed measuring methods. Based on this definition, let us assoiate x 1 with x. We have x 1/x = (1 v 1/) 1. For the extreme ase in whih v 1 =, we get x 1 = 0 and also x 1/x = 0, therefore in the worst ase this method does not aroximate x. Let us eliminate this extreme as a ossibility by assuming that there is a maximum known seed for the observer s system, i.e., v 1 /k, k 1. Then for the observer-light relative true seed v it holds that /k v + /k [(k 1)/k] v [(k + 1)/k]. For examle, if k = 2, then /2 v (3/2). If the observer s method always assumes for the relative seed a v =, then v /v = /v /(/2) = 2 and v /v = /v /[(3/2)] = 2/3, that is, 2/3 v /v 2, whih gives a 2-aroximation in the worst ase. We now show, that in the absene of any sort of additional information there is a method that guarantees a better worst-ase aroximation fator than the method of assuming that v =. To do this we just have to solve for v the equation v max/v = [(k + 1)/k]/v = v /[(k 1)/k] = v /v min, where v min and v max are the extreme observer-light relative true seeds. So, we have k + 1 = v k v k k 1 v = k2 1. (11) k For examle, if k 2 (meaning that v 1 /2, known by the observer) then the best worst-ase aroximation

7 is obtained if the observer always assumes that his relative seed against light is ( 3/2) On the other hand, for large k, as e.g. k 10 3 (meaning that v 1 /10 3 ), the otimum is obtained by assuming relative seed ( /10 3 ) , therefore, for low observer s seeds, assuming a relative seed ratially equal to is the best worst-ase aroximation that one an hoe for. Exat Comutation of the True Events Assume now that the light soure emits simultaneously a beam towards the observer and a beam traveling right as in the Moving Light event. The latter beam is now to be refleted on a mirror moving also right at seed v m and starting at distane x 0 from the light soure. Let x be the true distane traveled by the beam (until it is refleted on the mirror) and t be the true time of this motion. It holds that x = t = [/( v m)]x 0. See Figure 2 for an illustration. time T 2 t T 1 0 dt vo vo vo vo x 0 x x vm vm vm vm sae Figure 2: A light soure emits two beams simultaneously, one towards the observer and the other towards the mirror. Both the observer and the mirror are moving right at seeds v o and v m, resetively. By measuring the relative distane x until refletion of the beam moving right and the loal time-differene T 2 T 1 of the two reeived beams, the obsever an infer the absolute values of all quantities involved. Consider the following measuring method alied by the observer. The observer knows that his true seed v o is direted towards the moving light beam and that the latter is moving away from him. The observer is equied with rods to measure a distane traveled by the beam relative to him and also with a loal erfet lok to measure ereived time-differenes of inoming light signals. The distane that the observer will measure is x = x v ot = (1 v o/)x = [( v o)/]x and given his knowledge of the form of this equation and of x and he knows that x = v o x, (12) but he annot yet determine the value of x due to his ignorane of his own true seed v o. Assume now that the observer an additionally measure the loally ereived timega dt of the reeived beams. Then it will hold that ( 1 + ) vo x0, (13) + v o v m whih by relaing x 0 = [( v m)/]x gives x = + vo dt, (14) 2 where again dt is known, but not v o. Now, by ombining Eq. 12 and 14 we get that v o = 1 2 x dt. (15) Given the assumed knowledge (inluding the diretion of movement), the observer knows Eq. 15, therefore he an diretly substitute the known values of x, dt, and in it and omute his own absolute seed v o. Then he an substitute the known values of v o, x, and in Eq. 12 in order to omute the true distane x traveled by the (right) beam and finally t = x/ to omute the true total time of the beam s motion. Theorem 7. By ombining measurements of erfet relative distane and the time-differene of an inoming light air, an observer knowing the