Papers by Tommaso Toffoli
Cellular Automata Machines, 1987
This chapter contains sections titled: Noise-driven diffusion, Expansion and thermalization, Self... more This chapter contains sections titled: Noise-driven diffusion, Expansion and thermalization, Self-diffusion, Mean free path, A tour de force, A tuneable noise source, Diffusion-limited aggregation
International Colloquium on Automata, Languages and Programming, Jul 14, 1980
This chapter contains sections titled: The game of “life”, Echoing, Tracing, How to breed gliders
The theory of revervible computing is based on invertible primitives and composition rules that p... more The theory of revervible computing is based on invertible primitives and composition rules that premme invertibility. With these constraints, one can still satisfactorily deal with both junctional and structural aspects of computing processes, at the same time, one attains a closer correspondence between the behavior of abstract computing systemis and the microscopic physical laws (which ae presumed to be strictly reverN;ble) that underly' any concrete implementation of such systems. Here, we integrate into a comprehensive picture a variety r. c ancepts and results. According to a physical interpretation, the central r':0!., of this paper is that it i3 ideallypossible to build sequential circuits with cea .1ternal power dissipation. Even when these circuits are interfaced with o.-wextional ones, power dissipation at the Interface would be at moat pv'.P),Wonal t, the number of input/output lines, rather than to the numbe.-of lcgic gat(as in conventional computers.
Cellular Automata and Discrete Complex Systems, 2016
Hadn't this question already been answered? We all know about computation-universal Turing Machin... more Hadn't this question already been answered? We all know about computation-universal Turing Machines. And we know that any such machine can simulate a space-time dynamics not unlike von Neumann's cellular automaton, which is computation-and constructionuniversal and among other things can play host to self-replicating machines. And that self-replication sprinkled with a bit of randomness should inexorably lead to descent with variation, competition, and thence to evolution and all that. And note that the state of the art has much advanced in the fifty years since. "So?" Enrico Fermi would have asked, "Where are they?" It turns out that life is by its very nature a marginal, fragile, and ephemeral kind of phenomenon. For a substrate or a "culture medium" to be able to support it, computation-and construction-universality are necessary-but by no means sufficient! Most automata (including, I suspect, Conway's very game of Life) will go through their entire life course without ever originating anything like life. What questions, then, should we ask of a prospective mediumbe it a Turing machine, a cellular automaton, or some other kind of automaton-that will probe its capabilities to originate and/or sustain some form of life?
arXiv (Cornell University), Mar 24, 2011
The dynamics of a physical system is linked to its phase-space geometry by Noether's theorem, whi... more The dynamics of a physical system is linked to its phase-space geometry by Noether's theorem, which holds under standard hypotheses including continuity. Does an analogous theorem hold for discrete systems? As a testbed, we take the Ising spin model with both ferromagnetic and antiferromagnetic bonds. We show that-and why-energy not only acts as a generator of the dynamics for this family of systems, but is also conserved when the dynamics is time-invariant.
The MIT Press eBooks, 1987
Springer eBooks, 1978
Starting from models of particular interest for the physicist, there has evolved in the course of... more Starting from models of particular interest for the physicist, there has evolved in the course of years, with different emphasis and varying degrees of abstraction, a vast literature of mathematical structures known collectively as “general dynamical systems [1].” In this area of investigation, the analysis of particular systems has often led to the study of more general media, i.e., of “host” systems in which a variety of “guest” systems can be embedded. Great conceptual economy and, at the same time, great flexibility, is achieved by considering media that are arbitrarily extended and uniform. For example, while a differential equation of the form x = -f(x) describes a single, isolated oscillator, partial differential equations such as those of the electromagnetic field in a homogeneous medium may be made to describe, with a suitable choice of initial conditions, systems of waves of many kinds, including particle-like “packets” of waves. In the discrete domain, a particular sequential process realized by a specialized piece of hardware, may be reproduced by assigning appropriate initial conditions (stored program) to a more uniformly structured, general-purpose computer.
