Reversible cellular automaton

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A reversible cellular automaton is a cellular automaton in which every configuration has a unique predecessor. That is, it is a regular grid of cells, each containing a state drawn from a finite set of states, with a rule for updating all cells simultaneously based on the states of their neighbors, such that the previous state of any cell before an update can be determined uniquely from the updated states of all the cells. The time-reversed dynamics of a reversible cellular automaton can always be described by another cellular automaton rule, possibly on a much larger neighborhood.

Several methods are known for defining cellular automata rules that are reversible; these include the block cellular automaton method, in which each update partitions the cells into blocks and applies an invertible function separately to each block, and the second-order cellular automaton method, in which the update rule combines states from two previous steps of the automaton. However, for cellular automata that are not defined by these methods, on arrays of two or more dimensions, testing reversibility is undecidable.[1]

Reversible cellular automata form a natural model of reversible computing, a technology that could lead to ultra-low-power computing devices. Quantum cellular automata, one way of performing computations using the principles of quantum mechanics, are often required to be reversible. Additionally, many problems in physical modeling, such as the motion of particles in an ideal gas or the Ising model of alignment of magnetic charges, are naturally reversible and can be simulated by reversible cellular automata.

Properties related to reversibility may also be used to study cellular automata that are not reversible on their entire configuration space, but that have a subset of the configuration space as an attractor that all initially random configurations converge towards. As Stephen Wolfram writes, "once on an attractor, any system—even if it does not have reversible underlying rules—must in some sense show approximate reversibility."[2]

Examples[edit]

One-dimensional automata[edit]

The simplest possible cellular automata have a one-dimensional array of cells, each of which can hold a binary value (either 0 or 1). If the update rule for a cell causes it to always remain in the same state, then the automaton is reversible: the previous state of a cell can be recovered from its current state. Similarly, if the update rule causes every cell to change its state from 0 to 1 and vice versa, or if it causes a cell to copy the state from a fixed neighboring cell, or if it causes it to copy a state and then reverse its value, it is necessarily reversible. Toffoli & Margolus (1990) call these types of reversible cellular automata, in which each neighborhood has only one cell, "trivial". Despite its simplicity, the update rule that causes each cell to copy the state of a neighboring cell is important in the theory of symbolic dynamics, where it is known as the shift map.[3]

A little less trivially, suppose that each state of a cellular automaton is an ordered pair (r,l) consisting of a right part r and a left part l, each drawn from a finite set of possible values. Define a transition function that sets the left part of a cell to be the left part of its left neighbor and the right part of a cell to be the right part of its right neighbor. That is, if the left neighbor's state is (a,b) and the right neighbor's state is (c,d), the new state of a cell is the result of combining these states using a pairwise operation × defined by the equation (a,b) × (c,d) = (a,d). Then this automaton is reversible: the values on the left side of each pair move rightwards and the values on the right side migrate leftwards, so the prior state of each cell can be recovered by looking for these values in neighboring cells. The operation × used to combine pairs of states in this automaton forms an algebraic structure known as a rectangular band.[4]

Multiplication of decimal numbers by two or by five can be performed by a one-dimensional reversible cellular automaton with ten states per cell (the ten decimal digits). More generally, multiplication or division of doubly infinite digit sequences in any radix b, by a multiplier or divisor x all of whose prime factors are also prime factors of b, is an operation that forms a cellular automaton because it depends only on a bounded number of nearby digits, and is reversible because of the existence of multiplicative inverses.[5] Multiplication by other values (for instance, multiplication of decimal numbers by three) remains reversible, but does not define a cellular automaton, because there is no fixed bound on the number of digits in the initial value that are needed to determine a single digit in the result.

It is tempting to think of Rule 90 and other cellular automata based on the exclusive or function as being reversible, as the use of the exclusive or makes the transition rule locally invertible. However, it is not a reversible cellular automaton rule, because in Rule 90 every configuration has exactly four predecessors, whereas reversible rules are required to have exactly one predecessor per configuration.[6]

Critters[edit]

Gliders escape from a central random seed region in the critters block cellular automaton rule.

