Growth Functions, Rates and Classes of String-Based Multiway Systems
GGrowth Functions, Rates and Classes of
String-Based Multiway Systems
Yorick Zeschke Junior Research Affiliate, Wolfram Physics ProjectFebruary 9, 2021
In context of the Wolfram Physics Project [11], a certain class of abstract rewritesystems known as “multiway systems” have played an important role in discretemodels of spacetime and quantum mechanics. However, as abstract mathemati-cal entities, these rewrite systems are interesting in their own right. This paperundertakes the effort to establish computational properties of multiway systems.Specifically, we investigate growth rates and growth classes of string-based multiwaysystems. After introducing the concepts of “growth functions”, “growth rates” and“growth classes” to quantify a system’s state-space growth over “time” (successivesteps of evolution) on different levels of precision, we use them to show that mul-tiway systems can, in a specific sense, grow slower than all computable functionswhile never exceeding the growth rate of exponential functions. In addition, we startdeveloping a classification scheme for multiway systems based on their growth class.Furthermore, we find that multiway growth functions are not trivially regular butinstead “computationally diverse”, meaning that they are capable of computing orapproximating various commonly encountered mathematical functions. We discussseveral implications of these properties as well as their physical relevance. Apartfrom that, we present and exemplify methods for explicitly constructing multiwaysystems to yield desired growth functions.
Contents a r X i v : . [ c s . D M ] F e b Introduction and Overview
In 2019, Stephen Wolfram et. al. launched the Wolfram Physics Project [11] as a new at-tempt to find a fundamental theory of physics (see [25] and [24] for a general overview and [21]for a technical introduction, as well as the glossary of [7] as a reference for terminology usedin this paper). The Wolfram Models, discrete spacetime formalisms generalising models firstintroduced by Wolfram in [22], have been found to be significantly meaningful in a theoreti-cal physics context, showing various connections to known theories of relativity, gravity andquantum mechanics [6][5][7].In the project, a type of abstract rewriting systems (see definition 1 in [5], as well as [3]and [4] for more complete references) equipped with causal relations between their elementshas been called “multiway systems” (definition 10 in [5]) and shown to be connected to manyphysical properties of our universe. Additionally, various links between multiway systems andgroup theory [1], homotopic type theory [2], category theory [7], numerics of partial differentialequations [13] and the theory of theorem proving ([22] p. 775 ff.), as well as the study of com-plex systems, computational complexity and emergence ([21] p. 204, 939) have been established,showing that these systems are relevant and interesting from a mathematics or computer scienceperspective as well. While in the physical framework of the actual Wolfram Model (see section 2in [6] for a formal definition), hypergraph-based multiway systems have been used, we considerstring-based multiway systems instead since they are more fundamental because of their simplerstructure. Most likely, our results generalise easily to hypergraph-based multiway systems. Inany case, they yield significant new insights into the general principles underlying these systems.Although our investigations are rather theoretical and aim at laying a mathematical founda-tion for understanding the structure of multiway systems in themselves, we comment on severalpotential applications in the Wolfram Physics Project in section 4 and demonstrate our generalresult by computational simulations of specific examples. Our visualisations have been made us-ing Mathematica and all code for simulating multiway systems is available in the Wolfram Func-tions Repository [12]. Readers interested in running simulations and visualisations themselvesmay find the computational quick-start guide and the documentation of the MultiwaySystem resource function [8] to be useful references.The subsequent subsections start by formally defining what we mean by multiway systemgrowth functions, rates and classes. Next, we investigate the boundaries of possible growthrates and find that multiway systems are, simply put, bounded in the speed but unbounded inthe slowness of their growth rate (theorem 2.1). After that, we show that the growth classes ofmultiway systems we defined cover the entire set of multiway systems and apart from one triviallyempty class, all of them contain infinitely many multiway systems (theorem 3.1). To do this, wedefine arithmetic-like operations equipping the set of multiway systems with a semiring structure.Combing the two theorems, we conclude various interesting properties of the “computationaldiversity” and “-complexity” of multiway systems, showing that their growth functions constitutean interesting domain of further research.
Consider a string-based multiway system M (definition 10 in [5]), represented as a triplet( R, s init , Σ) where Σ is a finite alphabet, R = { r → t , . . . , r n → t n } is a set of string replace-ment rules over Σ and s init ∈ Σ ∗ is the initial string . We define the “state-set of generation n ” as the set of all new (previously nonexistent) states added to the multiway system in its n -th generation. These states are precisely the nodes of the states graph (c. f. section 5.3 in This means that the underlying “objects” or “elements” of the abstract rewriting systems are hypergraphs. A computational essay can be found at . In the following, Σ ∗ denotes the set of all words over the alphabet Σ. n . In [21] they are called“merged states”. Now, the “growth function ” g M ( n ) is simply the cardinality of the state-setof generation n . Figure 1: Both M = ( { “ A ” → “ AB ” } , “ AA ” , { A, B } ) and M = ( { “ A ” → “ AB ” , “ AB ” → “ A ” } , “ AA ” , { A, B } ) have the same growth function g ( n ) = n because cycles in the statesgraph do not lead to new states. In general, it is very hard or even undecidable (see section 3.3) to prove that some multiwaysystem has a certain growth function. It is also not obvious that the growth functions of multiwaysystems should be elementary functions or “simple” by any other definition. Examples of systemswhere the growth function is hard to describe were already given in “A New Kind of Science”([22] pp. 204 ff.). Therefore, we will approximate the growth functions of multiway systems bycontinuous, strictly monotonically increasing, unbounded (and hence bijective on R ≥ ) functionswhich can be analysed more easily. This way, many similar growth functions will be consideredmembers of the same equivalence class. We will say that the corresponding multiway systemshave the same “growth rate”. To formalize the notion of approximating functions, we use the asymptotic growth classes fromcomplexity theory, defined in the following way: For a function f : A → B where A = B = R ≥ or A = B = R , O ( f ) is defined as the set of functions g : A → B for which lim sup x →∞ | g ( x ) f ( x ) | exists and is a real number . The subset of O ( f ) for which the limit superior is zero is denoted o ( f ). Similarly, Ω( f ) := { g : A → B | f ∈ O ( g ) } and ω ( f ) := { g : A → B | f ∈ o ( g ) } . Finally,Θ( f ) := Ω( f ) ∩O ( f ). It is straightforward to show that f ∼ Θ g ⇐⇒ f ∈ Θ( g ) is an equivalencerelation. Thus, we may speak of functions that are “asymptotically equal”.As mentioned above, we want to approximate growth functions by bijective functions for thesubsequent mathematical analysis. For some multiway growth function a , we will define thesequences a and a as its tightest upper and lower bounds which are monotonically increasing,even if a itself is not monotonic at all. From these sequences, we will then construct twoequivalence classes of continuous functions which are all asymptotically equal to and hence“close approximations” of a or a respectively. Two representatives of these classes will be called“tight bounds” and, since we are generally concerned with unbounded growth functions , bothbijective on R ≥ . We use the terms “sequence” and “function” interchangeably for f : N + → N + . An equivalent definition is g ∈ O ( f ) ⇐⇒ ∃ C ∈ R + : ∃ x ∈ A : ∀ x > x : C | f ( x ) | ≥ | g ( x ) | . Bounded growth functions will be discussed shortly. R or R ≥ . How-ever, since the multiway growth function is always a function on N + , we consider its linearinterpolation, a continuous function from R ≥ to R ≥ which is equal to the sequence for nat-ural arguments and always bounded by consecutive values of the sequence (see definition 2.2),instead. Definition 1.1.
Let M be a multiway system and g M its growth function. We call M “finite”if ∃ n ∈ N + : g M ( n ) = 0 (as this implies that at a certain point, no further states will be added).We call M “bounded” if ∃ b ∈ N + : ∀ n ∈ N + : g M ( n ) ≤ b and M is not finite. Systems which areneither finite nor bounded are called “unbounded”. Figure 2: States graphs of the finite system M = ( { “ A ” → “ BC ” , “ B ” → “ C ” , “ C ” → “ B ” } , “ A ” , { A, B } ) and first six steps of the bounded system M = ( { “ A ” → “ AA ” } , “ A ” , { A } ).Notice that ∀ n ∈ N + : g M ( n ) = 1, despite the fact that the rule can be applied in many differentpositions, because we are only considering merged states. Definition 1.2.
