Logic: Lecture 22, 12 November 2015
-----------------------------------
 
View a finite word as encoding a sequence of states rather than a
sequence of actions.

Example: Peterson's algorithm and mutual exclusion

- Fix a set of atomic propositions AP

- In each state, some subset of propositions are true

- For an atomic proposition a, P_a is the subset of positions
  where a is true

- P_a, P_b, ... no longer partition the set of position, unlike
  the word interpretation

- Or, choose Sigma = 2^AP, in which case each letter describes a
  state valuation

- Note: can absorb the transition label into the target state of
  each transition, so there is no loss of expressiveness

MSO can now be used to reason about computational systems:

Simplest case: 

  Finite state automata, each state is labelled by a subset of AP

  Path through automaton is a sequence of states

  Example assertions

     In every execution 

(Linear time) Temporal Logic (LTL)

  Syntax    : X, F, G, U

  System    : (S,sin,->,v)
              S = states, sin = initial state
              -> = transition relation (direct edges between states)
              v : S -> 2^AP is valuation function
	       
  Semantics :
    sigma = s0 s1 ... sn is a run of the sytem
      - i.e. s0 s1 ... sn is a path in the underlying dircted
             graph starting at s0 = sin

    sigma,i |= phi,  i is a position in {0,2,...,n}

    Define sigma,i |= phi by induction on phi

    sigma,i |= p in AP iff p in v(si)
    sigma,i |= ~phi iff not(sigma,i |= phi)
    sigma,i |= phi or psi iff sigma,i |= phi or sigma,i |= psi
    sigma,i |= X phi iff i < n and sigma,i+1 |= phi 
    sigma,i |= F phi iff for some i <= j <= n, sigma,j |= phi 
    sigma,i |= G phi iff for every i <= j <= n, sigma,j |= phi 
    sigma,i |= phi U psi iff for some i <= j <= n, sigma,j |= psi
                             and for every i <= k < j, sigma,k |= phi 

  Examples  :
    Every p followed by q

    G (p -> Fq)

    If a request is made, it remains valid till it is granted.

    G (r -> r U g)

  X, U can express X, F, G, U

    F phi <=> true U phi
    G phi <=> ~F ~phi

======================================================================