Conway's Game of Life, one of the most famous cellular automaton rules, is not reversible: for instance, it has many patterns that die out completely, so the configuration in which all cells are dead has many predecessors, and it also has Garden of Eden patterns with no predecessors. However, another rule called "Critters" by its inventors, Tommaso Toffoli and Norman Margolus, is reversible and has similar dynamic behavior to life.[7]

The Critters rule is a block cellular automaton in which, at each step, the cells of the automaton are partitioned into 2×2 blocks and each block is updated independently of the other blocks; at each step, the four cells within each block of the partition come from four different blocks of the previous step's partition. The transition function reverses the state of every cell in a block, except for a block with exactly two live cells which remains unchanged. Additionally, blocks with three live cells undergo a 180-degree rotation as well as the state reversal. Because all of these operations are individually reversible, the automaton defined by these rules is a reversible cellular automaton.[7]

In the Critters rule, initial states in which all cells take randomly chosen states remain unstructured throughout their evolution. However, when started with a smaller field of random cells centered within a larger region of dead cells, many small patterns similar to life's glider escape from the central random area and interact with each other. The Critters rule can also support more complex spaceships of varying speeds as well as oscillators with infinitely many different periods.[7]

Constructions[edit]

Several general methods are known for constructing cellular automaton rules that are automatically reversible.

Block cellular automata[edit]

The Margolus neighborhood for block cellular automata. The partition of the cells alternates between the set of 2 × 2 blocks indicated by the solid blue lines, and the set of blocks indicated by the dashed red lines.

A block cellular automaton is an automaton at which, in each time step, the cells of the automaton are partitioned into congruent subsets, and the same transformation is applied independently to each subset.[8] However, the partition into subsets may differ on different time steps. In a frequently used form of this design, called the Margolus neighborhood, the cells of the automaton form a square grid and are partitioned into larger 2 × 2 squares at each step, in such a way that the four cells in each 2 × 2 belong to four different 2 × 2 squares of the previous partition.[9] The Critters rule discussed above is an example of this type of automaton.

Designing reversible rules for block cellular automata, and determining whether a given rule is reversible, is easy: it is necessary and sufficient that the transformation applied to the individual blocks at each step of the automaton is itself reversible. The reverse of a block rule uses the same block structure, with a reversed sequence of partitions into blocks, and with the transition function for each block being the inverse function of the original rule.[8]

Simulation of irreversible automata[edit]

Toffoli (1977) showed how to embed any non-reversible d-dimensional cellular automaton rule into a reversible (d + 1)-dimensional rule. Each d-dimensional slice of the new reversible rule simulates a single time step of the original rule. In this way, Toffoli showed that many features of non-reversible cellular automata, such as the ability to simulate arbitrary Turing machines, could also be extended to reversible cellular automata.

As Toffoli conjectured and Hertling (1998) proved, the increase in dimension incurred by Toffoli's method is a necessary payment for its generality: under mild assumptions (such as the translation-invariance of the embedding), any embedding of a cellular automaton that has a Garden of Eden into a reversible cellular automaton must increase the dimension.

Morita (1995) describes another type of simulation that does not change the dimension. Morita's method can simulate the finite configurations of any non-reversible automaton in which there is a "quiescent" or "dead" state, such that if a cell and all its neighbors are quiescent then the cell remains quiescent in the next step. The simulation uses a reversible block cellular automaton of the same dimension as the original non-reversible automaton. The information that would be destroyed by the non-reversible steps of the simulated automaton is instead sent away from the configuration into the infinite quiescent region of the simulating automaton. This simulation does not update all cells of the simulated automaton simultaneously; rather, the time to simulate a single step is proportional to the size of the configuration being simulated. Nevertheless, the simulation accurately preserves the behavior of the simulated automaton, as if all of its cells were being updated simultaneously. Using this method it is possible to show that even one-dimensional reversible cellular automata are capable of universal computation.[10]

The past cells affecting the state of a cell at time t in a second-order cellular automaton
The Rule 18 one-dimensional cellular automaton (left) and the second-order cellular automaton derived from it (right). Each row of the image shows a configuration of the automaton, with time running downwards.

