Many Roads to Synchrony: Natural Time Scales and Their Algorithms

San ta F e Institute W orking P ap er 10-11-025 arXiv:1010.5545 [nlin.CD] The Man y Roads to Sync hron y: Natural Time Scales and Their Algorithms Ry an G. James, 1 , ∗ John R. Mahoney, 2 , † Christopher J. Ellison, 1 , ‡ and James P . Crutc hfield 1, 3 , § 1 Complexity Scienc es Center and Dep artment of Physics, University of California at Davis, One Shields Avenue, Davis, CA 95616 2 Scho ol of Natur al Scienc es, University of California, Merc e d, California, 95344 3 Santa F e Institute, 1399 Hyde Park R o ad, Santa F e, NM 87501 (Dated: Octob er 25, 2018) W e consider t wo important time scales—the Mark o v and cryptic orders—that monitor ho w an observ er sync hronizes to a finitary sto chastic process. W e sho w ho w to compute these orders exactly and that they are most efficiently calculated from the  -mac hine, a pro cess’s minimal unifilar mo del. Surprisingly , though the Mark o v order is a basic concept from stochastic pro cess theory , it is not a probabilistic property of a process. Rather, it is a top ological prop erty and, moreov er, it is not computable from any finite-state mo del other than the  -machine. Via an exhaustive survey , we close by demonstrating that infinite Marko v and infinite cryptic orders are a dominan t feature in the space of finite-memory pro cesses. W e draw out the roles play ed in statistical mec hanical spin systems b y these t w o complemen tary length scales. Keyw ords : Mark o v c hain, hidden Marko v mo del, Marko v order, cryptic order, synchronization,  -mac hine P ACS num bers: 02.50.-r 89.70.+c 05.45.Tp 02.50.Ey 02.50.Ga I. INTR ODUCTION Sto c hastic pro cesses are frequen tly c haracterized b y the spatial and temp oral length scales ov er whic h cor- relations exist. In physics, the range of correlations is a structural prop erty giving, for example, the distance o ver which significan t energetic coupling exists among a system’s degrees of freedom [1]. In time series analy- sis, knowing the temp oral scale of correlations is key to successful forecasting [2]. In biosequence analysis, the deca y of correlations along DNA base pairs determines in some measure the difficulty faced by a replicating en- zyme as it “decides” to b egin transcribing a gene [3]. In m ultiagent systems, one of an agen t’s first goals is to de- tect useful states in its environmen t [4]. The common elemen t in these is that the correlation scale determines ho w quic kly an observer—analyst, forecaster, enzyme, or agen t— synchr onizes to a pro cess; that is, how it comes to know a relev ant structure of the sto c hastic pro cess. W e recently show ed that there are a num b er of distinct, though related, length scales asso ciated with synchroniz- ing to stationary sto chastic pro cesses [5]. Here, we sho w that these length scales are top olo gic al , dep ending only ∗ rgjames@ucdavis.edu † jmahoney3@ucmerced.edu ‡ cellison@cse.ucdavis.edu § chaos@ucda vis.edu on the underlying graph top ology of a canonical repre- sen tation of the stochastic process. This rev eals deep ties b et w een the structure of a pro cess’s minimal sufficient statistic and sync hronization of an observer. W e also re- cen tly introduced another class of synchronization length scales based, not on state-based models, but on the con- v ergence of sequence statistics [6]. W e briefly compare these to the Marko v and cryptic orders. Sp ecifically , we in vestigate measures of sync hroniza- tion and their asso ciated lengths scales for hidden Mark ov mo dels (HMMs)—a particular class of pro cesses with an internal (hidden) Marko vian dynamic that pro- duces an observ ed sequence. W e fo cus on tw o such measures—the Marko v order and the cryptic order—and sho w through a series of incremental steps how they can b e efficien tly and accurately computed from the process’s minimal sufficient statistic, the  -machine. Our dev elopment pro ceeds as follows. After briefly outlining the required background in Sec. I I, we intro- duce the t wo primary measures of in terest in Sec. II I and demonstrate their calculation via naive methods in Sec. IV. Reflecting on a surprising finding in Sec. V, Sec. V B sho ws to how alleviate several weaknesses in the naiv e approach. Then, borrowing relev ant data struc- tures from formal language theory , Sec. V C resolves the last of the issues. T ogether these steps provide an ef- ficien t algorithm for exactly calculating the Marko v or- der when it is finite and for determining when it is infi- nite. Building on this new understanding, Sec. VI go es on to sho w how to compute the second time scale—the cryptic order—through similar means. W e then briefly 2 touc h up on other time scales and their relative b ounds in Sec. VI I. Leveraging the computational efficiency , we surv ey the Marko v and cryptic orders among  -machines in Sec. VI I I and conclude that infinite correlation is a dominate prop ert y in the space of memoryful stationary pro cesses. The implication is that most, if not all, ob- serv ers cannot sync hronize exactly [7]. T o illustrate how these time scales apply in practice, Sec. IX characterizes correlations in one-dimensional spin systems. Finally , w e conclude b y discussing ho w these time scales compare to other measures of in terest and b y suggesting applications where they and their algorithms will prov e useful. I I. BA CK GROUND W e assume the reader has introductory kno wledge of information theory and finite-state machines, such as that found in the first few chapters of Ref. [8] and Ref. [9], resp ectiv ely . Our developmen t makes particular use of  -mac hines, a natural representation of a pro cess that mak es many prop erties directly and easily calculable; for a review see Ref. [10]. A cursory understanding of sym- b olic dynamics, such as found in the first few chapters of Ref. [11] is useful for several results. W e denote subsequences in a time series as X a : b , where a < b , to refer to the random v ariable sequence X a X a +1 X a +2 · · · X b − 1 , which has length b − a . W e drop an index when it is infinite. F or example, the p ast X −∞ :0 is denoted X :0 and the futur e X 0: ∞ is denoted X 0: . W e generally use w to refer to a wor d —a sequence of sym- b ols drawn from an alphab et A . W e place tw o words, u and v , adjacent to each other to mean concatenation: w = uv . W e define a pr o c ess to b e a joint probabilit y distribution ov er X : = X :0 X 0: . A pr esentation of a given pro cess is any state-based represen tation that generates the pro cess. A pro- cess’s  -machine is its unique, minimal unifilar presen- tation [12]. The recurrent states of a pro cess’s  -mac hine are known as the c ausal states and, at time t , are de- noted S t . The causal states are the minimal sufficient statistic of X :0 ab out X 0: . F or a thorough treatment on presen tations see Ref. [5]. I I I. PROBLEM ST A TEMENT When confronted with a pro cess, one of the most nat- ural questions to ask is, Ho w m uch memory does it ha v e? Is it like a coin or a die, with no memory? Do es it al- ternate b etw een tw o v alues, requiring that the pro cess remem b er its phase? Do es it express patterns that are arbitrarily long, requiring an equally long memory? This t yp e of memory is quantified by the Mark o v order: R ≡ min { ` | Pr( X 0 | X − ` :0 ) = Pr( X 0 | X :0 ) } . (1) T o put it collo quially , how many prior observ ations must one remember to predict as w ell as remembering the in- finite past? Marko v chains ha v e R = 1 by their very definition. Hidden Marko v mo dels, though their inter- nal dynamics are Marko vian ( R = 1), their observed b e- ha vior can range from memoryless ( R = 0) to infinite ( R = ∞ ). A ma jor goal in the following is to sho w how to compute a pro cess’s R efficiently and accurately given its  -machine. In this v ein it is pruden t to recast Eq. (1) using causal states: Pr( X 0 | X − R :0 ) = Pr( X 0 | X :0 ) = ⇒ X :0 ∼  X − R :0 = ⇒ H[ S 0 | X − R :0 ] = 0 = ⇒ R = min { ` | H[ S 0 | X − ` :0 ] = 0 } = min { ` | H[ S ` | X 0: ` ] = 0 } . (2) In effect, since the past R observ ations predict just as w ell as the infinite past, the causal states are a function of length- R pasts. The second primary length scale w e discuss is the cryp- tic or der k χ [13]. Its definition builds from Eq. (2): k χ ≡ min { ` | H[ S ` | X 0: ] = 0 } . (3) The difference b etw een the t w o is that cryptic order is conditioned on the infinite future, as opp osed to a finite one. This pro vides our interpretation of the cryptic or- der: k χ is the num b er of causal states that cannot b e r etr o dicte d . That is, no matter ho w man y future sym b ols w e know, the first k χ in ternal states the pro cess visited cannot b e inferred. IV. NAIVE APPRO ACH T o illustrate a direct metho d of determining a pro cess’s Mark ov and cryptic orders, w e appeal to y et another form of their definitions [5]: R = min { ` | H[ X 0: ` ] = E + ` h µ } (4) k χ = min { ` | H[ X 0: ` , S ` ] = E + ` h µ } , (5) where E = I[ X :0 , X 0: ] is kno wn as the exc ess entr opy and h µ = H[ X 0 | X :0 ] is known as the entr opy r ate [14]. The in tuition for these is identical to those ab ov e: Once we reac h Marko v (cryptic) order, we predict as accurately as p ossible. It is worth noting that these definitions only hold for finitary ( E < ∞ ), stationary pro cesses. 