Let a : N + → N + be the growth function of an unbounded multiway system andlet a n := max( { a k | k ≤ n } ) and a n := max( { a k | k ≤ n ∧ ( ∀ l ≥ k : a l ≥ a k ) } ∪ { } ). We call twocontinuous functions f, g : R ≥ → R ≥ “tight bounds of a ” if f ∈ Θ( L N + ( a )) ∧ g ∈ Θ( L N + ( a ))where L N + denotes the linear interpolation over N + (according to definition 2.2). Figure 3: The growth function a n of M = ( { “ AB ” → “” , “ ABA ” → “ ABBAB ” , “ ABABBB ” → “ AAAAABA ” } , “ ABABAB ” , { A, B } ) together with a n , a n and a pair ( f, g ) of tight bounds.Note: Only f and g are continuous, the other lines are drawn for visual appearance. One might ask why we introduce an upper and a lower bound instead of approximating thegrowth function with a single function. Doing so would however be a poor approximation asthere are multiway systems for which even the tightest upper and lower bounds are never inthe same asymptotic growth class (compare figure 4). We will call these multiway systems“strongly oscillating” and all others (i. e. systems where all tight bounds are asymptoticallyequal) “regular”. Notice that for every regular multiway system, a pair of bijective tight boundsexists because its tight bounds will be in the asymptotic equivalence class of two unboundedstrictly monotonically increasing functions and tight bounds are continuous on R + . Stronglyoscillating systems on the other hand are much more difficult to analyse since one cannot easilycome up with criteria for measuring the rate of oscillation and it is not clear at all whether there4as to be any periodicity or regularity in the way in which they oscillate. Thus, for our basicinvestigations about the fundamental structure of multiway systems, we shall focus on regularsystems. Figure 4: For this special system, one can prove (see section 3.3 for an explanation) the tight bounds f, g to be in different asymptotic growth classes: f ∈ Θ( x ) and g ∈ Θ( x ). Note again that only f and g are continuous while a n is drawn as a line for visual appearance. Definition 1.2 suggests a natural way to define classes of multiway systems with “similar”growth functions by considering growth functions with approximately equal tight bounds asequivalent. Let f, g : N + → N + be functions and ( a , b ) , ( a , b ) be tight bounds of f and g respectively. We define ∼ R by f ∼ R g ⇐⇒ ( a ∼ Θ a ∧ b ∼ Θ b ). Since tight bounds alwaysexist, ∼ R is an equivalence relation because ∼ Θ is one. For some multiway system M withgrowth function g M , we call the equivalence class of ∼ R that g M falls into the “growth rate” of g M (or sometimes the growth rate of just M ).It is obvious that every multiway system has exactly one growth function and exactly onegrowth rate. The converse, i. e. that every function f : N + → N + is the growth function ofsome multiway system or, respectively, that every pair of bijective functions on R ≥ is a pairof tight bounds of some multiway growth function, is clearly not true as emphasized in lemma2.1. However, if we define much more general classes of growth functions which we will call“multiway growth classes”, we will see (in theorem 3.1) that they indeed partition the set of allmultiway systems into a finite set of infinite subsets. To further distinguish between types of multiway systems on a more abstract level and demon-strate which kinds of growth functions can be achieved, we want to define very broad classesof multiway systems whose growth functions show similar behavior on a large scale. We havealready distinguished between finite, bounded and unbounded systems, as well as dividing thelatter into regular and strongly oscillating systems. As outlined above, we will focus on regu-lar systems. To group these into sets of systems of similar behavior, we use commonly knownclasses of functions such as polynomial or exponential functions , intermediately (faster thanpolynomial and slower than exponential) growing functions and some others.More precisely: Let G pol be defined as the set of all continuous bijections f : R ≥ → R ≥ which satisfy f ∈ Ω( x n ) ∩ O ( x n +1 ) for some n ∈ N + and define G exp as { f : R ≥ → R ≥ | f ∈ Ω( a x ) ∩ O (( a + 1) x ) } for some a ∈ N > . Similarly, let G supexp be the set { f : R ≥ → R ≥ | ∀ g ∈ G exp : f ∈ ω ( g ) } where f must be continuous and bijective. Additionally, denote by G int theset of all continuous bijections f : R ≥ → R ≥ fulfilling ∀ g ∈ G pol , h ∈ G exp : f ∈ ω ( g ) ∩ o ( h ).Now, it is easy to define G invpol := { f | f − ∈ G pol } , G invexp := { f | f − ∈ G exp } , G invsupexp := { f | f − ∈ G supexp } and G invint := { f | f − ∈ G int } . Actually, we are talking about polynomially or, respectively, exponentially bounded functions which need notbe polynomials or simple exponential functions. R ≥ because afunction grows either slower than f : x (cid:55)→ x in which case its inverse grows faster than f , or itgrows faster (or equal to) f in which case it is contained in one of the first four classes. We calla multiway system a member of the growth class C i if its growth function has tight bounds f, g belonging to G i . However, as definition 1.2 is not applicable for finite or bounded multiwaysystems, we handle them separately.Let C fin and C bnd the sets of all finite and bounded multiway systems respectively. For allmultiway systems in C fin ∪ C bnd , the growth rate is defined to be (1 , , their growth functions are not very interestingfor our purposes. They might be useful for applications not directly related to the WolframPhysics Project, but in this paper, they will not be discussed in great detail.As every continuous bijection on R + belongs to exactly one of the G i , every multiway systemone can imagine is either strongly oscillating or in one of those classes (including C fin and C bnd ).We will furthermore show in theorem 3.1 that every of these classes (except C supexp which isempty by lemma 2.1) contains infinitely many multiway systems.Summarizing the previous section, we introduced the three main concepts of multiway growthfunctions, multiway growth rates and multiway growth classes. We will now present the firstimportant result of this paper, a theorem about the boundaries of possible growth rates, andspend the next section proving and illustrating it. Having defined multiway growth rates, we might ask ourselves, which growth rates are possible,i. e. how the equivalence classes of ∼ R are distributed in the set of all possible pairs of bijectivefunctions on R ≥ . First of all, it is quite easy to give an upper bound for growth rates that canbe achieved. In fact, no multiway system can grow faster than exponentially. Lemma 2.1.
Let ( f, g ) be the growth rate of some multiway system. There exists some constant c ∈ R for which f, g ∈ o ( e cx ). Proof.
Denote by s max ( n ) the maximum string length that states of generation n can have. Forevery multiway system M = ( R, s init , Σ), the set of rules remains constant during the wholeevolution, so s max can at most increase constantly, i. e. s max ∈ O ( n ). Since the number ofwords with length l is given by | Σ | l , the growth function g M ( n ) will never exceed | Σ | s max ( n ) = e ln( | Σ | ) s max ( n ) ∈ Θ( e cn ) and the claim follows.So what about a lower bound for multiway growth rates? Formally, a trivial multiway systemwith no rules and thus only one state has the lowest possible growth function by point-wisevalue comparison. In general, “terminating” or “constant” asymptotic growth functions of finiteor bounded multiway systems (which have the growth rate (1 , , but examples like this are not very illuminating. Therefore, wemight ask what the slowest growth rate faster than constant is, i. e. what the smallest (byasymptotic comparison) functions f, g ∈ ω (1) are, for which ( f, g ) is the growth rate of somemultiway system. It turns out however, that no such smallest growth rate exists which meansthat multiway system can, in a certain sense, grow arbitrarily slowly.To understand why this is the case and make the even stronger statement that multiwaysystems can grow slower than all computable functions (see corollary 2.1.2), we need to introducea couple of constructions. First of all, we will show how multiway systems can emulate Turing This definition applies only to regular multiway systems In fact, what seems to be “complex behavior” in a “New Kind of Science”-fashion (compare [22]), occurredmuch more frequently in our empirical investigations of finite systems but this might just indicate a lack ofunderstanding. “Asymptotic comparison” refers to the total ordering ≤ O defined by f ≤ O g ⇐⇒ f ∈ O ( g ) T ( n ) steps where T ( n ) is the number of operationsthat a certain Turing machine T carries out before halting when provided with the input n .Such a machine, with some additional constraints, will be called a “ T -halter” and T its “haltingfunction”.By adding some specific rules to the multiway system that emulates T , it will be possible toevaluate T indefinitely for increasing inputs n = 1 , , . . . . Additionally, the multiway systemwill be constructed in a way such that every time, the underlying Turing machine is “startedagain” on the next input, the number of new states per time step is increased by one. This way,we will obtain a growth function informally described by the sequence “ n occurs T ( n ) times”(see definition 2.1), for example the sequence “ n occurs n times”, which would be given by1 , , , , , , , , , , , . . . . We will then show that this growth function is approximated bythe inverse of the linear interpolation (see definition 2.2) over the summatory function of T (seelemma 2.2). From this we conclude the following theorem: Theorem 2.1.
