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Modeling Cellular Automata with Partial Differential Equations

Explore the connection between cellular automata and partial differential equations, capturing the smooth continuous nature of discrete systems in space and time. Discover how universal computation in cellular automata implies universal properties in PDEs. Bump functions on a lattice represent cell states, with bump height indicating color, showcasing a novel approach to modeling. Learn how to solve standard math problems, diagonalize matrices, and sort lists using dynamic systems and gradient flows. Uncover the unpredictability and undecidability in dynamical systems, with long-term questions about chaotic systems and the equivalence of GSMs to Turing machines. Investigate the boundaries of computability in physical systems through an optimization perspective on sorting algorithms.

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Modeling Cellular Automata with Partial Differential Equations

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  1. Alloy = FOL + transitive closure + sets + relations • bounded exhaustive search for counterexample • sound but not complete Alloy Model Alloy instance spec Alloy Analyzer property translate formula translate instance mapping scope SAT solver boolean formula boolean instance

  2. Alloy Case Studies • firewire configuration protocol • unison file sychronizer • IMPP presence protocol for instant messaging • query interface in COM • key distribution for multicast • intentional naming • Chord distributed hash table • role-based access control • web ontologies • air traffic control protocols • telephone switch feature configuration • proton beam scheduling

  3. Stephen Omohundro "Modelling Cellular Automata with Partial Differential Equations" (1984) • modeled a 2D 9-neighbor cellular automata (CA) with 10 PDEs • modeling discrete system (in space and time) as smooth continuous • computation universal CA implies universal PDEs ? • bump functions shifted on a lattice to represent state of cells • height of bump is color of cell • N(x, y, t) variable represents "now" state of CA, F represents future S1 . . . S8 shift N to represent the 8 neighboring cells

  4. R. W. Brockett "Dynamical Systems that Sort Lists, Diagonalize Matrices, and Solve Linear Programming Problems" (1988) • solve standard math problems with H, N are square symmetric matrices, [A, B] = AB - BA • describes a gradient flow on space of orthogonal matrices • use gradient flow property to diagonalize a symmetric matrix • solve linear programming when constraint set is a convex polytope • H can evolve to a sort the diagonals of a matrix

  5. Cristopher Moore "Unpredictability and Undecidability in Dynamical Systems" (1990) • can answer long-term questions about chaotic systems providing initial conditions are known precisely • identified dynamical systems that one cannot answer long-term questions about even if initial conditions known precisely • system evolution described as a Generalized Shift Map (GSM) • GSMs equivalent to Turing machines → computation universal • questions about the behavior of GSM systems undecidable • one such system: particle moving in a 3 dimensional potential  physical systems can be computers

  6. Sorting as Optimization Problem given a list of numbers define sorted: sorted is minimal fun exercise to show how each step of a sorting algorithm keeps this minimal

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