Second-order cellular automata[edit]

The second-order cellular automaton technique is a method of transforming any cellular automaton into a reversible cellular automaton, invented by Edward Fredkin and first published by several other authors in 1984.[11] In this technique, the state of each cell in the automaton at time t is a function both of its neighborhood at time t − 1 and of its own state at time t − 2. Specifically, the transition function of the automaton maps each neighborhood at time t − 1 to a permutation on the set of states, and then applies that permutation to the state at time t − 2. The reverse dynamics of the automaton may be computed by mapping each neighborhood to the inverse permutation and proceeding in the same way.[12]

In the case of automata with binary-valued states (zero or one), there are only two possible permutations on the states (the identity permutation and the permutation that swaps the two states), which may themselves be represented as the exclusive or of a state with a binary value. In this way, any conventional two-valued cellular automaton may be converted to a second-order cellular automaton rule by using the conventional automaton's transition function on the states at time t − 1, and then computing the exclusive or of these states with the states at time t − 2 to determine the states at time t. However, the behavior of the reversible cellular automaton determined in this way may not bear any resemblance to the behavior of the cellular automaton from which it was defined.[12]

Any second-order automaton may be transformed into a conventional cellular automaton, in which the transition function depends only on the single previous time step, by combining pairs of states from consecutive time steps of the second-order automaton into single states of a conventional cellular automaton.[12]

Conserved landscape[edit]

A one-dimensional cellular automaton found by Patt (1971), the simplest possible nontrivial one-dimensional two-state reversible cellular automaton, has a neighborhood consisting of four contiguous cells; a cell flips its state whenever it occupies the "?" position in the pattern "0?10". No two such patterns can overlap, so the same "landscape" surrounding the flipped cell continues to be present after the transition. Therefore, this automaton is its own inverse; it is reversible, but all cycles in its dynamics have period two. However, the same conserved landscape technique is also capable of more complex behavior, and in particular it can simulate any second-order cellular automaton.[13]

Theory[edit]

A cellular automaton consists of an array of cells, each one of which has a finite number of possible states, together with a rule for updating all cells simultaneously based only on the states of neighboring cells. A configuration of a cellular automaton is an assignment of a state to every cell of the automaton; the update rule of a cellular automaton forms a function from configurations to configurations, with the requirement that the updated value of any cell depends only on some finite neighborhood of the cell, and that the function is invariant under translations of the input array.

With these definitions, a cellular automaton is reversible when it satisfies any one of the following equivalent conditions:[14]

  1. Every configuration of the automaton has a unique predecessor that is mapped to it by the update rule.
  2. The update rule of the automaton is a bijection; that is, a function that is both one-to-one and onto.
  3. The update rule is an injective function, that is, there are no two configurations that both map to the same common configuration. This condition is obviously implied by the assumption that the update rule is a bijection. In the other direction, the Garden of Eden theorem for cellular automata implies that every injective update rule is bijective.[15]
  4. The time-reversed dynamics of the automaton can be described by another cellular automaton. Clearly, for this to be possible, the update rule must be bijective. In the other direction, if the update rule is bijective, then it has an inverse function that is also bijective, and the fact that this inverse function must be a cellular automaton rule can be proven using the Curtis–Hedlund–Lyndon theorem, a topological characterization of cellular automata rules as the translation-invariant functions that are continuous with respect to the Cantor topology on the space of configurations.[16]
  5. The update rule of the automaton is an automorphism of the shift dynamical system defined by the state space and the translations of the lattice of cells. That is, it is a homeomorphism that commutes with the shift map, as the Curtis–Hedlund–Lyndon theorem implies.[17]

Di Gregorio & Trautteur (1975) analyze several alternative definitions of reversibility for cellular automata. Most of these turn out to be equivalent either to injectivity or to surjectivity of the transition function of the automaton; however, there is one more alternative that does not match either of these two definitions. It applies to automata such as the Game of Life that have a quiescent or dead state. In such an automaton, one can define a configuration to be "finite" if it has only finitely many non-quiescent cells, and one can consider the class of automata for which every finite configuration has at least one finite predecessor. This class turns out to be distinct from both the surjective and injective automata, and in some subsequent research, automata with this property have been called invertible finite automata.[18]

Testing reversibility[edit]

It was first shown by Amoroso & Patt (1972) that the problem of testing reversibility of a given one-dimensional cellular automaton has an algorithmic solution. Alternative algorithms based on automata theory and de Bruijn graphs were given by Culik (1987) and Sutner (1991), respectively.