diretions of travel of himself and of the observed light beam an omute the absolute values of the distane x and time t traveled by the beam as well as his own absolute seed v o. The Future This study oens some interesting researh questions that are exeted to require areful onsideration and have the otential to insire more work on the toi. We distinguish three of them whih we now disuss. The first one is to rove or disrove that the model resented here (or slight variations of it) is aable of reresenting the real dynamis of all existing models of omutation, with a seial fous on models that merge omutation with a hysial environment. It is immediate that a Turing mahine is a seial ase of this model and that any olletion of Turing mahines interonneted through a stati synhronous network (as in traditional Networked Distributed Systems) an, for examle, be reresented by deloying those mahines in sae and using an aroriate number and tyes of states to reresent the ommuniation links between them. Regarding models of omutation faed with unontrolled environmental events, the adversarial ones may be imlemented as algorithmi adversaries having aess to various levels of knowledge related to the internal workings of the mahine(s) that they are trying to ush to its (their) worst-ase exeution(s). An intriguing question onerns models assuming randomization in the mahines and/or the environment. The model resented here is inherently deterministi. Is it true that desite this, the model an still reresent any degree of randomization (as an emergent roerty) as an effet of a system A (e.g., a mahine) measuring some other system B (e.g., the osition of a hoton refleted between two ositions) under inomlete knowledge and ossibly in a way of measurement that is indeendent of system B (e.g., the starting time and duration of the measurement are flutuating in a way that is indeendent of B)? It would also ertainly be valuable to investigate the aaity of this model to reresent the models of Quantum Comutation 22;23 and of Dynami Networks 24;25. The other (related) question is how good is suh a model in reresenting the real underlying dynamis of nature and as a onsequene the higher-level dynami systems that they generate. We here gave a first hint that it might be adequate to study hysial systems and roesses through urely information-based models. It remains to be seen to what extent the fats formally roved here orresond to the workings of the Universe and if not whether there are alternative information-based models that ould serve this urose.

8 Moreover, we here only alied the model to inertia systems. Immediate next goals are to aly it to aelerated states, to exlore adequate reresentations of the fundamental fores, and to resolve whether suh disrete models give just aroximations or an reresent the exat dynamis of hysial systems. In other words, are there true loal ontinua and infinities or assuming a disretization of saetime and of the events therein is suffiient? The third one is a aradox whih enloses an intriguing question and at the same time serves as an examle of the onsequenes of an information-based, and onfigurationbased in artiular, interretation of nature. If any system in the Universe is a onfiguration of states evolving (ossibly but not neessarily deterministially) in saetime, then it is reasonable to assume that the same must be true for organisms, like human beings. Now take a human A frozen at some true time t (e.g., today) and subset of sae S (e.g., his/her urrent loation). Assume that a onfiguration C of states in sae omletely desribes human A (inluding, e.g., his DNA, hysial ondition, memories, and exat next thought and/or ation that he would erform just after time t). If it is so, then it is reasonable to assume that C ould be somehow (at least in rinile) rerodued at some later time t > t, e.g., after A s death, giving a future human A time who at time t is in everything idential to A at time t. So, it is reasonable for A to be satisfied with this as a way of