Let T ∈ Ω( n ) be the halting function of some Turing machine. There is amultiway system with growth rate ( a, b ) such that a, b ∈ O ( T − ∗ ) where T ∗ ( x ) = L N + ( (cid:80) nk =1 T ( k )) and L N + denotes the linear interpolation over N + (see definition 2.2). Figure 5: Graphical illustration of theorem 2.1. The theorem asserts that there is a multiway system forwhich the growth rate ( f, g ) is asymptotically less than T − ∗ ( x ). Now let us formalize the proof outlined above. For some function T : N → N , we define a “ T -halter” to be a Turing machine T such that T executes precisely T ( n ) operations when giventhe input n , taking into account the input and output constraints depicted in figure 6a. Theseconstraints will later allow us to “enchain” the multiway systems corresponding to T -halters.Of course, neither is there a T -halter for arbitrary T nor must there be a unique T -halter for agiven T . However, and this is the part we care about, there is a T -halter such that T ∈ Ω( f )for any computable function f because we can just take the Turing machine that computes f and add some logic to write n + 1 after the computation is finished.Having defined T -halters, the next step is to show how multiway systems can emulate these(and all other Turing machines). Since we are talking about deterministic Turing machines,no branching shall occur in the corresponding multiway system, i. e. the system should haveexactly one state in generation n which corresponds to the state of the Turing machine after7 − H right of the symbol the head is currently on, followedby the current state number. Hence, there are four additional symbols (two underscores, an H and a number) used in the multiway system but not written on the machine’s tape (see figure6b). (a) T is always started on an empty tape containingonly n in unary representation. After halting, T is required to have written n + 1 in unary onthe tape and placed its head onto or left of thefirst digit. The number n + 1 must be precededand followed by at least one empty symbol. (b) Evolution of T (compare fig. 7a) next to theevolution of M . Figure 6
Using this representation, plain read/write operations and state changes would be straightfor-ward to implement as replacement rules, as we could just introduce a rule “ xHn ” → “ yHm ”for every combination of currently read symbol x and head state n . However, since the headmust move left or right after each such operation, those rules are not suitable. What is neededinstead to encode the operation “when x is read in state n , write y , change state to m andmove the head right”, is a rule of the form “ x Hnx ” → “ y x Hm ” for every possible valueof x . Similarly, a left move of the head is encoded as “ x x Hn ” → “ x Hmy ”. If n is not ahalting state (in which case we would not need any rules), exactly one of these two rule patternswill be applicable for every possible value of x or, respectively, every state transition arrowstarting at H in the state transition diagram.For a Turing machine with N states working on an alphabet of S symbols, that already givesa worst-case (no halting states) of N · S rules in the corresponding multiway system. However, N · S (worst-case) more rules have to be added to handle the literal “edge cases” in which thehead is next to one of the underscores bounding the tape. The two rule patterns, of which, asbefore, exactly one will match for every state transition arrow, are “ xHn ” → “ 0 Hmy ” for aleft move and “ xHn ” → “ y Hm ” for a right move (in both cases x is read and y written).Now, the resulting rule set captures all of the Turing machine’s properties and is able to extendthe tape to any required length by itself. As an initial state of the multiway system to emulatethe machine, any string of characters from the machines alphabet together with an “ Hs ” where s is the starting state and the two bounding underscores can be used.To illustrate this construction, consider the Turing machine T shown in figure 7a. It is, insome sense, the easiest possible T -halter for it does nothing more than increasing the number Writing symbols next to each other in this contexts simply denotes their concatenation to a string.
8n the tape by one and placing its head back at the beginning. Figure 6b shows the successivestates of the machine (and tape) next to a multiway system M emulating T . The rule set forthis specific instance is {"00H1" -> "0H21", "10H1" -> "1H21", "1H10" -> "10H1", "1H11" -> "11H1","01H2" -> "0H21", "11H2" -> "1H21", "0H20" -> "00H3", "0H21" -> "01H3","_0H1" -> "_0H21", "1H1_" -> "10H1_", "_1H2" -> "_0H21", "0H2_" -> "00H3_"} (a) Rule plot and state transition diagram of T .The arrow labels indicate “read, write, move”. (b) States graph of the multiway system con-structed from T (see below). Figure 7: Two different representations of the same computational system: a Turing machine and amultiway system.
Now it is clear that a multiway system emulating some T -halter T has exactly one statefor T ( n ) generations when started with the initial condition “ 1 H n − ” (1 n − denotes n − Hf ” → “ X ” for every halting state f where X is one fixed symbol not contained in T ’salphabet. These additional rules will cause the multiway state in generation T ( n ) + 1 to looklike “ w X n w ” where w and w are arbitrary words that might be created as byproductsin the working of T . This is due to the T -halter constraints depicted in figure 6a or, rather, the T -halter constraints were chosen precisely to cause such a configuration of the tape.Adding the rules “ X → “1 X ”, “ X → “ Y Y ” → “ Y
1” and “0 Y → “0 1 H
1” willcause exactly one state where the X has “moved” one position to the right for n generations(first rule), then add an underscore behind the n ones (second rule), “move back to the left”using the Y for n + 1 generations (third rule) and finally add an underscore at the left side,replacing Y by the starting state symbol “ H
1” of T (see figure 7b). Now, the whole process canstart again because the new underscores ensure a “fresh” new tape for T which now contains,by the T -halter constraints, n + 1 as the next input for T to continue with while everythingoutside the bounding underscores will be ignored.The resulting multiway system of this continued re-evaluation of T will run indefinitely, sub-sequently running instances of T with larger and larger values of n . Despite that, it still hasonly one state in all generations. In order to make the number of states increase exactly when n increases, i. e. some instance of T finished working, we add the rules “0 Y → “ Z ” and“ Z ” → “ ZZ ”. This way, the multiway states graph branches every time the system starts a newinstance of T into a main branch where the evaluation of T continues and a diverging branchwhere the second rule just creates longer and longer strings of Z ’s forever, constantly adding one If the head is left of the first digit, “ w X n +1 w ” works analogously. n − p ( n ) = 2( n + 1) + 1 steps (the head moves over n + 1symbols , including the new 1) before the n + 1-th iteration of T can start after the n -thiteration is done, there will be n states for T ( n ) + p ( n ) steps in the multiway system constructedabove, before the number of states increases by one. Let us generally investigate the sequencesobtained this way: Definition 2.1.
Let f : N + → N + be a function. The sequence “ n occurs f ( n ) times” is definedby A f ( (cid:80) nk =1 f ( k )) = A f ( m + (cid:80) nk =1 f ( k )) = n for all n, m ∈ N with n ≥ ∧ m < f ( n + 1). Definition 2.2.
Let f : N + → N + be a function and S ⊆ N + an infinite set. The “linearinterpolation of f over S ”, denoted L S ( f ), is defined as the polygonal chain starting at (0 , n, f ( n )) , n ∈ S ordered by n . (a) Example for definition 2.1: the sequence “ n oc-curs f ( n ) = 2 n times” ( A n ). (b) Example for definition 2.2: the linear interpola-tion of A n over N + , now a continuous functionfrom R ≥ to R ≥ . Figure 8: Plots for illustrating definitions 2.1 and 2.2.
Since definition 2.1 requires f to be always greater than zero, every natural number can berepresented as some sum over consecutive values of f plus a remainder and, as figure 8a shows,this definition indeed matches the informal description of “ n occurs f ( n ) times”. Notice aswell that the linear interpolation, despite being defined as a curve in R , can be regarded asa continuous function from R ≥ to R ≥ because for all n ∈ S , the function to be interpolatedassigns precisely one y -value and since S is an infinite subset of N , the linear interpolationfunction is defined everywhere on R ≥ .Now, to express some sequence A f explicitly, define the set of increase-indices of A f as I ( A f ) := { n ∈ N + | A f ( n − < A f ( n ) } . It follows that L I ( A f ) ( A f ) will always be strictlymonotonically increasing and unbounded. Therefore, its inverse function L I ( A f ) ( A f ) − existsand we can formulate the following lemma: Lemma 2.2.
For some function f : N + → N + , we have L I ( A f ) ( A f )( x ) = L N + ( (cid:80) nk =1 f ( k )) − ( x ). Proof.