  • Culik begins with the observation that a cellular automaton has an injective transition function if and only if the transition function is injective on the subsets of configurations that are periodic (repeating the same substring infinitely often in both directions). He defines a nondeterministic finite state transducer that performs the transition rule of the automaton on periodic strings, essentially by remembering the neighborhood of the automaton at the start of the string and entering an accepting state when that neighborhood concatenated to the end of the input would cause the nondeterministically chosen transitions to be correct. He swaps the input for the output of the transducer, producing a different transducer that (if the automaton is reversible) simulates the dynamics of its inverse. Finally, he applies previously known algorithms to test whether the resulting transducer maps each input to a single output.[19]
  • Sutner defines a directed graph (a type of de Bruijn graph) in which each vertex represents two assignments of states (possibly equal to each other) for the cells in a contiguous sequence of cells of length one less than the neighborhood size of the automaton. Each edge in this graph connects sequences of cells that overlap in all but one cell (together forming a single complete neighborhood), that have compatible pairs of state assignments, and for which both state assignments would lead to the same state at the central cell of their overlapping neighborhood in the next time step of the automaton. By performing a linear-time strong connectivity analysis of this graph, it is possible to determine which of its vertices belong to cycles. The transition rule is non-injective if and only if this graph contains a cycle in which at least one vertex has two differing state assignments.[20]

These methods take polynomial time, proportional to the square of the size of the state transition table of the input automaton.[21] A related algorithm of Hillman (1991) determines whether a given rule is surjective when applied to finite-length arrays of cells with periodic boundary conditions, and if so, for which lengths.

For a block cellular automaton, testing reversibility is also easy: the automaton is reversible if and only if the transition function on the blocks of the automaton is invertible, and in this case the reverse automaton has the same block structure with the inverse transition function.[8]

However, for cellular automata with other neighborhoods in two or more dimensions, the problem of testing reversibility is undecidable, meaning that there cannot exist an algorithm that always halts and always correctly answers the problem. The proof of this fact by Kari (1990) is based on the previously known undecidability of tiling the plane by Wang tiles, sets of square tiles with markings on their edges that constrain which pairs of tiles can fit edge-to-edge. Kari defines a cellular automaton from a set of Wang tiles, such that the automaton fails to be injective if and only if the given tile set can tile the entire plane. His construction uses the von Neumann neighborhood, and cells with large numbers of states. In the same paper, Kari also showed that it is undecidable to test whether a given cellular automaton rule of two or more dimensions is surjective (that is, whether it has a Garden of Eden).

Reverse neighborhood size[edit]

In a one-dimensional reversible cellular automaton with n states per cell, in which the neighborhood of a cell is an interval of m cells, the automaton representing the reverse dynamics has neighborhoods that consist of at most nm − 1m + 1 cells.[22] This bound is known to be tight for m = 2: there exist n-state reversible cellular automata with two-cell neighborhoods whose time-reversed dynamics forms a cellular automaton with neighborhood size exactly n − 1.

For any integer m there are only finitely many two-dimensional reversible m-state cellular automata with the von Neumann neighborhood. Therefore there is a well-defined function f(m) such that all reverses of m-state cellular automata with the von Neumann neighborhood use a neighborhood with radius at most f(m): simply let f(m) be the maximum, among all of the finitely many reversible m-state cellular automata, of the neighborhood size needed to represent the time-reversed dynamics of the automaton. However, because of Kari's undecidability result, there is no algorithm for computing f(m) and the values of this function must grow very quickly, more quickly than any computable function.[10]

Wolfram's classification[edit]

