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HW2sol - 1 1What do we get if we add all the redundant...

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Unformatted text preview: 1) 1What do we get if we add all the redundant parentheses back to [Em.(0 5 m < n /\ Vj.0 5 j M < m e a[0] S GU] A 0-le S a[m])) ‘3’ (Elm.((0 g m < n) /\ (Vj.(((0 g j) /\ 2) (3' < Ml) —> (WM 5 aLil) A (GM 5 almlDDD Give a definition of predicate functions P(x,y) and Q(x,y) such that [Vm.§|n.P(m,n) /\ Q(m,n) is false but (Vm.(§n.P(m, n)) /\ Ep tensive proof of the truth and falsity of the If P(3;,y) (:> a: < y and Q(;r,y) (1} .Q(m, 33)) is true. (You don’t have to provide an ex- predicates, but it should be obvious to the grader). at > y then for all In you can not find a n such that n1<n and n1>n but for all n1 you can find a n such that m<n and find a p such that n1>p (n=n1+1 and p=In-1). Question 3 Find a predicate equivalent to —(E Answer : —.(: {i} (Hm—{Wynn 5 y) V V241: 2 2)) 4:} (Va-(Vyrr 5 y) A —-(Vz.a: 2 2)) {i} (V$.(Ely.a: > y) A 32.3: < z) Question 4 Write out the definition of a predicate function LT(b,m, n, p) that is true iff every element of b[m],b[m + 1],...1 and b[n — 1] is < ever}.r element of b[n],b[n + 1]1 ..., and b[p.1]. Assume 0 5 m 5 n 5 p and that b is of size 5 p. (Note the elements in b[m..n — 1] don’t have to be E.g., if b[0..5] is 3, 2,6,4,9,8, then LT(b, 0, 2, 6) is true sorted, nor do the elements in b[n..p— 1]. m.(Vy.x 5 y) V V2.2: 2 z)that has no uses of —._ x.(Vy.;c 5 3;) V V2.3," 2 2) but LT(b,O,3,6) is false (because b[2] isn’t < b[3]). Answer : LT(b,m, mp) {i} (W (Vj ((m 5 i < n) /\ (n 5 j < p) —> (3;:[3'] < b[j])))) Question 5 Assume we’ve defined a function 000(3), m, n, as) = the number of occurrences of {c in the array segment b[m..n — 1]. (E.g., if b[O..5] is 4, —1, 2,4, 2,4, then occ(b,0, 6, 4) = 3, occ(b, 0, 1,4) = 1, and occ(b, 0, 0,4) = 0.) Use occ to define a predicate perm(b, 5', m,n) is true iff b[m..n — 1] and b’[m..n — 1] are permutations ofeach other. Answer : perm(b, 5', m,n) «e» (Va: (m 5 :1: < n —> ooc(b, m,n, b[acl) : occUJ’, m, n, b[xDD 6. [E- = 2*3 points] Explain why each of the following is not a proper state. (a) x = 1 A x = 2 A y = 4 (b) b[O] = 1 A b[2] = 3 [c] b[O] = 17 A b[1] = 12.5 (assume for a moment that we’ve added floating-point numbers to our expressions, and integers and floatingpoint values have different types]. a) In this case x has two values and for being a proper state each variable only can have one. b) You have to define the complete array to be a proper state, in this case a definition of b[l] is missing. c] Each element of the same array must be of the same type, in this case one is integer and the other is floatingpoint. 7) Let U1: 1': 1 A k : 2 A1; : 5 AND] : 18A b[1] : 12 A b[2] : 5, and let the expression 61 be (1': k? 1: : b[1']) 1. Draw a graphical representation of crl. 2. What is 01 [k »—> 0101' — 1)](81)? 2- 01 [k H 0106 H 1)](61) = Ulik H (671(k) H 01(0)](61) MUCH [2— 1)](81) ”HIGH 11(61) 01(9) 5 8) Let 21H:>5[0] =b[y/2] Ab[y/2] =1:sz Bay/\y = 2. 1. Write out the definition of a state equivalent to 1,!) (textually, as a conjunction of bindings). 2. 1Why would ¢[b[[}] 1—» 8] be a reasonable update for tb but 1,!)[b[8] »—> 0] would not ? 1. 1,15% b[0] =b[y/2] Ab[y/2] =$A$= 3$yAy = 2 @340] =5[1]/\b[1] =$A$=6Ay=2 @340] =b[1]/\b[1] =6Ax=6Ay= 2 @b[0]=6/\b[1] =6Azc=6Ay=2 2. 1,!)[b[0] »—> 8] is a reasonable update because 5 is defined, it’s a continuous and homogeneous array. 1b[b[8] »—> 0] is not a reasonable update because b[2]...b[7] aren’t defined. 9) Our expressions have a ”visible variable” property : For any states CI and T and expression 6, if 0(1)) 2 7(1)) for every variable 1.! that appears in 8, then 0(8) 2 1'03). For each of the following constructs, say whether adding it to our expressions would break the visible variable property. 1. A randbéto function that randomly returns 0 or 1. 2. A + —|— ’2: expression that changes the state by adding 1 to the value of U and then yields the new value of v. 1. Tandbéto function randomly yelds 0 or 1. So when we evaluate 0(1)) and 7(0) the value returned can be randomly U or 1. So the ”visible variable” property is broken. 2. If our states have the ”visible variable” property then incrementing the variable of the two states conserve the property. ...
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