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
S 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

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
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

5. "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

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