For readability, let T σ be the function (cid:80) nK =1 T ( k ), ϕ ( x ) := L I ( A f ) ( A f )( x ) and ψ ( x ) := L N + ( (cid:80) nk =1 f ( k ))( x ). By definition 2.2, the linear interpolation of a function equals that functionon the interpolation set, so ∀ n ∈ N + : ( ϕ ◦ ψ )( n ) = ϕ ( n (cid:88) k =1 f ( k )) = n (1) If the head starts at the left of the first digit instead, the formula is p ( n ) = 2( n + 2). x ∈ ( n, n + 1) , n ∈ N , the linear interpolation gives ψ ( x ) = ∆ y ∆ x ( x − n ) + ψ ( n ) = ψ ( n + 1) − ψ ( n ) n + 1 − n ( x − n ) + ψ ( n )= ( ψ ( n + 1) − ψ ( n ))( x − n ) + ψ ( n ) . (2)Letting y = ψ ( x ), we know that( ϕ ◦ ψ )( x ) = ϕ ( y ) = ϕ ( y ) − ϕ ( y ) y − y ( y − y ) + ϕ ( y ) (3)for some y , y ∈ I ( A f ) where y < y < y and y , y are the values in I ( A f ) closest to y . Since ψ is strictly monotonically increasing, y and y must be given by ψ ( n ) and ψ ( n + 1) respectively.Thus, c | r ( ϕ ◦ ψ )( x ) = ϕ ( ψ ( n + 1)) − ϕ ( ψ ( n )) ψ ( n + 1) − ψ ( n ) ( ψ ( x ) − ψ ( n )) + ϕ ( ψ ( n )) (4)= n + 1 − nψ ( n + 1) − ψ ( n ) ( ψ ( x ) − ψ ( n )) + n = ( ψ ( n + 1) − ψ ( n ))( x − n ) + ψ ( n ) − ψ ( n ) ψ ( n + 1) − ψ ( n ) + n by equation 2= x − n + n = x. So ϕ is a left-inverse of ψ on R ≥ . Analogously, it can be shown that ϕ is also a right-inverse of ψ , so indeed, L I ( A f ) ( A f )( x ) and L N + ( (cid:80) nk =1 f ( k ))( x ) are inverse functions.Putting it all together, we conclude from the previous Turing machine investigation that forevery halting function T ∈ Ω( n ), there is a multiway system which has the growth function g M ( n ) = A T + p ( n ) for some p ∈ Θ( n ). Additionally, as p “delays” the growth function evenmore , i. e. ∀ n ∈ N + : A T + p ( n ) ≤ A T ( n ), it follows that L I ( g M ) ( g M )( n ) ≤ L I ( A T ) ( A T )( n ) = L N + ( T σ ) − ( x ) ⇒ g M ( n ) ∈ O (cid:32) L N + ( n (cid:88) k =1 T ( k )) − (cid:33) (5)by lemma 2.2, which proves theorem 2.1. Computing linear interpolations and their inverse functions seems hard to do analytically becausein most cases, there are no elementary closed-form expressions describing them. Therefore, itmight seem difficult to actually apply theorem 2.1. However, since we are only interested ingrowth rates, we can use approximations to make calculations much more easy.
Lemma 2.3. If f : N + → N + is strictly increasing, g : R ≥ → R ≥ is continuous and bijectiveand ∀ n ∈ N + : g ( n ) = f ( n ), then L N + ( f ) − ∈ Θ( g − ). Proof.
Because g is continuous and takes the same values as f for natural arguments, we knowthat ∀ n ∈ N + : ∀ x ∈ ( n, n + 1) : f ( n ) ≤ g ( x ) ≤ f ( n + 1). Using the fact that the linearinterpolation equals the function for natural arguments, this equation becomes L N + ( f )( n ) ≤ g ( x ) ≤ L N + ( f )( n + 1). This implies L N + ( f ) − ( y ) ≤ g − ( y ) ≤ L N + ( f ) − ( y ) for values y ∈ ( y , y ) where y = f ( n ) and y = f ( n + 1). Expanding out gives L N + ( f ) − ( f ( n )) ≤ g − ( y ) ≤ L N + ( f ) − ( f ( n + 1)) ⇒ n ≤ g − ( y ) ≤ n + 1 (6)which means that the difference of g − ( y ) and L N + ( f ) − ( y ) is always bounded by 1. Therefore, g − ∈ Θ( L N + ( f ) − ). Since p ∈ Θ( n ), g M will even become strictly less than L N + ( T σ ) − very soon. In most practical cases, g M ismuch lower. T : N + → N + , n (cid:55)→ n + 3 n ) which could also describe a bijective function on R ≥ (like f ( x ) = 2 x + 3 x ),we can use the lemma above to simplify calculations: Since f is monotonically increasing andthe linear interpolation equals the summatory function for natural arguments, we have (cid:98) x (cid:99) (cid:88) k =0 f ( k ) ≤ (cid:90) x f ( t ) d t ≤ (cid:100) x (cid:101) (cid:88) k =1 f ( k ) ⇒ L N + ( T σ )( (cid:98) x (cid:99) ) ≤ F ( x ) ≤ L N + ( T σ )( (cid:100) x (cid:101) ) . (7)Therefore, lemma 2.3 tells us that we can approximate the inverse of the summatory functionused in theorem 2.1 just by computing the inverse of the integral of f . Especially in the case oflogarithms or exponential functions, solving integrals is much easier than computing sums, sothis lemma can be very useful.To demonstrate this and assist the proof of theorem 3.1, let us imagine we wanted to constructa multiway system of logarithmic growth rate. We can approach this problem by designing a T -halter for some T exp ∈ Θ(2 n ) and implementing the construction described in section 2.1 toget a multiway system with the inverse growth rate. As an example, take the Turing machine T exp shown in figure 9. It is started in state 1 which simply moves the head to the right end ofthe word on the tape and changes to state 2. In this state, the head moves left again, replacing2’s by 1’s until it encounters a 1, which it changes to a 2 and returns to state 1, repeating theprocess. It is easy to see that this is precisely the process of incrementing a binary numberwhere 1 corresponds to a zero and 2 to a one. The process is repeated until the head moves tothe left of the word, which, by then, consists only of 1’s since the previous string was the symbol2 repeated n times. When the head encounters the first blank symbol on the left, it writes onemore 1 to satisfy the T -halter constraint of incrementing the unary number, and then halts.The process is shown in figure 9 for n = 3. Figure 9: The rule plot, state transition diagram and one example evolution (starting from three ones)for the Turing machine T exp . b symbols to the left until it encounters the rightmost 1, andthen b symbols back after changing it. Since there are 2 n − b binary words of length n where therightmost 1 is at position b , the Turing machine takes n (cid:88) k =1 k · n − k = 2 n (cid:88) k =1 ( n − k )2 k = 2 (cid:32) n n (cid:88) k =1 k − n (cid:88) k =1 k k (cid:33) = 2 (cid:18) n (2 n +1 − − − ( n + 1)2 n +1 + n n +2 (2 − (cid:19) geometric series and [20]= 2( n n +1 − n − n n +1 + 2 n +1 − n n +2 )= 2( − n − n +1 ) = 2 n +2 − n − n and in state 1. Since the machine takes n + 2 stepsto move to the left again, write the new 1 and halt, as well as taking n steps to move the headto the right in the first place, the total number of states, including starting and halting state,simplifies to T exp ( n ) = 2 n + 3 + 2 n +2 − n − n +2 − . (9)In combination with lemma 2.3, another strategy for simplifying calculations is to give easilycomputable bounds for T . In this case, we use the fact that 2 n +1 < n +2 − < n +2 for all n ∈ N to obtain 2 x +1 < L N + ( T exp )( x ) < x +2 for all x ∈ R ≥ . Letting l ( x ) := L N + ( T exp )( x ) forreadability, this becomes2 x +1 < l ( x ) < x +2 ⇐⇒ (cid:90) x t +1 d t < (cid:90) x l ( t ) d t < (cid:90) x t +2 d t ⇐⇒ x − < (cid:90) x l ( t ) d t < x − . (10)Since the inverse of x (cid:55)→ a ln(2) (2 x −
1) is y (cid:55)→ log ( y ln(2) a + 1), and f ( x ) < g ( x ) ⇐⇒ f − ( x ) >g − ( x ), the equation is equivalent tolog (cid:18) y ln(2)2 + 1 (cid:19) > (cid:18)(cid:90) x l ( t ) d t (cid:19) − > log (cid:18) y ln(2)4 + 1 (cid:19) . (11)Now, notice thatlog ( y + c ) = log ( y ) + log (cid:18) cy (cid:19) x →∞ −→ log ( y ) (since 1 + cy x →∞ −→
1) (12)and log ( yc ) = log ( y ) + log ( c ) ∧ log ( y ) + log ( c ) x →∞ −→ log ( y ) . (13)From this, we know ( (cid:82) x l ( t ) d t ) − ∈ Θ(log ( x )) because the upper and lower bound asymptoti-cally equal log ( x ). By lemma 2.3 and equation 7, ( (cid:82) x l ( t ) d t ) − is also in Θ( L I ( A T exp ) ( A T exp ))and we can conclude that A T exp ∼ Θ log ( x ). Simulating the multiway system and measuring thegrowth function empirically supports this as figure 10 shows. In future examples, most steps ofthe argumentation presented here can be shortened. However, this method of estimation doesnot work in all cases because the inverse bounds might not accurate enough to be asymptoticallyequal . It is not true in general that f ∈ Θ( g ) implies f − ∈ Θ( g − ). As a counter-example, consider f ( x ) = ln( x ) and g ( x ) = 2 ln( x ). igure 10: The growth function of the multiway system emulating T exp is bounded by log ( x ) and log ( x ), demonstrating that it is in Θ(log ( x )). What we have seen in the above example is just a simple demonstration of the power of theorem2.1. Besides from helping us later to prove theorem 3.1, it tells us a lot about the abstractstructure of multiway growth function, their “growth spectrum”. By providing the followingtwo corollaries, theorem 2.1 gives us knowledge about what this spectrum of possible growthrates looks like, i. e. which kinds of growth rates are possible and which kinds are not. Inaddition to that, it establishes connections between multiway growth functions and other classesof functions, namely computable functions and primitive recursive functions.
