Alloy = FOL transitive closure sets relations

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
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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

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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

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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

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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

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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