3 These definitions lead to a simple wa y of determining a process’s Marko v and cryptic orders. T o compute the Mark ov order, we calculate the entrop y H[ X 0: ` ] of longer and longer blo cks of contiguous observ ations un til it b e- gins to gro w linearly . W e call this function of ` the blo ck entr opy curve . The first ` at which H[ X 0: ` ] matches its linear asymptote is the Marko v order. T o compute the cryptic order, w e p erform a similar test, but rather than calculating the entrop y of blo cks of observ ations alone, w e calculate the en tropy H[ X 0: ` , S ` ] of those blo cks along with the causal states that are induced by those obser- v ations. W e call this function of ` the blo ck-state entr opy curve . The cryptic order is the length at which the block- state entrop y curve reaches its asymptotic linear behav- ior. This view of the tw o orders is sho wn in Fig. 1. The data for the blo ck entrop y and block-state entrop y curv es sho wn there comes from the Phase-Slip Backtr ack (PSB) Pro cess sho w in Fig. 2. It is imp ortant to p oint out the weaknesses of this ap- proac h. They are at least fourfold, one must (i) kno w h µ exactly , (ii) kno w E exactly , (iii) b e able to differentiate the blo ck entropies b eing exactly on the asymptote from less than machine pr e cision away fr om the asymptote, and (iv) b e able to “guess” when R or k χ are infinite in order to terminate the calculation. The first tw o are not prohibitive. The entrop y rate h µ can b e computed exactly from any unifilar mo del of the pro cess, and so its calculation can b e done fairly easily [15]. Similarly , the excess entrop y E can b e computed if the joint dis- tribution ov er b oth a unifilar, gauge-free mo del of the pro cess and a unifilar, gauge-free mo del of the reverse of the pro cess is on hand [16]. The last tw o weaknesses do not hav e suc h direct solu- tions. How are we to know if our entrop y calculation at length ` is exactly equal to E + `h µ ? Or, instead, are the curv e and linear asymptote so close that finite-precision estimates cannot differentiate them? Comp ounding this, what if H[ X 0: ` ] has not equaled E + `h µ b y ` = 10 6 ? Can one assume that it ever will? P erhaps the pro cess is Marko v order R = 10 8 . These are the tw o particular w eaknesses that need to b e ov ercome. V. MARK OV ORDER IS TOPOLOGICAL W e start with the somewhat surprising observ ation that Marko v order is not a probabilistic property , as seemingly suggested by Eq. (1), but rather a top ological one. The first hint at this comes, though, in an empirical study . The question then b ecomes just ho w is this so. By w ay of answering it, we solve the fundamen tal problems noted with the naive approac h to Marko v order. Several examples serve to drive home the idea and illustrate the 0 1 2 3 4 0 1 2 3 4 E C µ R k χ E + `h µ H[ X 0: ` , S ` ] H[ X 0: ` ] FIG. 1. Blo ck entrop y and block-state entrop y for the PSB Pro cess of Fig. 2: The blo ck en tropy curve reac hes its asymp- totic b ehavior ( E + ` h µ ) at ` = 3, indicating a Marko v or- der R = 3. The blo ck-state entrop y curve reac hes the same asymptote at ` = 2 and so the process is cryptic order k χ = 2. A B C D 1 2 | 1 1 2 | 0 1 2 | 1 1 2 | 0 1 | 0 1 | 1 FIG. 2. The Phase-Slip Backtrac k (PSB) Pro cess: Edges are lab eled p | s where p is the probabilit y of an edge being follow ed and s is the sym b ol emitted up on trav ersing it. calculation metho ds. A. An Observ ation The first step forward in solving the tw o main prob- lems encountered in the naive Marko v order metho d is to take a step back. Rather than considering the par- ticular pro cess generated by the machine in Fig. 2, w e study the family of pro cesses generated when its transi- tion probabilities are v aried while the structure remains the same. This family is sho wn by the parametrized ma- c hine of Fig. 3. If we compute blo ck and block-state en- trop y curv es for a random ensem ble of pro cesses from this family , plot the deriv ativ e of those curves and subtract- 4 A B C D 1 − p | 1 p | 0 1 − q | 1 q | 0 1 | 0 1 | 1 FIG. 3. Phase-Slip Bac ktrack Pro cess with parametrized transition probabilities. k χ R block length ` 0 ∆ H [ X 0: ` ] − h µ ∆ H [ X 0: ` , S ` ] − h µ FIG. 4. Entrop y con vergence curves versus blo ck length ` for Fig. 3’s family of pro cesses with several dozen random v alues for p and q . The linear asymptotic b ehavior ( h µ ) has b een subtracted out of each curv e. (See inset.) The Marko v or- der R and cryptic order k χ are the lengths ` at which the blue (darker) and green (lighter) lines, resp ectively , reach zero. Th us, b oth orders are indep endent of the generating mac hine’s probabilit y parameters. ing out their asymptotic b ehavior, we arrive at the blo ck and blo c k-state entrop y conv ergence shown in Fig. 4. As it dramatically demonstrates, the Marko v and cryp- tic orders are indep endent of the transition probabilities in the machine’s structure. Thus, any pattern relev ant for prediction is enco ded by the  -machine’s top ology . B. Sync hronizing W ords On careful insp ection of Eq. (2), how ev er, it is not sur- prising that the Mark ov order is a top ological prop erty . A conditional entrop y H[ X | Y ] v anishes only if X is a de- terministic function of Y . In our case, H[ S R | X 0: R ] = 0 means that eac h length- R word determines a unique state of the mo del. W e say that each word of length R is syn- chr onizing [5]. (Later, we consider only pr efix-fr e e syn- c hronizing wor ds—those which hav e no initial subw ord that also synchronizes.) If one observes a pro cess ha ving no inkling as to which state its hidden Marko v mo del b e- gan in, then after observing R symbols the exact state will b e kno wn. This provides an impro ved metho d of determining the Mark ov order. Enumerate all words of increasing length noting whic h hav e synchronized and which hav e not. When all the w ords at the current length ha ve synchro- nized, then that length is the Marko v order R . This pro cedure has b een completed for the PSB Process in Fig. 5. It can b e verified that at lengths 0, 1, and 2 it is p ossible to still ha v e ambiguit y as to which state this system is in. F or example, if the tw o symbols 10 are ob- serv ed, the system ma y b e in either state C or state D . One more observ ation is required to disambiguate which it is. Therefore, as observ ed previously , the Mark o v order for this pro cess is R = 3. This metho d is improv ed by lexicographically enumerating words of increasing length until they sync hronize to a single state. The longest suc h w ord—a prefix-free synchronizing word—is the Marko v order R , since b y that point every shorter word will hav e sync hronized and, therefore, the causal states will b e de- termined uniquely by words of that length. This metho d addresses several weaknesses of the naive approac h. No w, neither E nor h µ are needed, nor do w e need to concern ourselves with the details of comparing nearly equal numerical v alues. How ever, the method re- lies on enumerating prefix-free synchronizing w ords, and it is quite p ossible for a pro cess to hav e an infinite num b er of prefix-free synchronizing words. In these situations, it is not feasible to enumerate them all, hoping to iden tify the longest. T o address this problem, w e turn to formal language theory [9]. C. State Subset Construction The