Question

6. [8 marks Consider the sequence 10, 11, ... defined by To = 0, 11 = 1, and ri = i-1 + 24-2 for all i > 2. Consider the following algorithm that attempts to compute the value In, given an n > 0. Algorithm 6 1: //pre: (n e Z) ^ (n > 0) 2: if n == 0 then 3: bro 1: else 5: a 0 6: 5+ 1 it1 //LoopInv: while i<n do ct a+b atb bc iti+1 end while 15: end if 16: //post: b == In Prove that Algorithm 6 satisfies the Partial Correctness property. You will need to provide (and prove) your own loop invariant LoopInv for line 8.

Answer 1

Loop invariaat - a afli-2) and b zf(i-1) the clearly holds just before the cloup The doy sets a=b zf (i-1), b'rath 2f (1-2) + F (i-1) zfi, broom which it is easy to Prove that maintains the dop invariaat. when the loop har liriched, we onust hare izantix So a f(i-2) = f(n-1) 8 b =f(i-1) =f(on) there is a weak paint in sus arguancat we can korow that isant at the cand of a the loop due to our loop invarigat colica q=f(i-2) and bzf(i-1) and isnti they hadd just before the loop since onz1 Caxo) a Precondition that needs to be added to the cartire function. it ix anciortioned duriay the dool since at the begiraray of cech iteration icon. fionally, after the loop we have isnt 1 and in colich forers Izaarl finee is an integer.