A well-known classification of cellular automata by Stephen Wolfram studies their behavior on random initial conditions. For a reversible cellular automaton, if the initial configuration is chosen uniformly at random among all possible configurations, then that same uniform randomness continues to hold for all subsequent states. Thus it would appear that most reversible cellular automata are of Wolfram's Class 3: automata in which almost all initial configurations evolve pseudo-randomly or chaotically. However, it is still possible to distinguish among different reversible cellular automata by analyzing the effect of local perturbations on the behavior of the automaton: making a change to the initial state of a reversible cellular automaton may cause changes to later states to remain only within a bounded region, to propagate irregularly but unboundedly, or to spread quickly, and Wolfram (1984) lists one-dimensional reversible cellular automaton rules exhibiting all three of these types of behavior.

Later work by Wolfram identifies the one-dimensional Rule 37R automaton as being particularly interesting in this respect: when run with a small seed of random cells centered within a larger neighborhood, with a finite number of cells with circular boundary conditions, it tends to fluctuate between ordered and chaotic states, while with the same initial conditions on an unbounded set of cells its configurations tend to organize themselves into several types of simple moving particles.[23]

Abstract algebra[edit]

Boykett (2004) defines a semicentral bigroupoid to be an algebraic structure consisting of a set S of elements and two operations and on pairs of elements, satisfying the two equational axioms:

  • for all elements a, b, and c in S, (ab) ← (bc) = b, and
  • for all elements a, b, and c in S, (ab) → (bc) = b.

For instance, this is true for the two operations in which operation returns its right argument and operation returns its left argument.

As Boykett argues, any one-dimensional reversible cellular automaton is equivalent to an automaton in rectangular form, in which the cells are offset a half unit at each time step, and in which both the forward and reverse evolution of the automaton have neighborhoods with just two cells, the cells a half unit away in each direction. If a reversible automaton has neighborhoods larger than two cells, it can be simulated by a reversible automaton with smaller neighborhoods and more states per cell, in which each cell of the simulating automaton simulates a contiguous block of cells in the simulated automaton. The two axioms of a semicentral bigroupoid are exactly the conditions required on the forward and reverse transition functions of these two-cell neighborhoods to be the reverses of each other. That is, every semicentral bigroupoid defines a reversible cellular automaton in rectangular form, in which the transition function of the automaton uses the operation to combine the two cells of its neighborhood, and in which the similarly defines the reverse dynamics of the automaton. Every one-dimensional reversible cellular automaton is equivalent to one in this form.

Boykett used this algebraic formulation as the basis for algorithms that exhaustively list all possible inequivalent reversible cellular automata.[24]

Conservation laws[edit]

When researchers design reversible cellular automata to simulate physical systems, they typically incorporate into the design the conservation laws of the system; for instance, a cellular automaton that simulates an ideal gas should conserve the number of gas particles and their total momentum, for otherwise it would not provide an accurate simulation. However, there has also been some research on the conservation laws that reversible cellular automata can have, independent of any intentional design. The typical type of conserved quantity measured in these studies takes the form of a sum, over all contiguous subsets of k cells of the automaton, of some numerical function of the states of the cells in each subset. Such a quantity is conserved if, whenever it takes a finite value, that value automatically remains constant through each time step of the automaton, and in this case it is called a kth-order invariant of the automaton.[25]

For instance, recall the one-dimensional cellular automaton defined as an example from a rectangular band, in which the cell states are pairs of values (a,b) drawn from sets L and R of left values and right values, the left value of each cell moves rightwards at each time step, and the right value of each cell moves leftwards. In this case, for each left or right value x of the band, one can define a conserved quantity, the total number of cells that have that value. If there are l left values and r right values, then there are l + r − 2 independent first-order-invariants, and any first-order invariant can be represented as a linear combination of these fundamental ones. The conserved quantities associated with left values flow uniformly to the right at a constant rate: that is, if the number of left values equal to x within some region C of the line takes a certain value at time 0, then it will take the same value for the shifted region C + t/2 at time t. Similarly, the conserved quantities associated with right values flow uniformly to the left.

