UNIVERSITY OF EDINBURGH COLLEGE OF SCIENCE AND ENGINEERING SCHOOL OF INFORMATICS INFORMATICS 1 COMPUTATION & LOGIC INSTRUCTIONS TO CANDIDATES

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1 UNIVERSITY OF EDINBURGH COLLEGE OF SCIENCE AND ENGINEERING SCHOOL OF INFORMATICS INFORMATICS COMPUTATION & LOGIC Sturdy st April 7 : to : INSTRUCTIONS TO CANDIDATES This is tke-home exercise. It will not e mrked nonymously. The exmintion, which will e mrked nonymously, will hve similr formt. Bring your completed script with you to clss on Thursdy 4th Novemer. You will mrk your pper in clss nd then sumit it for feedck on ny outstnding queries. In ddition to nswering these questions you my find it useful to crete your own vritions on these exmples. Plese give your student numer nd tutoril group elow. Student ID: Tutoril group: THIS EXAMINATION WILL BE MARKED ANONYMOUSLY

2 This is tke-home exercise. The exmintion will hve similr formt. Bring your completed script with you to clss on Thursdy 4th Novemer.. () The entilment B A C, is invlid. B A C A B C Vlid Invlid How is this invlidity shown y compring the two Venn digrms ove? [4 mrks] () For ech of the following entilments, complete the two Venn digrms to represent the ssumption nd conclusion, nd plce mrk in one of the check oxes provided to indicte whether the entiment is vlid. (You should use the sme encoding s in the exmple ove, where ech circle represents one of the propositions A, B, C.) i. (B C) A C A [8 mrks] (B C) A C A (B C) A C A Vlid Invlid ii. (B C) (B C) A [8 mrks] B C (B C) A (B C) (B C) A Vlid Invlid Pge of 6

3 . This question concerns the 56 possile truth vlutions of the following eight propositionl letters A, B, C, D, E, F, G, H. For ech of the following expressions, sy how mny of the 56 vlutions stisfy the expression, nd riefly explin your resoning. For exmple, the expression D is stisfied y hlf of the vlutions, tht is 8 of the 56, since for ech vlution tht mkes D true there is mtching vlution tht mke D flse. () A B [ mrk] () (A B) C [ mrks] (c) (A B) C [ mrks] (d) (A B) (B A) (C D) (D F ) (E F ) (F G) (G H) [5 mrks] (e) (A B) (B C) (C D) (D C) (E F ) (F G) (F H) [5 mrks] (f) (H D) (A B) (B D) (C E) (D F ) (F (H G) [5 mrks] Pge of 6

4 3. You re given the following inference rules: (Γ, vry over finite sets of expressions; A, B vry over expressions): Γ, A, A (I) Γ, A, B Γ, A B ( L) Γ A, B, Γ A B, ( R) Γ, A Γ, B Γ, A B ( L) Γ A, Γ B, Γ A B, ( R) Γ A, Γ, B Γ, A B ( L) Γ, A B, Γ A B, ( R) Γ A, Γ, A ( L) Γ, A Γ A, ( R) (Where A nd B re propositionl expressions, Γ, re sets of expressions, nd Γ, A refers to Γ {A}.) This question concerns the use of these rules to prove the following entilment. This is your gol. S (P Q), R (Q P ) Q (R S) () () Which of these rules hve conclusion mtching the gol ()? For ech such rule complete line in the tle elow showing the nme of the rule nd the expressions mtched with Γ,, A, B [ mrks] Rule Γ A B Pge 3 of 6

5 () Use the rules given to construct forml proof with the gol s conclusion, mking ny remining ssumptions s simple s possile. Lel ech step in your proof with the nme of the rule eing pplied. [ mrks] S (P Q), R (Q P ) Q (R S) Pge 4 of 6

6 4. () Wht is counterexmple to n entilment? [ mrks] () How cn resolution e used to determine whether clusl form is stisfile? [ mrks] (c) Use resolution either to show this entilment is vlid or to produce counterexmple: P (Q R) (P Q) (P R) P Q R Counterexmple? [ mrks] (d) Use resolution either to show this entilment is vlid or to produce counterexmple: P (R S), Q (R S) (P B) (R S) (P Q) P Q R S Counterexmple? [ mrks] (e) The resolution procedure is sound nd complete. Wht would it men to sy the resolution procedure ws i. not sound [ mrks] ii. not complete. [ mrks] Pge 5 of 6

7 5. Ech digrm shows n FSM. In ech cse give regulr expression for the lnguge ccepted y the FSM, mke mrk in the check ox ginst ech string tht it ccepts (nd no mrk ginst those strings it does not ccept), mke mrk in the DFA check ox if it is deterministic, nd drw n equivlent DFA if it is not. () () (c) (d) (e),,,,, 3 3 [4 mrks] [4 mrks] [4 mrks] [4 mrks] [4 mrks] Pge 6 of 6