rebirth. But if A is susiious enough he might realize that if an exat oy of him an be generated at a future time then it should be also ossible to generate it at the resent time but in a subset of sae S, suh that S and S are disjoint and annot influene eah other, giving a human A sae. A sae and A are in everything idential to eah other at time t, oexisting in time but not in sae. Then the aradox is this: If human A time is human A, then also human A sae must be human A (as A sae is in all resets idential to A time and both to A). But human A oexists in time with A sae and may never notie the latter s existene. Then how an human A sae be human A? Referenes [1] Turing, A. On omutable numbers, with an aliation to the entsheidungsroblem. Proeedings of the London mathematial soiety, 2(1): , [2] Karr, J. R., Sanghvi, J. C., Maklin, D. N., Gutshow, M. V., Jaobs, J. M., Bolival, B., Assad-Garia, N., Glass, J. I., & Covert, M. W. A whole-ell omutational model redits henotye from genotye. Cell, 150(2): , [3] Vogelsberger, M., Genel, S., Sringel, V., Torrey, P., Sijaki, D., Xu, D., Snyder, G., Bird, S., Nelson, D., & Hernquist, L. Proerties of galaxies rerodued by a hydrodynami simulation. Nature, 509(7499): , [4] Angluin, D., Asnes, J., Diamadi, Z., Fisher, M. J., & Peralta, R. Comutation in networks of assively mobile finite-state sensors. Distributed Comuting, 18(4): , [5] Soloveihik, D., Cook, M., Winfree, E., & Bruk, J. Comutation with finite stohasti hemial reation networks. natural omuting, 7(4): , [6] Doty, D. Timing in hemial reation networks. In Proeedings of SODA, ages , [7] Oerter, R. The theory of almost everything: The standard model, the unsung triumh of modern hysis. Penguin, [8] Einstein, A. Ist die trägheit eines körers von seinem energieinhalt abhängig? Annalen der Physik, 323(13): , [9] Watson, J. D., Crik, F. H., et al. Moleular struture of nulei aids. Nature, 171(4356): , [10] Darwin, C. & Bynum, W. F. The origin of seies by means of natural seletion: or, the reservation of favored raes in the struggle for life. AL Burt, [11] Woods, D., Chen, H.-L., Goodfriend, S., Dabby, N., Winfree, E., & Yin, P. Ative self-assembly of algorithmi shaes and atterns in olylogarithmi time. In Proeedings of ITCS, ages , [12] Lamort, L. Time, loks, and the ordering of events in a distributed system. Commun. ACM, 21(7): , [13] Thrun, S. Toward roboti ars. Commun. ACM, 53(4):99 106, [14] Bringsjord, S., Liato, J., Govindarajulu, N. S., Ghosh, R., & Sen, A. Real robots that ass human tests of selfonsiousness. In Robot and Human Interative Communiation (RO-MAN), th IEEE International Symosium on, ages IEEE, [15] Silver, D., Shrittwieser, J., Simonyan, K., Antonoglou, I., Huang, A., Guez, A., Hubert, T., Baker, L., Lai, M., Bolton, A., et al. Mastering the game of go without human knowledge. Nature, 550(7676): , [16] Von Neumann, J. & Burks, A. W. Theory of selfreroduing automata. IEEE Transations on Neural Networks, 5(1):3 14, [17] Wolfram, S. Cellular automata as models of omlexity. Nature, 311(5985): , [18] Galilei, G. Dialogue onerning the two hief world systems. University of California Press Berkeley, [19] Einstein, A. On the eletrodynamis of moving bodies [20] Einstein, A. The foundation of the generalised theory of relativity [21] Lorentz, H. A., Einstein, A., Minkowski, H., Weyl, H., & Sommerfeld, A. The rinile of relativity: A olletion of original memoirs on the seial and general theory of relativity. Courier Cororation, [22] Shor, P. W. Polynomial-time algorithms for rime fatorization and disrete logarithms on a quantum omuter. SIAM review, 41(2): , [23] Ladd, T. D., Jelezko, F., Laflamme, R., Nakamura, Y., Monroe, C., & O Brien, J. L. Quantum omuters. Nature, 464(7285):45 53, [24] Casteigts, A., Flohini, P., Quattroiohi, W., & Santoro, N. Time-varying grahs and dynami networks. International Journal of Parallel, Emergent and Distributed Systems, 27(5): , [25] Mihail, O. & Sirakis, P. G. Elements of the theory of dynami networks. Commun. ACM, 2018.