Any one-dimensional reversible cellular automaton may be placed into rectangular form, after which its transition rule may be factored into the action of an idempotent semicentral bigroupoid (a reversible rule for which regions of cells with a single state value change only at their boundaries) together with a permutation on the set of states. The first-order invariants for the idempotent lifting of the automaton rule (the modified rule formed by omitting the permutation) necessarily behave like the ones for a rectangular band: they have a basis of invariants that flow either leftwards or rightwards at a constant rate without interaction. The first-order invariants for the overall automaton are then exactly the invariants for the idempotent lifting that give equal weight to every pair of states that belong to the same orbit of the permutation. However, the permutation of states in the rule may cause these invariants to behave differently from in the idempotent lifting, flowing non-uniformly and with interactions.[26]

In physical systems, Noether's theorem provides an equivalence between conservation laws and symmetries of the system. However, for cellular automata this theorem does not directly apply, because instead of being governed by the energy of the system the behavior of the automaton is encoded into its rules, and the automaton is guaranteed to obey certain symmetries (translation invariance in both space and time) regardless of any conservation laws it might obey. Nevertheless, the conserved quantities of certain reversible systems behave similarly to energy in some respects. For instance, if different regions of the automaton have different average values of some conserved quantity, the automaton's rules may cause this quantity to dissipate, so that the distribution of the quantity is more uniform in later states. Using these conserved quantities as a stand-in for the energy of the system can allow it to be analyzed using methods from classical physics.[27]

Applications[edit]

Lattice gas automata[edit]

Main article: Lattice gas automaton

A lattice gas automaton is a cellular automaton designed to simulate the motion of particles in a fluid or an ideal gas. In such a system, gas particles move on straight lines with constant velocity, until undergoing elastic collision with other particles. Lattice gas automata simplify these models by only allowing a constant number of velocities (typically, only one speed and either four or six directions of motion) and by simplifying the types of collision that are possible.[28]

Specifically, the HPP lattice gas model consists of particles moving at unit velocity in the four axis-parallel directions. When two particles meet on the same line in opposite directions, they collide and are sent outwards from the collision point on the perpendicular line. However, although this system obeys the conservation laws of physical gases, and produces simulations whose appearance resembles the behavior of physical gases, it was found to obey unrealistic additional conservation laws (the total momentum within any single line is conserved) as well as other forms of anisotropy. An improvement to it, the FHP lattice gas model, has particles moving in six different directions, at 60 degree angles to each other, with the two outgoing particles from a collision deflected at 60 degree angles from the two incoming particles. The possibility of three-way collisions in the FHP model allows it to avoid the unphysical added conservation laws of the HPP model.[28]

Because the motion of the particles in these systems is reversible, they are typically implemented with reversible cellular automata. In particular, both the HPP and FHP lattice gas automata can be implemented with a two-state block cellular automaton using the Margolus neighborhood.[28]

Ising model[edit]

The Ising model is used to model the behavior of magnetic systems. It consists of an array of cells, the state of each of which represents a spin, either up or down. The energy of the system is measured by a function that depends on the number of neighboring pairs of cells that have the same spin as each other; therefore, if a cell has equal numbers of neighbors in the two states, it may flip its own state without changing the total energy. However, such a flip is energy-conserving only if no two adjacent cells flip at the same time.[29]

Cellular automaton models of this system divide the square lattice into two alternating subsets, and perform updates on one of the two subsets at a time. In each update, every cell that can flip does so. This defines a reversible cellular automaton which can be used to investigate the Ising model.[29]

Billiard ball computation and low-power computing[edit]

Fredkin & Toffoli (1982) proposed the billiard-ball computer as part of their investigations into reversible computing. A billiard-ball computer consists of a system of synchronized particles (the billiard balls) moving in tracks and guided by a fixed set obstacles. When the particles collide with each other or with the obstacles, they undergo an elastic collision much as real billiard balls would do. The input to the computer is encoded using the presence or absence of particles on certain input tracks, and its output is similarly encoded using the presence or absence of particles on output tracks. The tracks themselves may be envisioned as wires, and the particles as being Boolean signals transported on those wires. When a particle hits an obstacle, it reflects from it, causing the wire to change direction, and two particles on different tracks may collide, forming a logic gate at their collision point.[30]