The MIT Press eBooks, 1987
Natural Computing, Aug 8, 2012
The connections between symmetries and conserved quantities of a dynamical system brought to ligh... more The connections between symmetries and conserved quantities of a dynamical system brought to light by Noether’s theorem depend in an essential way on the symplectic nature of the underlying kinematics. In the discrete dynamics realm, a rather suggestive analogy for this structure is offered by second-order cellular automata. We ask to what extent the latter systems may enjoy properties analogous to those conferred, for continuous systems, by Noether’s theorem. For definiteness, as a second-order cellular automaton we use the Ising spin model with both ferromagnetic and antiferromagnetic bonds. We show that—and why—energy not only acts as a generator of the dynamics for this family of systems, but is also conserved when the dynamics is time-invariant. We then begin to explore the issue of whether, in these systems, it may hold as well that translation invariance entails momentum conservation.
... Whether my son plays "Pirates" or "Nuns" ... more ... Whether my son plays "Pirates" or "Nuns" is deter-mined not only by his inclinations, but also by what the creators of LEGO saw fit to put in their ... [3] Feynman, Richard, "There is plenty of room at the bottom," reprinted in Microminiaturization (HD Gilbert, ed.), Reinhold (1961). ...
IBM journal of research and development, 2004
A pedestrian's introduction to spacetime crystallography Ordinary crystallography deals with regu... more A pedestrian's introduction to spacetime crystallography Ordinary crystallography deals with regular, discrete, static arrangements in space. Of course, dynamic considerationsand thus the additional dimension of time-must be introduced when one studies the origin of crystals (since they are emergent structures) and their physical properties such as conductivity and compressibility. The space and time of the dynamics in which the crystal is embedded are assumed to be those of ordinary continuous mechanics. In this paper, we take as the starting point a spacetime crystal, that is, the spacetime structure underlying a discrete and regular dynamics. A dynamics of this kind can be viewed as a "crystalline computer." After considering transformations that leave this structure invariant, we turn to the possible states of this crystal, that is, the discrete spacetime histories that can take place in it and how they transform under different crystal transformations. This introduction to spacetime crystallography provides the rationale for making certain definitions and addressing specific issues; presents the novel features of this approach to crystallography by analogy and by contrast with conventional crystallography; and raises issues that have no counterpart there. Note: This paper is based on a talk presented in May 2003 at a symposium at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York, honoring Charles Bennett on the occasion of his sixtieth birthday. 1 We give an example of this when we cover cellular automata and lattice gases in Section 5.
Physica D: Nonlinear Phenomena, 1991
ABSTRACT This paper is a manifesto, a brief tutorial, and a call for experiments on programmable ... more ABSTRACT This paper is a manifesto, a brief tutorial, and a call for experiments on programmable matter machines.
Lecture Notes in Computer Science, 1980
The theory of reversible computing is based on invertib|e primitives and composition rules that p... more The theory of reversible computing is based on invertib|e primitives and composition rules that preserve invertibility. With these constraints, one can still satisfactorily deal with both functional and structural aspects of computing processes; at the same time, one attains a closer correspondence between the behavior of abstract computing systems and the microscopic physical laws (which are presumed to be strictly reversible) that underly any concrete implementation of such systems. According to a physical interpretation, the central result of this paper is that i¢ is ideally possible to build sequential c/rcuits with zero internal power dissipation. L Introduction This is an abridged version of a much longer report of the same title[27], to which the reader may turn for further details, most proofs, and extended references. Here, the numbering of formulas, figures, etc. reflects that of the original version.
Physica D: Nonlinear Phenomena, 1984
Physica lOD (1984) 117-127 North-Holland, Amsterdam CELLULAR AUTOMATA AS AN ALTERNATIVE TO (RATHE... more Physica lOD (1984) 117-127 North-Holland, Amsterdam CELLULAR AUTOMATA AS AN ALTERNATIVE TO (RATHER THAN AN APPROXIMATION OF) DIFFERENTIAL EQUATIONS IN MODELING PHYSICS* Tommaso TOFFOLI MIT Laboratory for Computer Science, 545 Technology ...
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Papers by Tommaso Toffoli