Tommaso Toffoli

Cellular automata as an alternative to (rather than an approximation of) differential equations in modeling physics

Abstract Cellular automata are models of distributed dynamical systems whose structure is particularly well suited to ultrafast, exact numerical simulation. On the other hand, they constitute a radical departure from the traditional partial-differential-equation approach to distributed dynamics. Here we discuss the problem of encoding the state-variables and evolution laws of a physical system into this new setting, and of giving suitable correspondence rules for interpreting the models behavior.

Ininvertible cellular automata: a review

Abstract In the light of recent developments in the theory of invertible cellular automata, we attempt to give a unified presentation of the subject and discuss its relevance to computer science and mathematical physics.

Physics and computation

Computing processes are ultimately abstractions of physical processes; thus, a comprehensive theory of computation must reflect in a stylized way aspects of the underlying physical world. On the other hand, physics itself may draw fresh insights and productive methodological tools from looking at the world as an ongoing computation. The terminformation mechanics seems appropriate for this unified approach to physics and computation.

Programmable matter: concepts and realization

Abstract This paper is a manifesto, a brief tutorial, and a call for experiments on programmable matter machines.

Bicontinuous extensions of invertible combinatorial functions

We discuss and solve the problem of constructing a diffeomorphic componentwise extension for an arbitrary invertible combinatorial function. Interpreted in physical terms, our solution constitutes a proof of the physical realizability of general computing mechanisms based onreversible primitives.

Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight

How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. A lower bound on the orthogonalization time, based on the energy spread DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide and can be attained by certain initial states if DeltaE=E. Yet, the problem remained open when DeltaE not equal E. We consider the unified bound that involves both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE not equal E. However, the bound remains tight: for any values of DeltaE and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit. These results establish the fundamental limit of the operation rate of any information processing system.

Three-dimensional rotations by three shears

Abstract We show that a rotation in three dimensions can be achieved by a composition of three shears, the first and third along a specified axis and the second along another given axis orthogonal to the first; this process is invertible. The resulting rotation algorithm is practical for the processing of fine-grained digital images, and is well adapted to the access constraints of common storage media such as dynamicRAMor magnetic disk. For a 2-D image, rotation by composition of three shears is well known. For 3-D, an obvious nine-shear decomposition has been mentioned in the literature. Our three-shear decomposition is a sizable improvement over that, and is the best that can be attained—just two shears wont do. Also, we give a brief summary of how the present three-shear decomposition approach generalizes to any linear transformations of unit determinant in any number of dimensions.

Programmable matter methods

Abstract Fine-grained, indefinitely-extended mesh architectures, which can aptly be termed ‘Programmable Matter’, play a complementary rather than competitive role vis-a-vis traditional architectures. Ideal areas of applicability are physical modeling, materials science, interactive modeling of complex objects, image processing, and pattern recognition. Moreover, these architectures are well suited to furnish a viable blueprint for computers built out of atomic-scale components, towards which technology is inexorably leading. In spite of a potentially huge performance gain over conventional computers, programmable matter is currently hard to exploit. In most cases, its raw computational resources do not directly match the structure of problems one might want to apply it to. Since much ad hoc programming is needed to attain a reasonable fraction of the theoretical performance, so far only niche applications have been explored. We are developing a modeling methodology that will give programmable matter a much broader scope of application. This methodology makes extensive use of synthetic dynamics inspired by physics (kinematic transformations, cellular automata and lattice gases, statistical–mechanical ensembles, simulated annealing, simulated staining, texture-locked loops, etc.), but harnesses these dynamics to data-processing tasks of a more general nature as are encountered in a variety of mundane applications. In particular, by means of suitable feedback, the massive pattern-generation resources of cellular automata machines are used to construct flexible pattern recognizers.

Almost every unit matrix is a ULU

Abstract We call an n × n matrix a shear if it is triangular with all ls on the diagonal, and a unit matrix if it has unit determinant. Earlier we had shown that, for n = 3, every orthogonal matrix (except for degenerate cases when one of the Euler angles equals π) can be written in the form U 0 LU 1 , where the U are upper shears and L is a lower shear. Then Strang showed that, for any n , every unit matrix can be written as L 0 U 0 L 1 U 1 . Here, we show that every unit matrix (except for a subset of measure zero) can be decomposed into the product of just three shears, U 0 LU 1 , and we present a canonical form for this decomposition. On the residual subset, such a decomposition is still possible (up to a sign) if one is allowed to suitably prepermute the rows of the matrix.

When–and how–can a cellular automaton be rewritten as a lattice gas?

Both cellular automata (CA) and lattice-gas automata (LG) provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics; it is customary to use the terms cellular automaton and lattice gas for a dynamic system itself as well as for its presentation. The two kinds of presentation share many traits but also display profound differences on issues ranging from decidability to modeling convenience and physical implementability. Following a conjecture by Toffoli and Margolus, it had been proved by Kari that any invertible CA, at least up to two dimensions, can be rewritten as an isomorphic LG. But until now it was not known whether this is possible in general for noninvertible CA-which comprise almost all CA and represent the bulk of examples in theory and applications. Even circumstantial evidence-whether in favor or against-was lacking. Here, for noninvertible CA, (a) we prove that an LG presentation is out of the question for the vanishingly small class of surjective ones. We then turn our attention to all the rest-noninvertible and nonsurjective-which comprise all the typical ones, including Conways Game of Life. For these (b) we prove by explicit construction that all the one-dimensional ones are representable as LG, and (c) we present and motivate the conjecture that this result extends to any number of dimensions. The tradeoff between dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale.