Transcript Alloy
• 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 mapping translate instance SAT solver boolean instance scope boolean formula 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 R. W. Brockett "Dynamical Systems that Sort Lists, Diagonalize Matrices, and Solve Linear Programming Problems" (1988) • solve standard math problems with H H , H , N 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 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 Sorting as Optimization Problem given a list of numbers define sorted: x1 , x2 ,, xn i [1, n 1] : xi xi 1 n 1 sorted sgn(xi xi 1 ) is minimal i 1 fun exercise to show how each step of a sorting algorithm keeps this minimal