Corollary 2.1.1.
For every computable function f : N → N , f ∈ ω (1) , there is a multiwaysystem with growth rate ( a, b ) such that a, b ∈ O ( f − ) for an asymptotic inverse f − .Proof. Since f is computable, there exists some Turing machine computing f ( n ) when given n . If we require the machine to read and write in- and output in unary coding, computing f ( n ) must take T ( n ) ≥ f ( n ) steps simply because writing the result takes that long. Now,let g : R ≥ → R ≥ be a bijective tight lower bound of T . As T ∈ ω (1), T is unbounded so g always exists. From g ( x ) ≤ T ( x ), it follows that g − ( x ) ≥ T − ( x ) and by theorem 2.1, there is amultiway system for which the growth function has tight bounds a, b ∈ O ( g − ) ⇒ a, b ∈ O ( f − )for some asymptotic inverse of f . Corollary 2.1.2.
For every computable function f : N → N , f ∈ ω (1) , there is a multiwaysystem with growth rate ( a, b ) such that a, b ∈ o ( L N + ( f )) .Proof. The function L N + ( f ) (for the upper bounding sequence f from definition 1.2) is alwaysgreater than or equal to f , asymptotically equal to f and computable (because equal to f )on the set of increase-indices I ( f ). Therefore, f − is a computable function on N + and so is g : x (cid:55)→ f − ( x ). Using corollary 2.1.1, this gives us a way to construct a multiway system witha growth rate in O ( g − ) = O ( (cid:112) L N + ( f )) which is definitely in o ( L N + ( f )).This quite remarkable fact also shows that for every multiway system growing faster than abounded function, a more slowly growing multiway system exists because the growth function ofevery multiway system is obviously computable. We might therefore say that multiway systemscan grow arbitrarily slowly, i. e. the set of regular multiway systems excluding constant andfinite systems is “open” in some sense. Remember however that they cannot grow arbitrarilyquickly as shown in lemma 2.1. An asymptotic inverse of a function f is some function f − satisfying ( f − ) − ∈ Θ( f ). Such functions arehelpful for describing f when it is not invertible in general. Computational Capabilities of Growth Functions
After marking out the boundaries of the space of possible growth rates, we shall investigateits underlying structure. First of all, we will see that it contains no “holes”, i. e. all of themultiway growth classes defined in section 1.3 (except C supexp which we have already shown tobe empty and just defined for completeness) are non-empty and, furthermore, contain infinitelymany systems. In addition, we will have some insights into which functions are “multiway-growth-computable” and “multiway-growth-approximable”. We say, a function f : N + → N + ismultiway-growth-computable if there is a multiway system M such that ∀ n ∈ N + : g M ( n ) = f ( n )and we call a function f : R ≥ → R ≥ multiway-growth-approximable if there is a multiwaysystem M such that f ∼ Θ L N ( g ).First, we will define two operations, “multiway addition” and “multiway multiplication” whichwill enable us to combine systems into more complex ones of which the growth function is com-putable immediately from the growth functions of the parts. These two fairly simple operationswill be sufficient for demonstrating that multiway growth functions are interesting from an alge-braic point of view as well as regarding questions of their computational capabilities (see section3.3). Still, some basic multiway systems have to be constructed without using these operationsas the building blocks of further systems. Combining the multiway operations and specificallyconstructed systems will then yield the following theorem and several other interesting results: Theorem 3.1.
The classes C pol , C int , C exp , C invpol , C invint , C invexp , C invsupexp , C fin and C bnd par-tition the set of regular multiway systems into infinite subsets. Let M = ( R , s , Σ ) , M = ( R , s , Σ ) and M = ( R , s , Σ ) be multiway systems. Addition-ally, let X be a unique (equal for all multiway systems) symbol not included in any multiwaysystems alphabet. Now, we define the “sum system” by M ⊕ M = ( R ∪ R ∪ { X → s i | s i ∈ S ( M ) ∪ S ( M ) } , “ X ” , Σ ∪ Σ ) where S ( M ) is the state-set of M in generation 2, i. e. allnodes with distance 1 to the initial state in the respective states graphs. The “product system”of M and M is now defined as M (cid:12) M = ( R ∪ R , s s , Σ ∪ Σ ) where s s denotes theconcatenation of s and s .To calculate the growth functions of systems obtained by these operations, we shall requirethe parts M and M to be “rule independent” meaning that their rules do not interfere witheach other. Formally, rule independence can be defined as the property that the states graphof M is isomorphic to the states graph of ( R ∪ R , s , Σ ∪ Σ ), that is M with all rules of M added, and vice versa . This property can always be achieved by requiring the underlyingalphabets to be disjoint as in this case it will be impossible that a given string matches rulesfrom R and R at the same time.If we recall the definition of growth function g M ( n ) as the number of nodes to which theshortest path from the initial state has length n , it is easy to see that the growth function of M ⊕ M is one in the first iteration as “ X ” is the only state. In further iterations, we canimagine a path of length n simply as a path of length 1 entering either the states graph of M or M , followed by a path of length n − X so for all other states, the usual rules will applyand the rules replacing X will have no effect. This concludes that the growth function of thesum system is precisely g M ⊕ M ( n ) = g M ( n ) + g M ( n ) with g M ⊕ M (1) := 1 . (14) This works because if adding all the rules of M to M does not change its behavior, then these rules will notinfluence M ’s states even if the states of M get appended to them. igure 11: States graphs of rule independent multiway systems and their sum system. The systemsused are ( { “ AB ” → “ BA ” , “ B ” → “ AAB ” } , “ AB ” , { A, B } ) and ( { “ CD ” → “ CDD ” , “ C ” → “ CD ” } , “ CDC ” , { C, D } ).Figure 12: States graphs of ( { “ A ” → “ AB ” , “ AB ” → “ BA ” } , “ A ” , { A, B } ), ( { “ C ” → “ D ” , “ D ” → “ E ” , “ D ” → “ F ” } , “ C ” , { C, D, E, F } ) and their product system. Instances of the first systemare highlighted in red and orange in the product system’s graph. For the product system, remember that in the states graph of some system M , two nodes u, v are connected by an edge if string u gets transformed into v by a rule from R . The analogousholds for M . Since the initial node of M (cid:12) M consists of a concatenation of s and s , anynode in the states graph of M (cid:12) M corresponds to some combination of a node of M and oneof M . Hence, the states graph of M (cid:12) M is the Cartesian product graph [19] of the statesgraphs of M and M . Note how this is only possible because M and M are rule independent16ince otherwise more edges could be added due to rule matches overlapping between the M -and M -parts of the string or rules of one system getting applied to states of the other one.To obtain the number of nodes reachable in this Cartesian product graph, we might withoutloss of generality first traverse a path of length k in the “pure M -part” (i. e. the second halfof the string is still s ) and then take n − k steps through the “pure M -part”. For the firstsubpath, we have g M ( k ) options by definition of the growth function. This gets multiplied bythe g M ( n − k ) choices for the second subpath. Since we can choose k freely, the resulting totalcount of nodes in the product graph is given by g M (cid:12) M ( n ) = n (cid:88) k =0 g M ( k ) · g M ( n − k ) = f ( n ) + g ( n ) + n − (cid:88) k =1 g M ( k ) · g M ( n − k ) (15)where we set g M (0) = g M (0) = 1 for convenience.For these two formulae, we have assumed the systems to be rule independent. For systemswhere this is not the case, they still give lower bounds of the combined systems growth ratesince, generally speaking, in any system, the number of edges and nodes of the states graph canonly remain constant or be increased when new rules are added. This might sound surprising asone could imagine “deletion rules” but rules that cause fewer rules being applied in the futuredo this only in newly added branches of the states graph (or not at all) not affecting the alreadyexistent graph. To make systems rule independent, we required the alphabets to be disjoint,however, this is not necessary as any multiway system can be emulated by a system over somebinary alphabet so we can always make the alphabets of M and M equal. Lemma 3.1.
For any multiway system M = ( R, s,
Σ), there is a multiway system M (cid:48) =( R (cid:48) , s (cid:48) , { a, b } ) where a and b are two distinct symbols, such that the states graphs of M and M (cid:48) are isomorphic. Proof.
To show this we will perform a “translation” from M to M (cid:48) , i. e. replace every symbolin Σ by a word over { a, b } using some bijection f : Σ → T ⊂ { a, b } ∗ . By altering not only s butalso all rules, any word w ∈ Σ ∗ matched by some rule in R will correspond to the translatedword in w (cid:48) ∈ T being matched by a rule in R (cid:48) . Additionally, one must ensure that no twowords in T can overlap since otherwise, the rules could match in more places than before. Sincethere exist non-overlapping codes of arbitrary length, we can use these as elements in T so therealways exists some f with the required properties. Thus, the actions of the rules on the stateswill be equal and isomorphic states graphs will be created.Now, let us consider the algebraic properties of our multiway operations. We declare twomultiway systems to be isomorphic (written M ∼ = M ), if and only if their states graphs areisomorphic. Isomorphic multiway systems always have equal growth functions. In the followinganalysis, we consider only the set of different equivalence classes of ∼ =, i. e. the set of all multiwaysystems up to isomorphism, and denote it by M .Let M , M , M ∈ M be multiway systems and, without loss of generality, rule-independent.It is easy to see that ⊕ is commutative and associative since set unions are. More interestingly,the system 0 M := ( {} , X, {} ) is a neutral element of ⊕ since M ⊕ M = ( R ∪ {} ∪ { X → s | s ∈ S ( M ) ∪ {}} , X, Σ ∪ {} )= ( R ∪ { X → s | s ∈ S ( M ) } , X, Σ ) (16)which is isomorphic to ( R , s , Σ ) because the X symbol is used only once, acting precisely as s would have and therefore keeping the states graph structure unchanged.The commutativity of (cid:12) is granted because we required M and M to be rule-independent,so the order in which their states are concatenated does not matter because no overlaps whererules could apply on the intersection of M - and M -states can be created. Similarly, (cid:12) is17ssociative, simply because string concatenation is. One can also prove both properties with thecommutativity and associativity of the Cartesian graph product. There also is a neutral elementof (cid:12) , namely 1 M := ( {} , “” , {} ) or actually any system with no rules since its initial state willjust be appended onto every state of the system one is multiplying with and the states graphwill not change.Also notice that (cid:12) distributes from the left over ⊕ : M (cid:12) ( M ⊕ M )= ( R , s , Σ ) (cid:12) ( R ∪ R ∪ { X → S ( M ) ∪ S ( M ) } , X, Σ ∪ Σ )= ( R ∪ R ∪ R ∪ { X → S ( M ) ∪ S ( M ) } , s X, Σ ∪ Σ ∪ Σ )= (( R ∪ R ∪ { X → S ( M ) } ) ∪ ( R ∪ R ∪ { X → S ( M ) } ) , s X, (Σ ∪ Σ ) ∪ (Σ ∪ Σ ))= ( R ∪ R ∪ { X → S ( M ) } , s , Σ ∪ Σ ) (cid:12) ( R ∪ R ∪ { X → S ( M ) } , X, Σ ∪ Σ )= (( R , s , Σ ) ⊕ ( R , s , Σ )) (cid:12) (( R , s , Σ ) ⊕ ( R , s , Σ ))= ( M ⊕ M ) (cid:12) ( M ⊕ M ) . (17)Thus, we get right distributivity from commutativity and conclude that (cid:12) distributes over ⊕ .Therefore, we can conclude that ( M , ⊕ , (cid:12) ) is a semiring, however with a weakened annihilationproperty which does not hold in general. As a consequence, their growth functions also form asemiring with weakened annihilation under the operations ( g + g )( n ) := g ( n ) + g ( n ) (with( g + g )(1) defined to be 1) and ( g ∗ g )( n ) = (cid:80) nk =0 g ( k ) · g ( n − k ) (with g (0) = g (0) := 1).This demonstrates the potential of multiway systems to generate quite diverse and intricategrowth functions as the two operations can be used to combine systems in various very interestingways. The next section elaborates on this. Let us now construct multiway systems in the various growth classes to prove the above theorem3.1. First, take the product system of a finite “chain”-system M N having one new state for N generations until terminating and a constant system M = ( { “ A ” → “ AA ” } , “ A ” , { “ A ” } ). Afterthe first N − M N (cid:12) M will have a constant growth function of value N as the first N terms in (cid:80) nk =0 g M ( k ) g M ( n − k ) are one and all others zero because M is finite. Whilethis system produces (asymptotically) constant growth functions, it is not suited for multiplyingarbitrary growth functions by constants. To achieve the latter, adding some system to itselfseveral times resolves the issue.The next system to consider, M N , is again given by the rule set { “ A ” → “ AB ” } and startedon a string of N copies of “ A ” denoted “ A N ”. For calculating its growth function, we canrepresent it differently as the product system of N instances of itself started on a single “ A ”and thus having the growth rate g M ( n ) = 1. From this point of view, we can write the growthfunction of M started on “ A N ” recursively as g NM ( n ) = n (cid:88) k =0 g N − M ( k ) · g M ( n − k ) = n (cid:88) k =0 g N − M ( k ) (18)One might recognize this as the sequence of ( N − n -th element is given by (cid:0) N − n − N − (cid:1) . Hence, g NM is clearly apolynomial of degree N − x N − .Now, consider M N = ( { “ Q ” → “ Qx i ” | i = 1 , . . . , N } , “ Q ”, { Q, x , . . . , x N } ) for distinctsymbols x i . In the n -th step of evolution, it has basically generated all words of length n overthe alphabet of all x i . Every node “ Qw ” (where w ∈ { x , . . . , x N } ∗ ) in the states graph has N outgoing edges to the nodes “ Qx i w ” for 1 ≤ i ≤ N . Thus, its growth function is precisely g NM ( n ) = N n allowing the possibility of multiway systems growing like all exponential functions.18 more sophisticated example is the system M N = ( (cid:83) i =1 ,...,N { “ T L ” → “ T x i R ” , “ RT ” → “ Lx i T ” , “ Rx i ” → “ x i R ” , “ x i L ” → “ Lx i ” } , “ T LT ” , { “ L ” , “ R ” , “ T ” , x , . . . , x N } ). Similarly tothe previous system, L and R work as generators for words over the alphabet { x , . . . , x N } however only on the left and right ends of the word respectively. In every word ever producedby the system, there are exactly two “ T ” symbols, one at the beginning and one at the end.After generating some new symbol between itself and the “ T ”, the generator “ L ” or “ R ” movesone step left or right respectively thereby not generating any new symbols as it is not next toa “ T ”. Since a new symbol is created every time a “ L ” or “ R ” reaches the respective “ T ”,the length of the word is increased every n steps where n the previous word length. Thus,their word-length is the sequence “ n occurs n times” denoted A n and asymptotically equal to( (cid:80) nk =0 k ) − = ( n ( n +1)2 ) − = √ n − ∈ Θ( √ n ) by lemma 2.2. But the system’s growth functionis given by the number of possible words, i. e. N n and hence asymptotically equal to N √ n .The growth function of the previous system is noteworthy because it grows “intermediately”i. e. faster than every polynomial function and slower than all exponential functions. Formally,one checks this by noticing that lim x →∞ ln( N √ x ) x = 0 (if the logarithm grows slower than x ,the function is subexponential) and lim x →∞ ln( N √ x )ln( x ) = ∞ (if the logarithm grows faster thanln( x ) times a constant, the function grows faster than x n for all n ). In the study of groupsand semigroups, which are related to multiway systems [23], it has long been an open problem,finally solved by Grigorchuk [9] to find groups of intermediate growth . For multiway systems,this turns out to be remarkably easy, supporting the claim that multiway growth functions arecomputationally diverse and powerful.Now that we have shown the existence of infinitely many systems in C fin , C bnd , C pol , C exp and C int , only the classes of inverse functions remain. As mentioned above, theorem 2.1 appears tobe very useful for this. In section 2.2, we have already shown the existence of a system with agrowth function asymptotically equal to log ( x ). The Turing machine from that example canbe generalized to perform counting in any number system base a yielding a halting functionasymptotically equal to n (cid:55)→ a n so that the same construction can be used to obtain multiwaysystems growing like log a x . It is not essential to go through the details here because all log a x are asymptotically equal (since they only differ by constants by the base-change law). It isalso clear that infinitely many systems in the class C invexp can be created because one could forexample use multiway addition to add systems with constant growth functions to logarithmicsystems. Notice how this shows generally that every class containing at least one system containsinfinitely many systems.By the same style of argument, we conclude that there are infinitely many systems in C invpol .Consider a weaker version M of the system of intermediate growth rate M for N = 1. Thissystem has one state forever but the length of its strings is still the sequence “ n occurs n times”.Now simply apply the Turing machine construction from section 2.1 to this system by addingthe rules “ LT ” → “ Z ”, “ T R ” → “ Z ” and “ Z ” → “ ZZ ” respectively to generate differentbranches of the states graph. As described in section 2.1, this yields the desired growth functionof g M ( n ) = A n ( n ) ∈ Θ( √ n ) ⇒ M ∈ C invpol .Using corollary 2.1.2, it is also obvious that there are infinitely many multiway systems in C invsupexp as one might, for example, just construct a multiway system growing slower thanthe inverse Ackermann function. It remains to show that the class of inverses of intermediatelygrowing functions C invint is non-empty. To show this, first note that there is a Turing machine T which computes (cid:98)√ n (cid:99) when given n in unary in polynomial time. If one feeds the result of thiscomputation in the T exp machine from section 2.2, one can construct a machine that halts after T ( n ) = p ( n )+2 (cid:98) √ n (cid:99) steps for some polynomial function p ( n ). Similar to the argumentation fromthe proof of lemma 2.3, we see that T σ is asymptotically equal to the integral of p ( x ) + 2 √ x givenby F ( x ) = q ( x ) + c √ x √ x where q is some polynomial (because (cid:82) √ x d x = √ x +1 ( √ x ln(2) − (2) ). Here, “growth” refers to the notion of “group growth rate” from group theory. igure 13: States graphs of exemplary systems M , M , M , M and M with growth functions asymp-totically equal to n (cid:55)→ , n (cid:55)→ , n (cid:55)→ n , n (cid:55)→ n and n (cid:55)→ √ n respectively. Since q is a polynomial, this function grows still intermediately which one can easily verifyby calculating lim x →∞ ln( F ( x ))ln( x ) and lim x →∞ ln( F ( x )) x . By theorem 2.1 and lemma 2.3, there isa multiway system M which has a growth function asymptotically equal to F − ( x ). Since F grows intermediately, this system is in C invint . Together with the previous paragraphs, thisprovides the proof for theorem 3.1. As we have shown the existence of systems with growth function g M in Θ(1), Θ( x n ), Θ( a x ),Θ( a √ x ), Θ( √ x ) and Θ(ln( x )), it is possible to find a multiway growth function asymptoticallyequal to any combination of these functions using point-wise addition and discrete convolution.This works because asymptotic equivalence is preserved under addition, summation and mul-tiplication ([10] section 2.2) so one can take the appropriate multiway systems for the atomicparts of the functions expression and combine them using the multiway sum and product. Allof the basic functions are monotonously increasing and addition as well as discrete convolutionpreserve this property (because all functions have values in R ≥ ). This proves the following20orollary to theorem 3.1: Corollary 3.1.1.