As Margolus (1984) showed, billiard-ball computers may be simulated using a two-state reversible block cellular automaton with the Margolus neighborhood and a very simple update rule: blocks with exactly one live cell rotate by 180º, blocks with two diagonally opposite live cells rotate by 90º, and all other blocks remain unchanged. In this simulation, isolated live cells behave like billiard balls, moving on diagonal trajectories, while connected groups of more than one live cell behave like the fixed obstacles of the billiard-ball computer. In an appendix, Margolus also showed that a three-state second-order cellular automaton using the two-dimensional Moore neighborhood could simulate billiard-ball computers.

The reason to study reversible universal models of computation such as the billiard-ball model is that they could theoretically lead to actual computer systems that consume very low quantities of energy. According to Landauer's principle, irreversible computational steps require a certain minimal amount of energy per step, but reversible steps can be performed with an amount of energy per step that is arbitrarily close to zero. However, in order to perform computation using less energy than Landauer's bound, it is not good enough for a cellular automaton to have a transition function that is globally reversible: what is required is that the local computation of the transition function also be done in a reversible way. For instance, reversible block cellular automata are always locally reversible: the behavior of each individual block involves the application of an invertible function with finitely many inputs and outputs. Toffoli & Margolus (1990) were the first to ask whether every reversible cellular automaton has a locally reversible update rule; Kari (1996) showed that for one- and two-dimensional automata the answer is positive, and Durand-Lose (2001) showed that any reversible cellular automaton could be simulated by a (possibly different) locally reversible cellular automaton. However, the question of whether every reversible transition function is locally reversible remains open for dimensions higher than two.[10]

Synchronization[edit]

The rectilinear shapes generated by the Tron rule

The "Tron" rule of Toffoli and Margolus is a reversible block cellular rule with the Margolus neighborhood in which, in a 2 × 2 block of cells that all have the same state, all cells change state, and in which all other blocks of cells remain unchanged. As Toffoli and Margolus argue, the evolution of patterns generated by this rule can be used as a clock to synchronize any other rule on the Margolus neighborhood, so that it obeys the same dynamics as the standard Margolus-neighborhood rule while running on an asynchronous cellular automaton.[31]

Encryption[edit]

Kari (1990) proposed using multidimensional reversible cellular automata as an encryption system. In Kari's proposal, the cellular automaton rule would be the encryption key, encryption would be performed by running the rule forward one step, and decryption would be performed by running it backward one step. Kari suggests that a system such as this may be used as a public-key cryptosystem: in principle, an attacker could not algorithmically determine the decryption key (the reverse rule) from a given encryption key (forward rule) because of the undecidability of testing reversibility, so the forward rule could be made public without compromising the security of the system. However, Kari did not specify which types of reversible cellular automaton should be used for such a system, or show how a cryptosystem using this approach would be able to generate encryption/decryption key pairs.

Chai, Cao & Zhou (2005) have proposed an alternative encryption system, in which the encryption key determines the local rule for each cell of a one-dimensional cellular automaton, and then a second-order automaton based on that rule is run for several rounds on an input to transform it into an encrypted output. The reversibility property of the automaton ensures that any encrypted message can be decrypted by running the same system in reverse. In this system, keys must be kept secret, because the same key is used both for encryption and decryption.

Quantum computing[edit]

Quantum cellular automata are arrays of automata whose states and state transitions obey the laws of quantum dynamics. Quantum cellular automata were suggested as a model of computation by Feynman (1982) and first formalized by Watrous (1995). Several competing notions of these automata remain under research, many of which require that the automata constructed in this way be reversible.[32]

Physical universality[edit]

Janzing (2010) asked whether it was possible for a cellular automaton to be physically universal, meaning that, for any bounded region of the automaton's cells, it should be possible to surround that region with cells whose states form an appropriate support scaffolding that causes the automaton to implement any arbitrary transformation on sets of states within the region. Such an automaton must be reversible, or at least locally injective, because automata without this property have Garden of Eden patterns, and it is not possible to implement a transformation that creates a Garden of Eden.