remaining problem is to find the longest prefix-free sync hronizing word without having to en umerate them all. This can b e accomplished with a standard algorithm from the theory of finite automata. W e construct an ob ject known as the p ower automaton (P A), so-named since its states are elements of the p ow er set of a given automaton’s states. Construction of the p ow er automaton b egins with a single state: the set of all states from the  -machine. This is the P A’s start state. Then recursively , for eac h state in the P A and each symbol, consider all  -mac hine states that can b e reached b y an y  -mac hine state within 5 0 0 1 A B C C D D A 0 1 0 A B C C D A D 0 1 1 A B C C D A B 1 0 0 A B D A B C D D 1 0 1 A B D A B C D A 1 1 0 A B D A B B C 1 1 1 A B D A B B B FIG. 5. All observ able words of length 3 for the PSB Pro- cess. Each w ord has b een annotated with the paths through whic h that wo rd inv okes synchron y . It is not until the obser- v ation of three sym b ols that in all cases there is only a single p ossible state. There are, ho w ever, some words whic h induce sync hron y more quic kly . the current P A state on the curren tly considered symbol. A new P A state consisting of the set of  -machine suc- cessor states is added, along with a directed edge from the curren t to the new P A state, lab eled with the current sym b ol. Once the successors to eac h P A state hav e b een determined, there will b e a subgraph of the P A that is isomorphic to the recurrent  -mac hine. This subgraph is the P A’s r e curr ent component. When the  -machine generates an ergo dic pro cess, this subgraph is the only strongly connected comp onent with no outgoing edges. The remainder of the P A consists of tr ansient states. Sync hronizing words are asso ciated with particular P A paths. Each path b egins in the start state and trav erses edges in P A’s transient p ortion. Even tually , the path con tinues to a P A recurren t state. Prefix-free sync hro- nizing w ords hav e paths that end as so on as they reac h a recurren t P A state. T o find the longest prefix-free syn- c hronizing word, we weigh t each edge in P A’s transient part with the v alue − 1 and each edge in the recurren t part with 0. With these modifications, the Bellman-F ord algorithm can b e employ ed to discov er the path of least w eight from the start state to any recurren t state. Due to the weigh ting, the path of least weigh t is the longest. The alternative Floyd-W arshall algorithm can also be used; see Ref. [17] for details regarding b oth. W e choose the Bellman-F ord algorithm for tw o reasons. First, it w orks on graphs with negativ e weigh t and, second, it detects negative-w eight cycles. A negative weigh t cycle here implies that the longest path is arbitrary (infinite) in length. This sp ecifies a complete metho d for computing the Mark ov order efficiently and accurately from a mo del of a pro cess. First, construct the p ow er automaton. Then, w eight the edges according to their status as transient or recurrent. Last, find the path of least weigh t from the start to a recurrent state. It runs in O ( A 2 2 N ) time, whic h is exp onential but finite. And, it dep ends only on integer calculations. In this wa y , it circumv ents all the computational difficulties encountered in the naiv e approac h. Thus, if one can infer an accurate mo del from observ ations of a system, the problem of computing that system’s Marko v order is solved. This method also provides a solution to weakness (iv) of the naiv e algorithm (Sec. IV). When finite, the Marko v order dep ends on the longest path through the transient states of the p ow er automaton, and for an n state re- curren t  -mac hine, there are at most f ( n ) : = 2 n − n − 1 transien t states (subtracting n recurren t states and also the empty set). Since lo ops in the transient structure imply infinite Marko v order, it follows that the longest p ossible path is one which visits each of the transient states. Thus, if the Marko v order has not b een found b y L = f ( n ), then it is safe to conclude that the Marko v order is infinite. Since, the Marko v order b ounds the cryptic order, the same b ound w orks for the cryptic or- der. It is an op en problem to find a tight upper b ound for the Marko v order in terms of the n umber of states and the num b er of symbols in the alphab et. D. Examples A v ariety of qualitatively different b ehaviors can b e exhibited by the Marko v order algorithm. Here, we il- lustrate the typical cases. Applying it to the PSB Pro- cess, the algorithm pro duces the fairly simple transien t structure consisting of three no des—P A states AB C D , AB , and C D —seen in Fig. 6. There are t wo longest paths starting from P A start state AB C D and ending in a recurrent no de: AB C D 1 → AB 0 → C D 1 → A , which is trav ersed with the word 101, and AB C D 1 → AB 0 → C D 0 → D , trav ersed with the word 100. This means that the longest prefix-free synchronizing words are 101 and 100, b oth of length three, and therefore PSB Pro cess’s Mark ov order is R = 3. The second pro cess we analyze is shown in Fig. 7. It has a sligh tly more complicated transient structure than that of the PSB Pro cess. Of particular note is the self- lo op on P A state AB . This loop exists because  -mac hine states A and B transition to each other on pro ducing a 0. As a consequence, we cannot determine the state un til 6 A B C D AB CD ABCD 0 1 0 1 0 1 1 0 1 0 0 1 FIG. 6. PSB Process pow er automaton. The longest path b eginning from state AB C D , trav ersing transient (red) edges, and ending in a recurrent (black) state is of length 3: AB C D 1 → AB 0 → C D 1 → A (or 0 → D ). observing a 1. This inability to synchronize on some w ords results in a non-Marko vian pro cess; that is, R = ∞ . The Bellman-F ord algorithm terminates as so on as it detects the corresp onding negative-w eight cycle. Our third example is the Nemo Pro cess, shown in Fig. 8. Its transient structure is particularly simple: a single state representing all the recurren t states. Since the recurrent states simply p ermute up on observing a 0, the word 0000 . . . nev er allo ws one to determine in whic h state the system is. This is indicated by the self-lo op on P A state AB C . This once again means that the pro- cess is non-Marko vian and has R = ∞ . This condition is detected by the algorithm as well. VI. CR YPTIC ORDER W e no w turn to calculate a pro cess’s cryptic order k χ . Recall that Eq. (3) inv olves a condition on the infinite fu- ture. With probability one, each infinite future synchro- nizes for exactly synchronizing  -machines [7]. W e can then consider the problem of calculating k χ to b e that of determining as muc h of a state history as p ossible, giv en a prefix-free sync hronizing w ord and the state to which it sync hronized. The maxim um n umber of states w e cannot retro dict is then the cryptic order. ABC A C AB A B C 0 1 0 1 0 1 0 0 1 0 1 FIG. 7. T ypical complications in the P A for a finite-state non-Mark o vian Pro cess. The signature is the lo op AB 0 → AB in the transien t structure. This means there is the possibility of an arbitrarily long series of observ ations that never syn- c hronize and that, in turn, cause Marko v order to diverge. Generically , lo ops in the transient structure can consist of more than one P A state. ABC A B C 0 1 1 0 0 0 1 FIG. 8. Like Fig. 7’s pro cess, the Nemo Pro cess here is non- Mark o vian. The Nemo Pro cess make this p erhaps clearer, ho w ev er, since the recurrent states p ermute into each other up on observing a 0. The transient structure makes this ex- plicit: AB C maps back to itself on a 0. A. Calculation Figure 9 depicts how the cryptic order is determined. Only the paths in Fig. 5 that surviv e all the w ay to sync hrony (at the Marko v order) are repro duced. F rom these, we determine how many symbols into each word 7 w e must parse (from the left) b efore the  -machine is in one state only . The maximum such length is the cryptic order k χ . As with the Marko v order, we need only consider prefix-free sync hronizing w ords. Ho wev er, we are again faced with the prosp ect that there may b e an infinite n umber of prefix-free synchronizing words. F ortunately , a b etter metho d is av ailable, and it to o b egins by con- structing the p o wer automaton. No w, we examine the “v eracity” of each transient edge. T ake as an example the