1 Complexity Theory Think about the accompanying choice issues. 1

(a) Which of the two issues is in P and which is NP-finished? [2 marks] (b) Describe a polynomial time calculation for the issue in P. [6 marks] (c) Prove that the other issue is truth be told NP-complete. [12 marks]

(a) Define exactly what we mean when we compose L1 ?P L2. [4 marks] (b) What is the distinction between NP-fulfillment and NP-hardness? [2 marks] (c) Let 3COL mean the accompanying choice issue. Given a diagram G = (V, E), is it 3-colourable? (I) Is 3COL in NP? Why? [2 marks] (ii) Show that 3SAT ?P 3COL. [10 marks] (iii) Argue that 3COL is NP-finished. [2 marks] 2 CST.2009.6.3 3 Computation Theory (a) What is implied by a state (or setup) of a register machine? [2 marks] (b) A register machine program Prog is said to circle at x ? N if, when begun with register R1 containing x and any remaining registers set to nothing, the arrangement of states Prog registers contains the equivalent non-stopped state at two distinct times. (I) At which x does the accompanying project circle? R HALT _ 0 [2 marks] (ii) Show that in the event that Prog circles at x, the calculation of Prog doesn't stop when begun with register R1 containing x and any remaining registers set to zero. Is the opposite evident? [4 marks] (iii) Consider the set S = {he, xi | Proge circles at x} of codes of sets of numbers (e, x) to such an extent that the register machine program Proge with file e circles at x. By adjusting the typical confirmation of undecidability of the ending issue, or in any case, show that S is an undecidable arrangement of numbers. [Answer: assuming M were a register machine that chose enrollment of S, first think about supplanting each HALT guidance (and each leap to a name with no guidance) with the program to some extent (i).] [12 marks] 3 (TURN OVER) CST.2009.6.4 4 Computation Theory (a) Define how it affects a subset S ? N to be a recursively enumerable arrangement of numbers. [2 marks] (b) Show that if S and S 0 are recursively enumerable arrangements of numbers, then, at that point, so are the accompanying sets (where hx, yi = 2x (2y + 1) ? 1). (I) S1 = {x | x ? S or x ? S 0 } (ii) S2 = {hx, x0 I | x ? S and x 0 ? S 0 } (iii) S3 = {x | hx, x0 I ? S for some x 0 ? N} (iv) S4 = {x | x ? S and x ? S 0 } Any standard outcomes about halfway recursive capacities you use ought to be plainly expressed, however need not be demonstrated. [16 marks] (c) Give an illustration of a subset S ? N that isn't recursively enumerable. [2 marks] 4 CST.2009.6.5 5 Foundations of Functional Programming (a) Define the Church numerals giving the encodings of zero 0, one 1 and an erratic number n. [3 marks] (b) Define ?-terms to play out the accompanying procedure on Church numerals. You may accept standard definitions for Booleans (valid, bogus, if, and, and additionally) and matches (match, fst, and snd). For each part, you might accept answers for the past pieces of the inquiry. You may not utilize a fixed-point combinator. (I) Test for nothing. [2 marks] (ii) Successor. [2 marks] (iii) Predecessor (where ancestor of zero will be zero). [4 marks] (iv) Less than or equivalent. [3 marks] (v) Equality. [2 marks] (vi) Successor modulus n (where succn n m = 0 if n = m + 1, and succn n m = m + 1 in any case). [2 marks] (vii)Modulus (e.g mod n m = m mod n).

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