If some function f : R ≥ → R ≥ there is a function g ∈ Θ( f ) expressibleas a finite combination of the functions x (cid:55)→ c, x (cid:55)→ x n , x (cid:55)→ a x , x (cid:55)→ a √ x , x (cid:55)→ √ x, x (cid:55)→ ln( x ) ( x, a, c ∈ N + ) using the operations point-wise addition and discrete convolution, then f ismultiway-growth-approximable. The corollary provides further insights into the multiway-growth-approximability and multiway-growth-computability of functions. Let M C and M A be the sets of multiway-growth-computableand multiway-growth-approximable functions respectively. Note that M C ⊂ M A and M C iscountably infinite, since the set of multiway systems is countably infinite whereas M A is un-countably infinite because it contains, for example, all constant functions from R ≥ to R ≥ .Using the above corollary, we notice that a variety of classes of functions are multiway-growth-approximable:1. For any function f ∈ M C , the function λ · f where λ ∈ N is a constant is also in M C .2. M A contains all polynomials with natural coefficients because they can be build by addingpowers x n multiplied by constants.3. For all polynomials p ( x ) in M A , M A contains functions asymptotically equal to x (cid:55)→ ln( x ) · p ( x ).Consequence 3 takes a little longer to prove but will be worth explaining in detail because similarmethods may be used to generalise it, for example showing that M A contains polylogarithmicfunctions. Proof.
Consider the product system of a polynomial and a logarithmic system. By equation 15and the fact that asymptotic equivalence is preserved under addition and multiplication, thesystem’s growth function is asymptotically equal to n a + ln( n ) + n (cid:88) k =1 k a ln( n − k ) = O ( n a ) + n − (cid:88) k =1 h ( k ) . (19)Consider some fixed input n for g . Let B k be the k -th Bernoulli number, B m ( x ) the periodiccontinuation of the m -th Bernoulli polynomial and choose m = n +1. Using the Euler-MaclaurinFormula ([17] pp. 501 ff.), we obtain n − (cid:88) k =1 h ( k ) = (cid:90) n − h ( x ) d x + h ( n ) + h (1)2 + S m + R m . (20)First of all, h ( n )+ h (1)2 simply evaluates to ln( n )2 ∈ O ( ∞ ). Next, it is easy to show inductively thatthe k -th derivative of h is of the form h ( k ) ( x ) = x a − k ( c ln( n − x ) + p ( x )) for a real constant c and a rational function p ( x ) ∈ O (1) as long as k ≥ a . The a + 1-th derivative is some rationalfunction in O ( x ). Thus, the remainder sum and integral satisfy S m = n +1 (cid:88) k =2 ( − k B k k ! ( h ( k ) ( n ) − h ( k ) (1)) ∼ Θ n +1 (cid:88) k =2 x a − k ln( n − x ) ∈ O ( x a − ln( n − x )) (21) R m = ( − n +2 ( n + 1)! (cid:90) n − f ( n +1) ( x ) B m +1 ( x ) d x ∼ Θ (cid:90) n − O ( 1 x ) d x ∈ O (ln( x )) . (22) This follows from the fact that every multiway system can be reduced to use only the alphabet { A, B } (lemma3.1) and then be expressed using the symbols “ A ” , “ B ” , “ → ” , “ { ” , “ } ” , “(” , “)” and “ , ” by writing down itssignature. However, the set of words over this finite 8-symbol alphabet is countably infinite. (cid:80) n − k =1 h ( k ) ∼ Θ (cid:82) n − h ( x ) d x . Using the fact that dd x ( x + ( n − x ) ln( n − x )) = − ln( n − x )and applying integration by parts, we obtain I a = (cid:90) x a ln( n − x ) d x = − x a ( x + ( n − x ) ln( n − x )) + (cid:90) ( x + ( n − x ) ln( n − x )) ax a − d x = T + JJ = a (cid:90) x a d x + na (cid:90) x a − ln( n − x ) d x − a (cid:90) x a ln( n − x ) d x = aa + 1 x a +1 + naI a − − aI a ⇒ ( a + 1) I a = − x a ( x + ( n − x ) ln( n − x )) + aa + 1 x a +1 + naI a − + C ⇒ (cid:90) n − x a ln( n − x ) d x = I a ( n − − I a (1) (23)= 1 a + 1 ( − ( n − a ( n − ·
0) + aa + 1 ( n − a +1 + naI a − ( n − a (1 + ( n −
1) ln( n − − aa + 1 1 a +1 − naI a − (1))= 1 a + 1 (cid:18) − ( n − a +1 a + 1 + 1 a + 1 + ( n −
1) ln( n −
1) + na ( I a − ( n − − I a − (1)) (cid:19) = 1 a + 1 (cid:18) − ( n − a +1 a + 1 + ( n −
1) ln( n −
1) + na (cid:90) n − x a − ln( n − x ) d x (cid:19) . (24)For a = 1 we know (cid:90) n − x a − ln( n − x ) d x = [ − ( x + ( n − x ) ln( n − x ))] n − = − ( n −
1) + (1 + ( n −
1) ln( n − − n + 2 + n ln( n − − ln( n − ∈ Θ( n ln( n − ⇒ (cid:90) n − x a ln( n − x ) d x = 12 ( 1 − ( n − n −
1) ln( n −
1) + n Θ( n ln( n − ⇒ (cid:90) n − x ln( n − x ) d x ∈ Θ( n ln( n − . (25)Now, by assuming (cid:82) n − x a ln( n − x ) d x ∈ Θ( n a +1 ln( n − (cid:90) n − x a ln( n − x ) d x = 1 a + 1 (cid:18) − ( n − a +1 a + 1 + ( n −
1) ln( n −
1) + na (cid:90) n − x a − ln( n − x ) d x (cid:19) = Θ( − ( n − a +1 ) + Θ(( n −
1) ln( n − an Θ( n a ln( n − ⇒ (cid:90) n − x a ln( n − x ) d x ∈ Θ( n a +1 ln( n − (cid:82) n − x a ln( n − x ) d x ∈ Θ( n a +1 ln( n − O ( n a ) + n − (cid:88) k =1 h ( k ) ∼ Θ n a +1 ln( n − ∼ Θ n a +1 ln( n ) (27)proving in fact, that for every x a -system, there exists a x a ln( x )-system.From these three properties, we might already conclude that a significant number of func-tions usually investigated in mathematical analysis can be approximated by multiway growthfunctions. This demonstrates the computational diversity of multiway growth functions whichis neither a trivial nor an expected property.We know that multiway systems themselves are capable of universal computation as they canemulate Turing machines but it is unknown so far which computations their growth functions22re able to perform. Thus, it may be considered remarkable that many common mathematicalfunctions are expressible (i. e. approximable) as multiway growth functions. Conversely, thiscould later allow to make statements about a multiway system’s complexity or structure byconsidering only its growth function. Maybe, multiway systems can even be used to makegeneral statements about the mathematical functions themselves since they give a new way oflooking at them.Notice however that multiway growth functions are strictly less powerful than computablefunctions in general due to the following lemma: Lemma 3.2.
Every multiway growth function is primitively recursive.