edge AB C 1 → A in Fig. 8. It states that up on pro- ducing a 1 from the sup erp osition of states A , B , and C , the system can only transition to state A . F or the cryptic order, w e now condition on the fact that we are in state A and ask what states could hav e transitioned to A on a 1. Upon insp ection, its clear that the system could hav e only transitioned from states A or C on a 1. The core of the cryptic order algorithm is to insp ect each transien t edge in the p ow er automaton in this manner, up dating the P A’s structure to “honestly” reflect the pro- cess’s dynamics. In this instance, we create a state AC that transitions to state A on a 1 instead of transitioning from AB C on a 1. After creating a state, the automaton must b e made consisten t. T o do this, subset construction is applied to include any newly added states. Generally , this creates new edges as well. And, these to o m ust be analyzed by the cryptic order algorithm. Once every edge has b een insp ected, some transien t structure will remain. Once again, the longest path is the key , and the same edge- w eighting metho d (Bellman-F ord) is emplo yed to find it and so give the cryptic order. B. Examples The w ays in whic h the cryptic order algorithm mo difies the p ow er automaton are diverse. Each example from Sec. V D ab ov e illustrates a different b ehavior. First, consider its b ehavior on the PSB Pro cess (Fig. 6), the final result of which is shown in Fig. 10. The edge C D 0 → D in Fig. 6 can b e remov ed since it do es not represent a path that is true. T o see wh y , note that to get to D on a 0, one must come from either state A or state C . Ho w ever, since w e are assuming C D , the pro cess must b e in either state C or D . The intersection of those tw o sets is state C and it is, therefore, the only p ossible state the system could hav e actual ly b een in. Th us, C D 0 → D is a misrepresen tation from the cryptic order persp ective and, in fact, it corresp onds to the edge C 0 → D , which already exists in the P A. So, the edge C D 0 → D is remov ed. 0 0 1 B C D A 0 1 0 A C D A D 0 1 1 A C D A B 1 0 0 A B B C D 1 0 1 D A D A 1 1 0 A B D A B B C 1 1 1 A B D A B B B FIG. 9. Key paths for determining cryptic order k χ : W e start with the paths in Fig. 5, except w e remo ve paths that do not surviv e to the end of the sy nc word. The surviving paths giv e us the cryptic order: They each identify a single state by length ` = 2 and so k χ = 2. This is not all, ho wev er. W e must maintain the path’s pro venance. The edges that came in to C D m ust b e redirected to C (add edges AB C D 0 → C and AB 0 → C ), since those are the edges that w ould ha ve b een trav ersed immediately prior to C D 0 → D . Note that these edges are later remov ed in this recursive algorithm and so do not app ear in Fig. 10. In the end, w e see that the longest path from a start state to the recurrent states is 2 and, therefore, k χ = 2, one less than the Marko v order R = 3. Next, consider the example from Fig. 7. The final out- put of the cryptic order algorithm is sho wn in Fig. 11. This pro cess’s P A consists of tw o ma jor branches: One with a maximum depth of 2 and the other containing a lo op. The cryptic order algorithm discov ers that the branc h with a lo op is completely retro dictable. AB 1 → C is actually B 1 → C , and this creates edges AB 0 → B and AB C 0 → B , again to main tain prov enance. The first of these newly added edges is also retro dictable: AB 0 → B can only b e A 0 → B . The second, AB C 0 → B , is in fact AC 0 → B . Along this branch of the transien t struc- ture, we are thus only unable to retro dict the word 01, of which the 1 can b e retro dicted, simply leaving us with AC 0 → B . The previous branc h is more easily analyzed, lea ving us with B C 1 → AC 0 → B , the later part of whic h w as already in the P A from analyzing the other branch. This leav es a longest path of length 2, making k χ = 2. Th us, we see that this pro cess is an example with infinite 8 A B C D AB AC ABD 1 1 0 1 0 1 0 0 1 FIG. 10. Cryptic order algorithm applied to the PSB Pro cess: The p o w er automaton in F ig. 6 suggests that the word 11 could originate in any of A , B , C , or D . Careful insp ection of the recurrent structure, though, shows that C cannot b e the originator of 11, whereas the other three states can. The cryptic order algorithm accounts for such constraints. The longest path from a transient state to a recurrent state is AB D 1 → AB 1 → B and, therefore, k χ = 2. Mark ov order, but finite cryptic order. The last example to consider is the Nemo Pro cess. Re- call that it is infinite Mark ov, as observed in Fig. 8. Ap- plying the cryptic order algorithm results in the structure sho wn in Fig. 12. In this case, the transien t structure gro ws under the algorithm. The edge AB C 1 → A , con- necting the transient to the recurrent structure in the p o w er automaton, is mo dified by the algorithm since B cannot transition to A on a 1. The state AC is created and connected to A . Completing the p o wer automaton structure from this state results in states AB and B C b e- ing added, forming the cycle AC 0 → AB 0 → B C 0 → AC . The algorithm terminates when the cycle is detected in this w ay . The cycle is v alid as far as the cryptic order is concerned: Eac h of its states can b e transitioned to from the recurrent state asso ciated with the prior state in the cycle. The cycle results in an arbitrarily long path and, therefore, k χ = ∞ . A B C A C BC 1 0 0 0 1 0 1 FIG. 11. Cryptic order analysis of Fig. 7’s pro cess: The transien t structure branch shown there— AB C 0 → ( AB 0 → AB ) ∗ 1 → C , with the arbitrarily long synchronizing w ord 00 ∗ 1—can b e perfectly retro dicted. Moreov er, only a frag- men t of the left branch of the transient structure remains. This fragmen t has a length of 2, and so k χ = 2. VI I. OTHER NA TURAL TIME SCALES P aralleling the interpretation of the Marko v and cryp- tic orders as the blo ck lengths at whic h an asso ciated in- formation measure reac hes its asymptotic b eha vior, this section briefly defines several new time scales associated with the multiv ariate information measures recently in- tro duced in Ref. [6] to dissect the information in a single measuremen t. The first order k I is the length at which the m ulti- v ariate m utual information I [ X 0 ; X 1 ; . . . ; X N − 1 ] reaches its asymptotic b ehavior. Unfortunately , no b ounds are kno wn for this order. The next collection of time scales—denoted k R , k B , k Q , and k W —are the lengths at which the residual en- trop y r µ , b ound information b µ , enigmatic information, and local exogenous information each reach their resp ec- tiv e asymptotes [6]. F urthermore, these four orders are equal, due to the linear interdependence of their resp ec- tiv e measures. It turns out that there are low er and upp er b ounds for these with resp ect to the Marko v order, whic h can be easily explained. Consider Fig. 8 in Ref. [6]: By definition H[ X :0 ] can b e replaced with H[ X − R :0 ] and, if the pro cess is stationary , H[ X 1: ] with H[ X 1: R +1 ]. It is therefore reasonable that one requires at least R sym b ols and most 2 R symbols to accurately dissect H[ X 0 ]. In fact, numerical surveys that we hav e carried out agree 9 A C AB BC A B C 0 1 0 0 1 0 0 0 1 FIG. 12. Cryptic order analysis of the Nemo Pro cess: Its p o w er automaton (Fig. 8) contains the edge AB C 1 → A . Ho w ev er, upon closer insp ection only states A and C can tran- sition to A on a 1. This creates the AC state. When emitting a 0, AC b ecomes AB and on a second 0 that b ecomes B C . A third 0 completes the cycle. The edges indicate legitimate transitions as well: States that actually lead to AC on a 0 are B C and those that lead to B C are AB , and so on. This leads to a cycle in the cryptic order algorithm’s calculated transien t structure. Therefore, one concludes that k χ = ∞ . with these limits. Finally , a sequel analyzes the elusiv e information σ µ , sho wing that the Marko v order R equals the length k σ µ at whic h the present measurement block X 0: ` renders the past and future conditionally indep endent. While we ha ve defined these orders and pro vided b ounds, it remains to be seen if there exist efficien t meth- o ds to calculate them, let alone top ological interpreta- tions for each. VI I I. SUR VEY W e illustrate the abov e results and