property

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