9 Omitted Details Proof of Theorem 1 Theorem 1. If the observer an base his deision only on dt, x, and x, then deiding whether e and e are simultaneous is imossible. Proof. We break this roof into two arts. 1. If the observer an base his deision only on the loally ereived dt, then deiding whether e and e are simultaneous is imossible. We rove this for the ase in whih the observer knows that he is stationary and has aess to the true time differene T 2 T 1 0 (via a erfet lok; see disussion in Simultaneity setion on erfet and imerfet loks). Assume, for the sake of ontradition, that the observer an orretly deide simultaneity. Consider a ase in whih the observer lies to the left of both events, with e being the losest and e the furthest (or olloated). Assume first that e, e are olloated and simultaneous. Then the observer will reeive them simultaneously, i.e. measure 0, and given that the observer orretly deides, he must answer yes. Now onsider the ase in whih d > d, e ours first and reisely when its hoton asses through the loation u of e, e ours and therefore the two hotons travel together and arrive at the observer simultaneously. This means that the observer will measure again 0. But the observer is orret meaning that he must deide no in this ase, as e and e were not simultaneous. This leads us to the ontraditory onlusion that a orret observer must assoiate exatly the same observation with both deisions (whih annot be both true at the same time), therefore suh a orret observer annot have existed in the first lae. We next assume that the observer additionally knows = x x. 2. If the observer an base his deision only on dt and, then deiding whether e and e are simultaneous is imossible. Assume that the observer is loated between the two events (the same imossibility an be obtained even if the observer has both events to the same side and even if the observer knows the exat values of x and x ). If the events were simultaneous then when one of the hotons arrives at the observer the other must be at distane from the observer, therefore the measured dt must be equal to / in this ase. Now assume that x x, i.e., = x x and onsider the ase in whih e ours first and when its hoton is at a distane x from the observer, suh that x x =, e s hoton starts. This ase is equivalent to the ase in whih two events our simultaneously at distanes x and x from the observer, where we define = x x. By the same argument as above, the observer must measure a / = / = /, that is he reeives the same measurement dt in a simultaneous and a non-simultaneous ase, where in both ases the known by the observer is the same. We again onlude that suh an observer annot distinguish between the two ases. Proof of Theorem 3 Theorem 3 (No Internal Doler Effet). In the Refleted Light Pair method it will always hold that t indeendently of the hosen sequene (and number) of refletions, of the starting diretion of the light air, of the starting and ending osition of the air, and of the veloity v of system S (for v < ). Proof. Assume first that the light air ( 1, 2) starts in the diretion of system s S veloity v. Then the initial sae-distane between the two beams will be ( v)t. Whenever a air with sae-distane x moving in v s diretion is refleted, its udated distane after refletion will be x( + v)/( v) and whenever moving in the oosite diretion than that of v, its udated distane after refletion will be x( v)/( + v). Therefore every odd refletion of ( 1, 2) (in the ase onsidered, making the diretion of the air to beome oosite than v s) will give udated distane x oosite = [( + v)/( v)]( v)t = ( + v)t and every even refletion will give x same = [( v)/( + v)]( + v)t = ( v)t. Exatly the same sae-distanes hold when the air is initially emitted in the oosite diretion than v s (where it simly starts with x oosite = ( + v)t instead of x same = ( v)t ). Finally, observe that there are only two ways in whih any observer (indeendently of his osition) an absorb and measure the air (and end the exeriment): 1. When ( 1, 2) has the same diretion as v reeive it from his bak (from the oosite diretion than that of his own motion). In this ase, the loally ereived time-differene will be xsamet v = v v t = t. 2. When ( 1, 2) has the oosite diretion as v reeive it from his front (from the same diretion as that of his own motion). In this ase, the loally ereived time-differene will be x oositet = +v t = t. +v +v Therefore, in both ases it holds that t and this measurement is indeendent of all the quantities listed in the statement of the theorem. Proof of Proosition 2 Proosition 2. For any suh single-beam travel initiated by the observer at some time t 0 and then absorbed bak at some later time t f at the original osition on S from whih it was emitted, the total travel time t total = t f t 0 is indeendent of the diretion of v (but not of its magnitude) and is given by t total = 1 ( 1 2 v + 1 ) x total. + v Proof. No matter how many times the beam is refleted and no matter the lengths of the individual travels (from emission to refletion, between refletions, and form refletion to absortion), given that the beam starts and ends at the same osition of S it must hold that any distane x (relative to S) traveled away form the observer in any diretion must eventually be traveled bak. Therefore, any subinterval x of S traveled must by the end of the exeriment have been traveled in both diretions (where we should emhasize that x orresonds to length of matter as measured on the system S). It follows that one diretion of travel must have traveled x in time x/( v) (the one in the - unknown - diretion of v) and the other diretion in time x/( + v) (the one in the oosite diretion than v s). Therefore, indeendently of v s diretion, the total travel time of length x must always be x/( v) + x/( + v). Then if x total denotes the