algorithms, and their usefulness, by empirically answ ering several sim- ple, but comp elling questions ab out the space of finitary pro cesses. In particular, ho w typical are infinite Marko v order and infinite cryptic order? Restricting ourselves to top ological  -mac hines—those  -mac hines with a distinct set of allo wed transitions and 3 4 5 6 7 8 9 10 11 12 13 · · · ∞ Marko v order R 0 1 2 3 4 5 6 7 8 9 10 11 12 13 . . . ∞ cryptic order k χ FIG. 13. Distribution of Marko v order R and cryptic or- der k χ for all 1 , 132 , 613 six-state, binary-alphab et, exactly- sync hronizing  -machines. Marker size is prop ortional to the n um b er of  -mac hines within this class at the same ( R , k χ ). equiprobable transition probabilities—we enumerate all binary-alphab et pro cesses with a given num b er of states to which one can exactly synchronize. Ref. [18] details their definition, the enumeration algorithm, and how it giv es a view of the space of structured sto chastic pro- cesses. F or eac h of these  -mac hines, we compute its Mark ov and cryptic orders. The result for all of the 1 , 132 , 613 six-state  -machines is shown in Fig. 13. The num b er of  -machines that share a ( R , k χ ) pair is enco ded b y the size of the circle at that ( R , k χ ). The v ast ma jority of pro cesses—in fact, 98%—are non-Marko vian at this state-size (6 states). F urthermore, most (85%) of those non-Mark ovian processes are also ∞ -cryptic. How- ev er, this do es not imply that sync hronization is difficult; quite the contrary: synchronization o ccurs exp onentially quic kly [19]. What this do es mean is that with gro wing state size it b ecomes predominately likely that a given pro cess has particular sequences whic h will not induce state synchronization. Also of interest are the “forbidden” ( R , k χ ) pairs within the space of 6-state top ological  -machines. F or example,  -mac hines with k χ = 4 , 5 , 8 , 10 , 11 do not o ccur with R = 13. Also, pro cesses with infinite Marko v order and finite cryptic order app ear to hav e a maximum cryp- tic order of k χ = 11, despite the fact that larger finite cryptic orders exist for finite Marko v-order pro cesses. IX. SPIN CHAINS AND BEYOND Although our primary goal w as to precisely define length scales, sev eral b eing new, and to presen t efficien t calculation metho ds for them, it will b e helpful to briefly 10 dra w out the ph ysical meaning of Mark ov and cryptic orders by analyzing their role in spin c hains and related systems. (A sequel will delve into this topic in greater depth.) T o start, recall that Ref. [20] show ed that the Marko v order R of an  -mac hine represen ting a (one-dimensional) Ising spin system is upp er b ounded by the in terac- tion range sp ecified in a system’s Hamiltonian. Con- sider first the ferromagnetic, one-dimensional, nearest- neigh b or Ising mo del at different temp eratures T . The  -mac hines for this family of systems are sho wn in Fig. 14. As just noted, since the system has nearest-neighbor in- teractions, the Marko v order should b e R = 1. This is straigh tforward to see from the first  -machine, which is for a finite temperature T . Without an observ ation there are tw o p ossible causal states the system could b e in: ↑ or ↓ . Once a single spin has b een observed, ho wev er, the causal state is known exactly . This c hanges markedly at the temp erature limits, though. At T = 0, the sys- tem is in a ground configuration of either all up spins or all down spins. Without an external field to break this symmetry an observ ation must b e made to determine in whic h of these states it is and so the Mark ov order is still R = 1. In the presence of an external field, how ever, there is only a single ground state—that aligned with the field—and no observ ation is required to know in which state the system is. Thus, R = 0. At T = ∞ the system collapses to a single causal state where the next spin is en tirely determined by thermal fluctuations and so the Mark ov order is R = 0. As a second case, consider the an tiferromagnetic, one- dimensional, nearest-neigh b or Ising model, which is simi- lar enough that it mak es for a useful con trast; see Fig. 15. The finite temp erature and high temp erature limits are iden tical to the ferromagnetic case, but the low temp er- ature case differs. At T = 0 the spin system forms a p erfect crystal of alternating spins and so one m ust make a single observ ation to know in which spatial-phase the crystal is. Then, the entire structure is known exactly . Th us, the Marko v order is R = 1. This situation is not a broken symmetry as in ferromagnetic low-temperature case. Even with a nonzero external field, an observ ation is still required to know in whic h causal state the system is. Ov erall, now that we can directly determine intrin- sic lengths in configurations, w e see that the coupling range sp ecified b y a Hamiltonian need not be an in trin- sic prop erty of realized configurations. The simple ex- tremes ab ov e make this easy to understand. At infinite temp erature each system configuration is equally likely: the Hamiltonian range has no effect on which configura- tions are realized. At zero temp erature only the ground states are expressed and these need not explore all the p ossible configurations allow ed by the Hamiltonian. Both of these situations mask the coupling range sp ecified b y the Hamiltonian. Due to this, the Mark ov order R cap- tures the effective coupling range and need not match that sp ecified b y the Hamiltonian. In Ising spin chains, the cryptic order equals the Mark ov order. (This is due most directly to the fact that spin blo cks are in one-to-one c orresp ondence with the  -mac hine causal states. In addition, one must add the ca veat that the  -machine b e ergo dic.) This equality need not b e the case, ho w ever, even in simple physical systems. W e note how restrictive Hamiltonian-sp ecified dynamics are via t w o (again, 1D) examples of infinite Marko v order, but finite cryptic order, that arise from finite sp ecifica- tion. In the first class of systems, even though one starts with strictly local in teractions—configurations with finite Mark ov order sp ecified by a Hamiltonian with finite cou- pling range—a 1D system can anneal to one with effec- tiv ely infinite-range interactions, as shown in Ref. [21]. (See the  -mac hine in Fig. 2 there.) In this particu- lar case, the annealed state is non-Mark o vian, exhibiting infinite-range structure and Marko v order R = ∞ . No- tably , the annealed configurations for this example ha ve finite cryptic order k χ = 4. F or the second class of sys- tems we just briefly note that these un usual length-scale prop erties are not restricted to classical systems. They also arise in quantum systems. See the analyses in Refs. [22] and [23]. Finally , since the results here emphasize prop erties in- trinsic to realized configurations, let’s turn the question around. Given a single typical instance from the ensem- ble of allow ed configurations, ho w m uch can b e inferred ab out the Hamiltonian? Though the top ological tech- niques describ ed ab ov e do not provide coupling ampli- tudes and the like, they do give the maximum range of effectiv e interactions. What do es one do, though, with- out a Hamiltonian or some other system sp ecification? It turns out that a v ariety of metho ds exist for inferring hidden Marko v mo dels from a sample. And, since any hidden Mark o v model can b e con verted to an  -machine [16], from there the Marko v and cryptic orders can b e directly computed. And so, the ab ov e metho ds can b e applied to a wide range of theoretically mo deled or ex- p erimen tally realized physical systems. X. CONCLUSION W e began by defining tw o different measures of mem- ory in complex systems. The first, the Mark ov order R , is the length of time one must observe a system in order to mak e accurate predictions of its b eha vior. The second, the cryptic order k χ , quantifies the ability to retro dict 11 0 < T < ∞ ↑ ↓ p | ↑ 1 − p | ↓ 1 − p | ↑ p | ↓ T = 0 ↑ ↓ 1 | ↑ 1 | ↓ T = ∞ ↑↓ 1 2 | ↑ 1 2 | ↓ FIG. 14.  -Machines for a one-dimensional ferromagnetic Ising mo del as a function of temp erature T , where p = 1 2 (1 + tanh β ), the external field B = 0, and J = k B = 1. a system’s internal dynamics. W e show ed that despite their statistical nature, these time scales are top ological prop erties—prop erties of the sync hronizing words of a pro cess’s  -mac hine. W e demonstrated how to compute these length scales for hidden Marko v mo dels, most of which can b e moti- v ated in terms of the synchronization prop erties of the underlying pro cess. In terestingly , w e found that one of the most fundamen tal and imp ortan t prop erties—the Mark ov order R —is computable using only the pro cess’s  -mac hine. When calculated with non-  -mac hines, the algorithms yield related quan tities, such as the synchro- nization order. F or more details, see the App endices. In addition, the  -machine provides an exact metho d for computing the cryptic order. F rom these results, we con- structed very efficient algorithms for their calculation. In the empirical setting, we no w see that one should first infer the  -mac hine and then, from it, calculate the Mark ov and cryptic orders. There are a num b er of meth- o ds of inferring an  -mac hine from data; e.g., Ref. [24, and citations therein]. In the theoretical setting, given some formal description of a pro cess—such as a Hamil- tonian or general hidden Marko v mo del—one can ana- lytically calculate a process’s  -mac hine. In an y case, as so on as one has the  -machine the preceding gives exact results. T o appreciate what is typical ab out these length scales, w e surv ey ed the range of Marko v and cryptic orders in 0 < T < ∞ ↑ ↓ 1 − p | ↑ p | ↓ p | ↑ 1 − p | ↓ T = 0 ↑ ↓ 1 | ↓ 1 | ↑ T = ∞ ↑↓ 1 2 | ↑ 1 2 | ↓ FIG. 15.  -Machines for a one-dimensional antiferromag- netic Ising mo del as a function of temp erature T . Here, p = 1 2 (1 + tanh β ), the external field B = 0, and J = k B = 1. the space of all structured binary pro cesses represented b y  -mac hines with six states. The main result w as rather surprising, infinite Marko v and cryptic orders dominate. Th us, the top ological analysis leads one to conclude that sync hronization, even to finite-state stochastic processes, is generically difficult. How ever, from a probabilistic view it is exp onentially fast [19, 25]. A wa y to resolv e this seeming contradiction is to conjecture that the top ologi- cal properties are driven b y sequences whose relativ e pro- p ortion v anishes with increasing length. The survey also rev ealed a v ariety of interesting ancillary prop erties that p ose a num b er of op en questions, presumably combina- toric and group-theoretic in nature. W e closed analyzing the role these scales play in classi- cal (and briefly quantum) spin systems, drawing out the ph ysical interpretations. W e emphasized, in particular, the difference b etw een the interaction range sp ecified by a Hamiltonian and the effectiv e range of correlation in realized spin configurations. This led us to prop ose cal- culating the orders to put constraints on spin systems whose Hamiltonians are unknown. Finally , app endices prov e k ey claims ab ov e, discuss other related measures of synchronization, survey the sync hronization time and synchronization entrop y , and pro vide step-by-step details for each algorithm. 12 A CKNOWLEDGMENTS W e thank Ben Johnson and Nick T rav ers for man y in- v aluable discussions. This w ork was partially supp orted b y ARO grants W911NF-12-1-0234 and by W911NF-12- 1-0288 and b y the Defense Adv anced Researc h Pro jects Agency (DARP A) Physical Intelligence pro ject. The views, opinions, and findings contained here are those of the authors and should not b e interpreted as repre- sen ting the official views or p olicies, either expressed or implied, of ARO, D ARP A, or the Departmen t of Defense. [1] L. E. Reichl. A Mo dern Course in Statistic al Mechanics . Univ ersit y of T exas Press, Austin, T exas, 1980. [2] A. S. W eigend and N. A. Gershenfeld, editors. Time Se- ries Pr e diction: F or e c asting The F utur e And Understand- ing The Past , SFI Studies in the Sciences of Complexit y , Reading, Massac h usetts, 1993. Addison-W esley . [3] A. Ra j and A. v an Oudenaarden. Nature, nurture, or c hance: Sto chastic gene expression and its consequences. Cel l , 135:216–226, 2008. [4] J. P . Crutchfield and D. P . F eldman. Sync hronizing to the environmen t: Information theoretic limits on agent learning. A dv. Complex Sys. , 4(2):251–264, 2001. [5] J. P . Crutchfield, C. J. Ellison, R. G. James, and J. R. Mahoney . Sync hronization and Contro l in Intrinsic and Designed Computation: An Information-Theoretic Anal- ysis of Comp eting Models of Sto chastic Computation. Chaos , 20(3):037105, July 2010. [6] R. G. James, C. J. Ellison, and J. P . Crutchfield. Anatom y of a Bit: Information in a Time Series Ob- serv ation. Chaos , 21(3):1–15, May 2011. [7] N. F. T rav ers and J. P . Crutchfield. Exact sync hroniza- tion for finite-state sources. J. Stat. Phys. , 145(5):1181– 1201, 2011. [8] T. M. Cov er and J. A. Thomas. Elements of Information The ory . Wiley-In terscience, New Y ork, second edition, 2006. [9] J. E. Hop croft, R. Motw ani, and J. D. Ullman. Introduc- tion to automata theory , languages, and computation, 2nd edition, Marc h 2001. [10] J. P . Crutchfield. Betw een order and c haos. Natur e Physics , 8(Jan uary):17–24, 2012. [11] D. Lind and B. Marcus. An intro duction to Symb olic Dy- namics and Co ding . Cambridge Universit y Press, 1999. [12] Unifilar is known as “deterministic” in the finite au- tomata literature. Here, w e av oid the latter term so that confusion do es not arise due to the stochastic nature of the mo dels being used. It is referred to as “right- resolving” in sym b olic dynamics [11]. [13] J. R. Mahoney , C. J. Ellison, and J. P . Crutchfield. In- formation accessibilit y and cryptic processes. J. Phys. A: Math. The o. , 42(36):362002, 2009. [14] J. P . Crutchfield and D. P . F eldman. Regularities Unseen, Randomness Observed: Levels of En tropy Con vergence. Chaos , 13(1):25–54, 2003. [15] C. R. Shalizi and J. P . Crutc hfield. Computational Me- c hanics: Pattern and Prediction, Structure and Simplic- it y . J. Stat. Phys. , 104:817–879, 2001. [16] C. J. Ellison, J. R. Mahoney , and J. P . Crutchfield. Pre- diction, Retro diction, and The Amount of Information Stored in the Present. J. Stat. Phys. , 136(6):1005–1034, 2009. [17] T. H. Cormen, C. E. Leiserson, R. L. Riv est, and C. Stein. Intr o duction to algorithms . The MIT Press, Cambridge, Massac h usetts, 3rd edition, Septem b er 2009. [18] B. D. Johnson, J. P . Crutchfield, C. J. Ellison, and C. S. Mctague. En umerating Finitary Pro cesses. arXiv:1011.0036 . [19] N. T rav ers and J. P . Crutchfield. Exact synchronization for finite-state sources. J. Stat. Phys. , 145(5):1181–1201, 2011. [20] D. P . F eldman and J. P . Crutchfield. Discov ering non- critical organization: Statistical mechanical, information theoretic, and computational views of patterns in one- dimensional spin systems. Santa F e Institute Working Pap er 98-04-026 , 1998. [21] D. P . V arn and J. P . Crutchfield. F rom finite to infinite range order via annealing: The causal architecture of deformation faulting in annealed close-pack ed crystals. Phys. L ett. A , 234(4):299–307, 2004. [22] K. Wiesner and J. P . Crutchfield. Infinite correlation in measured quantum pro cesses. In Santa F e Institute Working Pap er 06-11-043; arxiv.or g quant-ph/0611143 , 2007. [23] K. Wiesner and J. P . Crutc hfield. Computation in sofic quantum dynamical systems. Natur al Computing , 7(1):317–327, 2007. [24] C. C. Strelioff and J. P . Crutchfield. Ba yesian structural inference for hidden pro cesses. 2013. San ta F e Institute W orking P ap er 13-09-027, arXiv:1309.1392 [stat. ML]. [25] N. T rav ers and J. P . Crutchfield. Asymptotic syn- c hronization for finite-state sources. J. Stat. Phys. , 145(5):1202–1223, 2011. App endix A: Definitions Here, we provide additional results on length scales and synchronization and prov e a num b er of claims made in the main text. First, w e lay out the definitions needed and then give several key results that follow. Building on these, we delineate the central algorithms and con- clude with a comparison of sync hronization time and syn- c hronization entrop y , notions perhaps more familiar than other measures used and their length scales. 