CIS 4930/6930: Programs, Functions, Strange Loops, and Consciousness Programs and Functions Assignme

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CIS 4930/6930: Programs, Functions, Strange Loops, and Consciousness Programs and Functions Assignment 3 1. (5 pts.) If f = [while p do g], which one of the following, according to the Iteration Recursion Lemma (IRL), is functionally equivalent to [while p do g]? Circle ONE only. a. [if p then f;g end_if]

b. [if p then g o f end_if]

d. [if p then f else g end_if_else] g. [if p then g;f end_if]

c. [if p then g else f end_if_else]

e. [if p then g end_if]

h. [if p then f end_if]

f. [if p then f end_if; g]

i. (none of the above)

2. (14 pts.) Match each assertion of functional correctness below to the appropriate Correctness Condition(s) among the following. (Note: Correctness Condition(s) may be appropriate for none, one, or more than one assertion.)

A. (f = gohok)

E. term(f,S), pog  (f = g), ¬(pog)  (f = fog)

B. p  (f = g), ¬p  (f = h)

F. term(f,S), p  (f = fog) ¬p  (f = I)

C. (f = kohog)

G. term(f,S), gop  (f = g), ¬(gop)  (f = gof)

D. p  (f = g), ¬p  (f = I)

H. term(f,S), p  (f = gof) ¬p  (f = I)

__E__ a. f = [S] where S = repeat g until p _D___ b. f = [if p then g] _C___ c. f = [g;h;k] __E__ d. f = [S] where S = g; if ¬p then repeat g until p end_if _B___ e. f = [if p then g else h] ___E_ f. f = [S] where S = g; while ¬p do g __F__ g. f = [S] where S = if p then g; while p do g end_if 3. (12 pts.) Determine (but do not formally verify) the function of the following program:

x := y-2 while x<>5 do x := x-1 end_while Give your answer in the simplified form: (p  x,y := ?,?) where p is a Boolean predicate which specifies the domain of the program function. (Hint: Use heuristics to identify the function of the while loop and then use the Axiom of Replacement and function composition to determine the program function.) program function: f = ( y ≥ 7  x,y := 5, y) while loop: (x ≥ 5  x := 5) This study source was downloaded by 100000861003072 from CourseHero.com on 04-16-2023 13:33:42 GMT -05:00

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2 replacement x = y-2 Y≥7 4. Suppose that you wish to prove that the program, M, below computes the given intended function, f, by showing (1) term(f,M), (2) p  ( f = f o g ), and (3) ¬p  ( f=I ). f = (k≥0  k,t := 0,t2k | k<0  I)

M: while k>0 do t := 2*t k := k-1 end_while

a. (4 pts.) Express “g” as a concurrent assignment function in terms of variables k and t.

g: ___G(k,t := k -1 , 2*t) _______ (Y < 0) => (f = I) (Y > 0 ) => (f = f o g) b. (2 pts.) What is “p” from correctness conditions (2) and (3) above? (Give the actual expression that p represents.)

p: ___(k > 0)___ c. (4 pts.) It can be shown that wp(M, true) = true. Does this imply term(f,M) holds?

Answer either “yes” or “no” and then briefly justify your answer.

Yes, Since wp implies the weakest loop that will guarantee termination, then the term(f,M) will hold, as M should terminate for all states.

d. (8 pts.) Prove that correctness condition (2) holds by completing the following proof template.

Proof that (k>0)  ( f = f o g ): i. (k>0)  [ f = ( k,t := 0,t2k ) ] ii. (k>0)  [ f o g = ( k,t := 0,t2k ) o ( k,t := k -1 , 2*t = ( k,t := 0,(2 * t)2 k - 1 ) = ( k,t := 0,t2 k ) ] Therefore, Correctness Condition (2) holds.

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