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

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

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 10 coupled non-linear smooth PDEs common trick to discretize continuous systems for approximation on a computer universal Turing machine that simulates a computer = can have a universal CA – universal PDEs PDEs computation universal – support self-reproducing configurations just like Turing machines could one use this fact to show that finding a closed form to a PDE is equivalent to solving the halting problem? height of bumps represent state of individual cell

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 technically understood this one the least, because it touched so many different domains explaining what this system could do given appropriate choice of H(0) and N inscrutable proof of the gradient flow

"Unpredictability and Undecidability in Dynamical Systems" (1990) 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 more than random – they are highly complex butterfly effect

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 find a utility function that is optimal when the list is sorted