13 1. Minimal Synchronizing W ords F or the sync hronization problem, we consider an ob- serv er who b egins with a correct mo del (a presentation) of a pro cess. The observer, how ever, has no knowledge of the pro cess’s internal state. The challenge is to an- alyze how an observ er’s knowledge of the internal state c hanges as more and more measurements are observed. A t first glance, one might say that the observ er’s kno wledge should never decrease with additional mea- suremen ts, corresp onding to a never-increasing state un- certain ty , but this is generically not true. In fact, it is p ossible for the observ er’s knowledge (measured in bits) to oscillate with each new measurement. The crux of the issue is that additional measurements are b eing used to inquire ab out the curr ent state rather than the state at some fixed moment in time. It is helpful to iden tify the set of words that take the observ er from the condition of total ignorance to exactly kno wing a pro cess’s state. First, we introduce what we mean by sync hronization in terms of lack of state uncer- tain ty . Second, we define the set of minimal synchroniz- ing words. Definition 1. A wor d w of length L is synchronizing if the Shannon entr opy over the internal state, c onditione d on w , is zer o: S y nc ( w ) ⇔ H [ S ` | X 0: ` = w ] = 0 , (A1) wher e S ync ( w ) is Bo ole an function. Definition 2. A pr esentation ’s set of minimal synchro- nizing words is the set of synchr onizing wor ds that have no synchr onizing pr efix: L sync ≡ { w | S y nc ( w ) and ¬ S y nc ( u ) for all u : w = uv } . Remark. L sync is a pr efix-fr e e, r e gular language. If e ach wor d is asso ciate d with its pr ob ability of b eing ob- serve d, we obtain a pr efix-fr e e c o de enc o ding e ach p ath to synchr ony—a wor d in L sync —with the asso ciate d pr ob a- bility of synchr onizing via that p ath. These c o des ar e gen- er al ly nonoptimal in the familiar information-the or etic sense. 2. Sync hronization Order According to Sec. A 1, one is synchronized to a pro- cess’s presentation after seeing word w if there is com- plete certain t y in the state. W e now expand this view sligh tly to ask ab out sync hronization o ver all w ords of a particular length. Equiv alently , w e examine sync hroniza- tion to an ensemble of pro cess realizations. Definition 3. The synchronization order k S [5] is the minimum length for which every al lowe d wor d is a syn- chr onizing wor d: k S ≡ min { ` | H [ S ` | X 0: ` ] = 0 } . (A2) As for the Markov and cryptic or ders, k S is c onsider e d ∞ when the c ondition do es not hold for any finite ` . App endix B: Results W e now pro vide several results related to these length scales that shed light on their nature, introducing con- nections and simplifications that mak e their computation tractable. Prop osition 1. The synchr onization or der is: k S = max { R, k χ } . (B1) Pro of. First, note that: H [ S ` | X 0: ` ] = H [ X 0: ` , S ` ] − H [ X 0: ` ] . (B2) Sinc e the blo ck-state entr opy upp er b ounds the blo ck entr opy, the c onditional entr opy ab ove c an only r e ach its asymptotic value onc e b oth terms have individual ly r e ache d their asymptotic b ehavior. The latter ar e c on- tr ol le d by k χ and R , r esp e ctively. This result reduces the apparent div ersity of length scales, even tually allowing one to calculate the Marko v order via the synchronization order, whic h itself is di- rectly computable. Prop osition 2. F or  -machines: R = k S . (B3) Pro of. Applying the c ausal e quivalenc e r elation ∼  to Def. 1 we find: Pr( X 0: | X :0 ) = Pr( X 0: | X − ` :0 ) = ⇒ X :0 ∼  X − R :0 . (B4) This further implies that the c ausal states S ar e c om- pletely determine d by X − R :0 : H [ S 0 | X − R :0 ] = 0 . (B5) 14 This statement is e quivalent to the Markov criterion. Remark. This pr ovides an alternate pr o of that the cryp- tic or der k χ is b ounde d ab ove by the Markov or der R in an  -machine via a simple shift in indic es: H [ S 0 | X − R :0 ] = 0 (B6) = ⇒ H [ S R | X 0: R ] = 0 (B7) = ⇒ H [ S R | X 0: ] = 0 . (B8) This prop osition gives indirect access to the Marko v or- der via a particular presen tation—the  -machine. Since the Marko v order is not defined as a prop ert y of a pre- sen tation it would generally b e unobtainable, but due to unique prop erties of the  -machine, it can b e accessed through the synchronization order. There is a sub class of  -machines to which one syn- c hronizes in finite time; these are the exact  -mac hines of Ref. [7]. Prop osition 3. Given an exact  -machine with finite Markov or der R , the subshift of finite typ e that underlies it has a “step” [11] e qual to R . Corollary 1. Given an exactly synchr onizing  -machine, the underlying sofic system is a subshift of finite typ e if and only if R is finite. Remark. A pr o c ess with infinite Markov or der c an have a pr esentation whose underlying sofic system is a subshift of finite typ e. These results draw out a connection with length scales of sofic systems from sym b olic dynamics [11]. Subshifts of finite type hav e a probability-agnostic length scale analog of the Marko v order known as the “step”. In the case of  -mac hine presentations, they are in fact equal. W e will no w prov e that tw o of the lengths defined— the cryptic and sync hronization orders—are top ological. That is, they are prop erties of the presentation’s graph top ology and are indep enden t of transition probabilities, so long as changes to the probabilities do not remov e transitions and do not cause states to merge. Addition- ally , due to Prop. 2, the Marko v order is top ological. All three are topological since they depend only on the length at which a conditional entrop y v anishes, not on how it v anishes. Theorem 1. Synchr onization or der k S is a top olo gic al pr op erty of a pr esentation. Pro of. Be ginning fr om Def. 3, ther e is length ` = k S at which: H [ S ` | X 0: ` ] = X w ∈A ` Pr ( w ) H [ S ` | X 0: ` = w ] = 0 . Thus, H [ S ` | X 0: ` = w ] = 0 for al l w ∈ A ` , the set of length- ` wor ds with p ositive pr ob ability. Sinc e every wor d of length ` is synchr onizing, ` is c ertainly gr e ater than the synchr onization or der. As synchr onizing wor ds ar e synchr onizing r e gar d less of their pr ob ability of o c curring, the synchr onization or der k S is top olo gic al. Corollary 2. Markov or der R is a top olo gic al pr op erty of an  -machine. Pro of. Sinc e k S is a top olo gic al pr op erty by Thm. 1 and sinc e an  -machine’s R = k S by Pr op. 2, the Markov or der is top olo gic al. Theorem 2. Cryptic or der k χ is a top olo gic al pr op erty of a pr esentation. Pro of. Be ginning fr om Def. 3, ther e is a length ` = k χ at which: 0 = H [ S ` | X 0: ] (1) = X x 0: ∈A ∞ Pr ( x 0: ) H [ S ` | X 0: = x 0: ] (2) = X w ∈L sync Pr ( w, σ w ) H [ S ` | X 0: | w | = w , S | w | = σ w ] . Her e, step (1) simply exp ands the c onditional entr opy. Step (2) is true pr ovide d that the sum is over minimal synchr onizing wor ds and σ w is the state to which one syn- chr onizes via w . This final sum is zer o only if the sum vanishes term-by-term. Thus, given a wor d that synchr o- nizes and the state to which it synchr onizes, e ach term pr ovides a cryptic-order candidate —the numb er of states that c ould not b e r etr o dicte d fr om that state and wor d. Final ly, the longest such cryptic or der c andidate is the cryptic or der for the pr esentation. Restated, the cryptic order k χ is top ological as it de- p ends only on the minimal synchronizing words, which are top ological b y definition. App endix C: Algorithms W e are now ready to turn to computing the v arious sync hronization length scales giv en a presentation. While all of the algorithms to follow hav e compute times that are exp onen tial in the num b er of machine states, we find them to b e very efficient in practice. This is particularly the case when compared to naive algorithms to compute these prop erties. F or example, computing synchroniza- tion, Marko v, or cryptic orders by testing successively longer blo cks of symbols is exp onen tial in the length of the longest block tested. W orse, in the case of non- Mark ovian and ∞ -cryptic pro cesses the naiv e algorithm 15 will not halt. In addition, the naive implemen tation of Thm. 2 given in the pro of to compute the cryptic order has a compute time of O (2 2 N ), whereas the one presen ted b elo w is a simple exp onential of N . Unsurprisingly , given the results provided in Sec. B, w e b egin with the minimal synchronizing words as they are the underpinnings of the synchronization and cryp- tic orders. The algorithms make use of standard pro ce- dures. Most textbo oks on algorithms provide the neces- sary background; see, for example, Ref. [17]. 