Schaeffer (2014) constructed a reversible cellular automaton that is physically universal in this sense. Schaeffer's automaton is a block cellular automaton with two states and the Margolis neighborhood, closely related to the automata for the billiard ball model and for the HPP lattice gas. However, the billiard ball model is not physically universal, as it can be used to construct impenetrable walls preventing the state within some region from being read and transformed. In Schaeffer's model, every pattern eventually decomposes into particles moving diagonally in four directions. Thus, his automaton is not Turing complete. However, before the whole system stabilizes in this way, it is possible for a system of Boolean circuits constructed as part of the scaffolding surrounding the given region to decode the particles coming from the region, apply an arbitrary function to the decoded state, and then re-encode the function value as a system of particles that converge back on the initial region and collide with each other to build a copy of the transformed state.[33]

Notes[edit]

  1. ^ Kari (1990).
  2. ^ Wolfram (2002), p. 1018.
  3. ^ Blanchard, Devaney & Keen (2004), p. 38: "The shift map is without doubt the fundamental object in symbolic dynamics."
  4. ^ Boykett (2004).
  5. ^ Wolfram (2002), p. 1093.
  6. ^ Sutner (1991).
  7. ^ a b c Toffoli & Margolus (1987), section 12.8.2, "Critters", pp. 132–134; Margolus (1999); Marotta (2005).
  8. ^ a b c Toffoli & Margolus (1987), Section 14.5, "Partitioning technique", pp. 150–153; Schiff (2008), Section 4.2.1, "Partitioning Cellular Automata", pp. 115–116.
  9. ^ Toffoli & Margolus (1987), Chapter 12, "The Margolus Neighborhood", pp. 119–138.
  10. ^ a b c Kari (2005).
  11. ^ Margolus (1984); Vichniac (1984); Wolfram (1984).
  12. ^ a b c Toffoli & Margolus (1987), Section 14.2, "Second-order technique", pp. 147–149. Wolfram (2002), pp. 437ff. McIntosh (2009).
  13. ^ Toffoli & Margolus (1990), section 5.3, "Conserved-landscape permutations", pp. 237–238.
  14. ^ In the one-dimensional case, several of these equivalences were already presented, in the language of dynamical systems rather than cellular automata, by Hedlund (1969), Theorem 4.1. For higher dimensions, see Richardson (1972) and Di Gregorio & Trautteur (1975).
  15. ^ Myhill (1963).
  16. ^ Richardson (1972).
  17. ^ Hedlund (1969).
  18. ^ Moraal (2000).
  19. ^ Culik cites a 1979 automata theory textbook for this result, but see Béal et al. (2003) for more recent developments on testing functionality of transducers efficiently.
  20. ^ Sutner (1991).
  21. ^ Neither Amoroso & Patt (1972) nor Culik (1987) state their algorithms' time complexities explicitly, but Sutner (1991) does, and this bound can also be found e.g. in Czeizler & Kari (2007).
  22. ^ Kari (1992); Czeizler (2004); Czeizler & Kari (2007).
  23. ^ Wolfram (2002), pp. 454–457.
  24. ^ See Hillman (1991) and Seck Tuoh Mora et al. (2005) for closely related work on the enumeration of width-2 reversible cellular automata.
  25. ^ Hattori & Takesue (1991); Fukś (2008).
  26. ^ Boykett, Kari & Taati (2008).
  27. ^ Pomeau (1984); Takesue (1990); Capobianco & Toffoli (2011).
  28. ^ a b c Toffoli & Margolus (1987), Chapter 16, "Fluid dynamics", pp. 172–184.
  29. ^ a b Toffoli & Margolus (1987), Chapter 17.2, "Ising systems", pp. 186–190.
  30. ^ Durand-Lose (2002).
  31. ^ Toffoli & Margolus (1987), Section 12.8.3, "Asynchronous computation", pp. 134–136.
  32. ^ Meyer (1996); Schumacher & Werner (2004); Shepherd, Franz & Werner (2006); Nagaj & Wocjan (2008).
  33. ^ See also "A Physically Universal Cellular Automaton", Shtetl-Optimized, Scott Aaronson, June 26, 2014.

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