2)

BIG-STEP OPERATIONAL SEMANTICS consider this definition of conditional arithmetic in terms of a big-step operational semantics. Syntax of arithmetic (A) and boolean (B) expressions, and natural number (n) and boolean (b) values: nu=0|1|2|3... A :=nplus(A1, A2) | minus(A1, A2) | times(A1, A2) | div(A1, A2) | if(B, A1, A2) b::= true false B::=band(B1, B2) or(B1, B2) | zero?(A) Big-step operational semantics of arithmetic expressions (An): nin An A2 n2ni + 112 = n An A, n2 11 - 12 = n ni >n2 plus(A1, A2) in minus(A1, A2) in A ni Agn2 ni X n2 = n An A2 n2 ni = n2 = n 1270 times(A1, A2) in div(A1, A2) in B true Alin B false A2 n2 if(B, A1, A2) in if(B, A1, A2) 2 Big-step operational semantics of boolean expressions (Bb): true true false false B1 true B2b Bi false and(B1, B2) and(B1, B2) false Bi false B2 16 Bi true or(B1, B2) 1b or(B1, B2) true AO Ann70 zero?(A) true zero?(A) V false For the natural number division ni + n2 returns only the whole number dividend and drops the remainder, so that 7 = 2 is 3 for example. . Multiple Choice). Which of the following evaluations of times(if(zero?(minus(1,1)),3,1), 2) can be derived by the operational semantics? (a) times(if(zero?(minus (2.plus(1,1))), 3.1), 2) 3 (b) times(if(zero?(minus (2.plus(1,1))), 3.1), 2) 1 (c) times(if(zero?(minus (2.plus(1,1))), 3.1), 2) 2 (d) times(if(zero?(minus(2.plus(1,1))), 3.1), 2) 6 (e) times(if(zero?(minus(2.plus(1,1))), 3. 1),2) true (f) times(if(zero?(minus (2.plus(1,1))), 3.1), 2) false (This or That). An arithmetic expression A returns if there is some number n such that An, and diverges if there is no such n. For example, div(1.0) and minus(0.1) both diverge, and div(0.1) and minus(1,0) both return (since div(0.1) 40 and minus(1,0) 11). Similarly, a boolean expression B returns if there is some boolean value b = true or b = false such that Bb, and diverges otherwise. For each of the following arithmetic and boolean expressions, say if that expres- sion returns or diverges. (a) minus(plus(3,1), 2) (b) plus(3, minus(1,2)) (c) if(zero?(minus(2, 2)),0, div(3, minus(2, 2))) (d) and (zero?(div(0,0)), false) (e) and(false, zero?(div(0,0)))