1. Minimal Synchronizing W ords W e construct a deterministic finite automaton (DF A) that recognizes L sync of a giv en presentation M = ( Q, E ), where Q are the states and E are the edges. This is done as follows: Algorithm 1. 1. Be gin with the r e curr ent pr esentation M . 2. Construct M ’s p ower automaton 2 M , pr o ducing a DF A T = 2 M . 3. Set the no de in T that c orr esp onds to al l M ’s states as T ’s start state. 4. R emove al l e dges b etwe en singleton states of T . (These ar e the e dges fr om M .) 5. Set al l singleton states of T as ac c epting states. No w, we enumerate L sync via an ordered breadth-first tra versal of T , outputting each accepted word. 2. Sync hronization Order Thanks to Eq. (1) we see that k S is the shortest length ` that encompasses all of L sync . This is, trivially , the longest word in L sync . With this, computing the syn- c hronization order reduces to: Algorithm 2. 1. If L sync is infinite, r eturn ∞ . 2. Enumer ate e ach wor d in L sync and r eturn the length of the longest wor d. The test in the first step can b e done simply b y running a lo op-detection algorithm on DF A T . If there is a lo op, then L sync is infinite. 3. Mark ov Order Due to Thm. 2, a pro cess’s Mark ov order can b e com- puted by finding the sync hronization order of the pro- cess’s  -machine. If one do es not hav e the  -machine for a pro cess, but rather some other unifilar presen tation, it is still p ossible in some cases to obtain the Marko v order through the synchronization order. That is, the algorithms for k S and k χ pro vide prob es into the presen- tation’s length scales. It can b e the case that R is accessi- ble to those prob es, if k χ < k S , but it is only guaranteed to b e accessible in the case of  -machines. Note, there exist techniques for constructing the  -mac hine from any presen tation [16]. 4. Cryptic Order In the following algorithm T refers to the p ow er au- tomaton of the machine M . T ’s states— p , q , and r —are elemen ts of the p ow er set of the states of M . By the pr e- de c essors of a state q along edge p → xq we refer to the set p 0 = { m | ( m → xn ) ∈ M and m ∈ p and n ∈ q } . These are the states m ∈ p that actually transition to a state n ∈ q on symbol x . By subset c onstruction b elo w we refer to the standard NF A-to-DF A con version algorithm [9]. Algorithm 3. 1. Construct the p ower automaton T = 2 M via subset c onstruction. 2. Push e ach e dge p → xq in T to a queue. 3. While queue is not empty: (a) Pop e dge p → xq in the queue. (b) If e dge is in pr o c esse d list: i. R estart lo op, p opping the next e dge fr om the queue. (c) Find the pr e de c essors p 0 of q along p → xq . (d) If p 0 6 = p : i. R emove e dge p → xq fr om T . (e) If | p 0 | > 1 : i. Perform subset c onstruction on p 0 (im- plicitly, this adds the e dge p 0 → xq to T ). ii. Push e ach e dge cr e ate d in the prior step into the queue. iii. F or e ach r → y p in T : A. A dd e dge r → y p 0 to T . B. A dd e dge r → y p 0 to the queue. iv. A dd p → xq and p 0 → xq to the pr o c esse d list. 16 The result is an automaton T 0 . The longest path in T 0 through transient states ending in a recurrent state is the cryptic order. Skipping previously processed edges is imp ortant since for some top ologies the algorithm can en ter a cycle where it will remov e and then later add the same e dge, ad infinitum. There are three simple additions to this algorithm that result in a sizable decrease in running time. The first is to store the edges to b e pro cessed in a priority queue, suc h that an edge p → xq is popp e d b efore an edge r → y s if | q | < | s | , or if | q | = | s | , then | p | < | r | . The second optimization is to trim dangling states after each pass through the outer lo op. A dangling state is a state p suc h that there is no path from p to the recurrent states. The last metho d for improving sp eed is to not add edges b et w een recurrent states to the queue in step 2. This algorithm for computing the cryptic order only holds for unifilar presentations. App endix D: Statistical Measures of Synchronization 1. The Synchronization Distribution T aking a sligh tly more general view than the syn- c hronization order, we consider statistical prop erties of sync hronization, rather than just the absolute length at whic h an ensemble will all b e synchronized. In this vein, w e define a distribution that gives the probability for a w ord to first synchronize at length ` . Definition 4. The synchronization distribution S gives the pr ob ability of synchr onizing to a pr esentation at length ` : S ( ` ) ≡ X w ∈L sync Pr ( w ) δ ( | w | − ` ) . (D1) wher e δ is the Kr one cker delta function. Remark. S is normalize d: P ∞ ` =0 S ( ` ) = 1 . W e now dra w out t wo particular quan tities from this distribution—quan tities that hav e observ able meaning for a presentation. Definition 5. The sync hronization time τ [5] is the av- er age numb er of observations ne e de d to synchr onize to a pr esentation: τ ≡ X w ∈L sync | w | Pr ( w ) (D2) = E ` [ S ( ` )] , (D3) wher e the se c ond e quality shows that τ is also e qual to the exp e ctation value of the synchr onization distribution. The synchronization time is useful for understanding ho w long it takes on aver age to synchronize to a mo del. This is in con trast to the Mark ov order whic h is the mini- mal longest-sync hronization-time across all presen tations of a pro cess. Definition 6. The synchronization entrop y H sync is the unc ertainty in the synchr onization distribution: H sync ≡ H [ S ( ` )] . (D4) Remark. Note that this is quite distinct fr om the syn- c hronization information S of R ef. [14]: S = ∞ X ` =1 H [ S ` | X 0: ` ] . The synchronization entrop y , in contrast, measures the flatness of the synchronization distribution. And, since the synchronization distribution decays exp onen- tially with length, the fatter the tail, the higher the un- certain ty in synchronization. 2. The Synchronization Distribution There are tw o metho ds to compute the synchronization distribution. The first requires an  -mac hine with finite recurren t and transient comp onents. The second, only a finite recurren t comp onent. W e present the former case first. Algorithm 4. 1. Perform an or der e d br e adth-first tr aversal of the fi- nite  -machine. 2. While tr aversing, ke ep tr ack of the wor d induc e d by the p ath and the pr o duct of the pr ob abilities along that p ath. 3. When a r e curr ent no de is r e ache d, stop that p artic- ular thr e ad of the tr aversal. 4. Sum the pr ob abilities of al l wor ds with the same length. This algorithm pro duces each minimal synchronizing w ord and its probability in lexicographic order. Then w ords of each length can b e group ed and their probabil- ities summed to get the synchronization distribution. The second algorithm is used when an  -mac hine with finite transient structure is not av ailable: Algorithm 5. 17 1. Pr o duc e al l the minimal synchr onizing wor ds fr om the DF A given in Algorithm 1. 2. F or e ach minimal synchr onizing wor d, c ompute its pr ob ability using the r e curr ent  -machine [14]. 3. Sum the pr ob abilities of al l wor ds with the same length. 1 2 3 4 5 6 7 8 9 10 synchronization time τ 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 synchronization entropy H sync FIG. 16. Distribution of synchronization time τ and syn- c hronization entrop y H sync for all 1 , 388 four-state, binary- alphab et, exactly-sync hronizing  -mac hines with uniform out- going transition probabilities. Individual histograms for each prop ert y are sho wn ab o v e and to the righ t. Once the distribution is computed using one of the ab o v e algorithms, it is trivial to compute the Shannon en tropy and mean of the distribution to get the sync hro- nization entrop y and synchronization time, resp ectively . 3. Results Finally , w e survey the distribution of synchronization times τ and synchronization en tropies H sync for all 1 , 388 four-state, binary-alphab et, exact  -machines with uni- form outgoing transition probabilities [18]. See Fig. 16. It is interesting to note that there is structure in the dis- tribution in the form of veils. How ever, the veils are not the entiret y of the distribution, there are man y mac hines that fall elsewhere.