Proof.
Given some multiway system, one can compute the growth function g ( n ) in the followingway: One uses two lists S n and T n ( S = {} , T = { s } ) to store all states the system has haduntil generation n and all states of generation n . In every iteration, the length of T n is the valueof g ( n ). In the n + 1-th step, the algorithm iterates through all rules and searches through thecharacters of all strings in T n to check if any rule applies. If this happens, the string with somepart replaced will be added to T n +1 if it is not in S n or T n +1 already (only new states get added).After all possible such operations are done, T n +1 contains all new states of the system and weset S n +1 to be S n ∪ T n +1 . When repeated, this process simulates the system’s evolution andthus yields the correct g ( n ). All loops required can be implemented using DO-loops. If stringsare treated as lists of characters (numbers), the string replacement and substring matchingoperations can be implemented using only list insertions, deletions and searches through thelist. No further data structures and no comparisons are needed. Hence, the entire program isprimitively recursive by [14].As primitive recursive functions still contain some superexponential functions, multiway growthfunctions are also strictly weaker than those. Still, multiway growth functions can approximatea lot of elementary functions so they might even be stronger than elementary arithmetic (EA)while probably weaker than EA+ (EA and the axiom that the superexponential function istotal). It remains an open question to find out how M A or M C can be characterized elegantly.Besides their use for investigating multiway-growth-computability, the tools obtained in thispaper allow us to elegantly prove some statements about undecidability of multiway-growth-related questions. For example, deciding if a given multiway system is finite is undecidable asa system could simulate some arbitrary Turing machine and have zero new states when themachine halts, reducing the question to the halting problem. Additionally, even for an infinitemultiway system, deciding whether its growth function is equal to some conjectured function isundecidable in general since the system could be the sum of some usual system and a Turingmachine emulator which becomes, for example, exponentially growing after the machine halts buthas only one state before that. This observation makes it especially important in the context ofthe Wolfram Physics Project to not only use empirical (computed) observations about a system’sgrowth function, rate or class but also take into account the system’s rule when conjecturingabout it.This “trick” of integrating a system which grows very differently once some Turing machinehalts into some larger system was also the strategy used to construct the strongly oscillatingsystem in figure 4. Specifically, we first construct a system similar to M N from section 3.2,defined as M N,M = ( (cid:83) i =1 ,...,N { “ RA ” → “ x i R ” } , “ A M ” , { “ R ” , A, x , . . . , x N } . When started ona string of M A s, R moves to the right while replacing the A it just moved over by any of the N x i . Hence, the states of the system after N steps are precisely the words of length N over { x , . . . , x N } since R moved N steps to the right. When R reaches the right end, the systemterminates. Thus, this system has M N new states after M steps and 0 after that.The oscillating system from figure 4 is now the sum of a linearly growing system and a versionthe logarithm system from section 2.2. However, the customized logarithm system does notincrease its number of states after T exp halts but just triggers an instance of M ,N on the string23f ones T exp has written. Since after roughly 2 n steps, the n -th version of T exp halts, n ones arewritten on the tape so about n steps later, the system has 4 n states for one generation and then“collapses” into one state again. The smallest monotonically increasing upper bound for thisgrowth function is one that stays 4 n = (2 n ) for roughly 2 n steps and then increases to 4 n +1 .Denote this sequence a ( n ). Now log (cid:112) a ( n ) is approximately the sequence “ n occurs 2 n times”which is in Θ(log ( n )). Thus, log (cid:112) a ( n ) ∈ Θ(log ( n )) ⇐⇒ (cid:112) a ( n ) ∈ Θ( n ) ⇐⇒ a ( n ) ∈ Θ( n ). Since the whole system consists of this and an added linear system, the total growthrate has lower and upper tight bounds of Θ( x ) and Θ( x ) respectively. If one generalizes themethods used in this paper, they might be used perform a kind of “multiway system engineering”i. e. they could help to construct systems for specific purposes. Our main results may be summarised as follows:1. This paper introduced the formalisms of multiway growth functions, rates and classeswhich have large potential for mathematically investigating multiway systems.2. In theorem 2.1, we showed that multiway systems can grow slower than all computablefunctions while never exceeding exponential functions. Not only is this asymmetry verysuspicious of a more general underlying principle but the theorem also demonstrates thatmultiway growth functions cannot be trivial and must have some computational complexityassociated with them.3. This gets supported by the fact that multiway systems are capable of simulating (approx-imating) quite an extend of known functions while being contained in the set of primitiverecursive functions. It remains entirely unclear, which status the sets M A and M C haveamong other well-known sets of functions but our theorem 3.1 starts a characterisation bysubdividing them into nontrivial classes.4. Additionally, we have exemplarily demonstrated some basic systems which can be com-bined to yield systems giving a wanted growth function. This “multiway engineering” couldbe generalised and turn out to be useful for getting intuition about multiway systems aswell as potentially constructing (counter-)examples to empirically grounded conjecturesexisting in the Wolfram Physics Project.5. Another very interesting foundation for further research are the arithmetic-like operationson multiway systems which we have shown to equip the set of all multiway systems withan almost-semiring structure.These results could be applied in various ways:1. The most obvious next step is generalising our theorems to hypergraphs to make themmeaningful for the actual Wolfram Model. However, this should not be difficult to do.Generalising the algebraic structure of the set of multiway systems with the operations wedefined seems much more interesting, especially for hypergraph-based multiway systemssince it becomes relevant in the theoretical physics context of multiway systems as thereare various recent findings about a connection of Wolfram Models, category theory andquantum mechanics [7]. Additionally, more general forms of our results could be obtainedin context of the connection between multiway systems and the foundations of homotopytype theory [2].2. More specifically, the changes in structure of branchial space (see glossary of [7]) over timeand thus the growth functions of multiway systems are related to the states of quantum24ystems and measurements of these [5]. Our upper bound on multiway system growth ratemay be used to give upper bounds on entanglement speed and maximum possible infor-mation entropy in Wolfram Models. Similarly, the fact that slowness of multiway growthrates is “unbounded” could be used to investigate very slowly developing quantum sys-tems which might be especially stable and hence useful for quantum computation, but thatis mere speculation. More straightforward is the application of multiway growth classesto estimating the complexity of quantum computational algorithms or make predictionsabout quantum supremacy using the Wolfram model (c. f. [16] and [7]).3. Another potential physical application is early-universe cosmology. Since there seems tobe an empirical connection [15], formalising which would also be an important project,between the growth rates of physical and branchial space i. e. in our context string-length (or hypergraph size) and multiway growth functions, our boundaries on growthfunctions and especially the “unboundedness” of its slowness may be related to the physicalexpansion of the early universe in the view of the Wolfram Model’s formalism.4. Building on our classification scheme, for example by allowing combined classes like “poly-nomial times inverse intermediate”-functions, trying to quantify and find regularity in theoscillations of a system’s growth function (consider the system in figure 3 as an example)or analysing strongly oscillating systems like the one depicted in figure 4.5. Related to the regularity of multiway systems, one can ask whether determining whichgrowth rate (or growth class) a given multiway system has from its rules is possible ingeneral. Maybe, methods from automated theorem proving could be used for this. Anotherquestion is whether there are multiway systems which show no regularity in their growthfunctions. The latter would be important for “A New Kind of Science”-related researchsince a considerable part of Wolfram’s work concerns the complexity and irregularity ofsuch computational systems [22].The preceding points are just a few possibilities showing how much potential the investigationof multiway systems and their growth functions, rates and classes has. This paper marks onlythe beginning of many further research projects. However, while we lay a very basic foundation,we succeed in doing so as our results are formally proven and computationally applicable. First and foremost, I would like to thank Stephen Wolfram for suggesting this project and givingimportant advice concerning the general directions of research and methodology. Speaking ofmethodology, I have to mention my mentor Xerxes D. Arsiwalla for whose guidance and feedbackI am very grateful. Many thanks also to Jonathan Gorard for repeatedly proof-reading this paperand assisting the proof that multiway systems form a semiring with weakened annihilationproperty (section 3.1). Additionally, I highly appreciated the encouragement and support ofPeter Barendse who helped to kickstart this research project at the Wolfram Summer Camp2020 and Paul Siewert whose critique and explanations were very useful for formalising theproofs presented in this paper. 25 eferences [1] D. Aggarwal. [Wolfram Winter School 2021] Multiway Systems as Cayley Graphs . url : https://community.wolfram.com/groups/-/m/t/2162902 .[2] X. D. Arsiwalla. [Wolfram Summer School 2020] Homotopic Foundations of Wolfram Mod-els . url : https://community.wolfram.com/groups/-/m/t/2032113 .[3] F. Baader and T. Nipkow. Term Rewriting and All That . USA: Cambridge UniversityPress, 1998. isbn : 0521455200.[4] N. 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Finally We May Have a Path to the Fundamental Theory of Physics... and It’sBeautiful . 2020. url : https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/