639.sagews

Create a "power series ring" over the indeterminate x (which has the side effect of binding the Python variable x to an object representing the symbolic variable x). The argument default_prec specifies a default value for how many terms in the power series are kept before Sage just writes O(x^n) to represent all terms in the power series for some power (n) of x onwards.

R.<x> = PowerSeriesRing(QQ, default_prec=500)

fay and fan are the ordinary generating functions (ogfs) for the numbers of successful and unsuccessful excecutions of an action, parameterised by the number of action failures that occur in the execution, i.e. fay = Sum_n ny(a,n).x^n. We want the coefficient of x^0 for fay (written as [1]fay) to be 1, and [x^n]fay = 0 for n>0. Similarly we have [1]fan = 0, [x]fan = 1, and [x^n]fan = 0 for n>1. fa is the Python dictionary structure associating 'y' with fay and 'n' with fan.

fay = 1
fan = x
fa = {'y':fay,'n':fan}

Suppose b_1 and b_2 are plan bodies, fdict = {'y': fb_1^y,'n': fb_2^n} where fb_1^y is the ogf for the sequence [removed], fb_1^n is the ogf for the sequence [removed], and gdict is defined similarly. Then seq(fdict,gdict) is the dictionary {'y': fb_1b_2^y, 'n': fb_1b_2^n} where fb_1b_2^y is the ogf for the sequence [removed] and fb_1b_2^n is the sequence [removed] seql performs a similar 'sequential composition' of 'y' and 'n' ogf dictionaries, but takes a list of dictionaries as an argument (i.e. it is for the sequential composition of a list of plan bodies. repeat does the same thing for a single plan body repeated in a sequence a given number of times.

def seq(fdict,gdict) :
  return {'y':fdict['y']*gdict['y'], 'n':fdict['n'] + fdict['y']*gdict['n']}

def seql(fdicts) :
  if fdicts == []:
     raise ValueError("seql called on empty list")
  elif fdicts[1:] == []:
     return fdicts[0]
  else:
     return seq(fdicts[0],seql(fdicts[1:]))

def repeat(n,fdict):
  return seql([fdict for i in range(n)])

fp(depth) and fg(depth) compute the 'y' and 'n' ogfs for a plan and goal (respectively) at depth d. The definition of fp(depth) is derived solely from the structure of our assumed uniform plan body and the definitions of seq and repeat above. The definition of fg(depth) is based on a combinatorial analysis (not given here) that allows the 'y' and 'n' ogfs to be expressed as products of the 'y' and 'n' ogfs for fp(depth-1).

def fp(depth):
  if depth < 0:
    raise ValueError("fp called with depth < 0")
  elif depth == 0:
    return repeat(l,fa)
  else:
    return seq(repeat(l,fa), repeat(k,seq(fg(depth),repeat(l,fa))))

def fg(depth):
  if depth < 1:
    raise ValueError("fg called with depth < 1")
  else:
    fpdict = fp(depth-1)
    return {'y':j*sum([binomial(j-1,i)*factorial(i)*(fpdict['n']**i)*fpdict['y']
                       for i in range(j)]),
            'n':factorial(j)*fpdict['n']**j}

Test cases - set the parameters d,j,k and l to the desired values. For larger values the power series ring's 'default_prec' parameter may need to be adjusted. The coefficients in the ogfs for fg(d) show how many successful and failed executions there are with a given number of action failures. The ogfs for the partial sums (calculated using a result from generatingfunctionology) show how many successful and failed executions there are with *up to* a given number of action failures. These should converge to the values we got in our previous analysis that didn't consider how many action failures there were in an execution.

d=3
j=3
k=3
l=1

fgd = fg(d)
print "fg(",d,") is:\n"
print fgd

ogfs_for_partial_sums = {'y':fgd['y']/(1-x),'n':fgd['n']/(1-x)}
print "\nThe partial sum ogfs are:\n"
print ogfs_for_partial_sums

WARNING: Output truncated!

full_output.txt

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