Foundations of Computation
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1 The Austalian National University Semester 2, 2018 Research School of Comuter Science Assignment 1 Dirk Pattinson Foundations of Comutation Released: Tue Aug Due: Tue Se (any time) Mode: individual submissions only Submit: hard coy with cover sheet, ground floor CSIT building Question 1 Truth Tables [ credits] a) For each of the following roositions, use truth tables to determine whether the roosition is valid, or is a contradiction, or is a contingency In each case, show the entire truth table and state clearly whether the roosition is valid, or is a contradiction, or is a contingency 1 2 a b a b 3 ( ) 1 is a contradiction T F F F T F 2 a b a b is a contingency a b a b a b a b a b T T T T T T F T F F F T T F F F F F F T 3 ( ) is valid ( ) T T T T T F T T F T F T F F T T b) Consider the truth value assignment v that assigns the following truth values to the atomic roositions, and r: v() = F, v() = T, v(r) = T Which of the following formulae evaluate to T under the assignment v, ie when the truth values of, and r are given according to v? Show the line of the truth table that justifies your answer 1 ( ) (r ) 2 ( ) ( r) 3 ( ) r 4 ( r) 1 and 4 evaluate to true 1
2 For 1: r r (r ) ( ) (r ) F T T F T T F T For 4: r r r r ( r F T T T F F F T c) Consider the boolean function given by the following truth table: x y z f(x, y, z) F F F T F F T F F T F F F T T T x y z f(x, y, z) T F F F T F T T T T F T T T T T Give a formula (in variables x, y and z) that reresents the boolean function given above Briefly argue why the formula indeed reresents the boolean function We look at those rows of the truth table where the function evaluates to T and obtain: ( x y z) ( x y z) (x y z) (x y z) (x y z) This formula reresents the boolean function, because: every row in the truth table for which the formula evaluates to T corresonds recisely to a (conjunctive) term in the formula that evaluates to T under the assignment in the corresonding row taking the disjunction (or) covers recisely all ossibilities where the function evaluates to T (Parts b and c were art of the 2017 exam) Question 2 First Order Secification [10 credits] This uestion discusses a tram network, which consists of a number of segments Segments consist of a number of routes Each route is used to service a number of stos The following redicates are given: S(x) - x runs to schedule L(x) - x runs late B(x, y) - x belongs to y T (x) - x is serviced within 2 minutes of the scheduled time The following sentences describe the timeliness of the tram network and its comonents Translate each of them into First-Order Logic: a) The tram route r runs to schedule if every sto belonging to the route is serviced within 2 minutes of the scheduled time 2
3 b) The tram segment s runs late if some tram route in the segment does not run to schedule c) The tram network n runs to schedule unless some segment belonging to the network runs late 1 ( sb(s, r) T (s)) S(r) 2 ( rb(r, s) S(r)) L(s) 3 ( sb(s, n) L(s)) S(n) Question 3 Natural Deduction [ credits] In the following uestions, rove the derived rules using natural deduction You may only use the rules given in the Aendix Do not use algebraic laws, or any of the derived rules obtained in lectures Number each line and include justifications for each ste in your roofs a) Give a natural deduction roof of the derived rule 1 2 ( ) 3 ( ) -I, F E, 4, 5 8 PC, I, ( ) -I, 9 11 ( ) -E, 1, 2 3, 4 10 b) Give a natural deduction roof of a b (a b) 3
4 1 a b 2 a -E, 1 3 b -E, 1 4 a b 5 a 6 F -E, 5, 2 7 b 8 F -E, 7, 3 9 F -E, 4, 5 6, (a b) PC, a b (a b) -I, 1 10 c) Give a natural deduction roof of the derived rule 1 ( yq(y) T (y)) ( yq(y) T (y)) xq(x) T (x) 2 a Q(a) 3 T (a) 4 Q(a) T (a) -I, 2, 3 5 yq(y) T (y) -I, 4 6 F E, 5, 1 7 T (a) PC, Q(a) T (a) -I, xq(x) T (x) -I, 8 d) Give a natural deduction roof of x(p (x) yp (y)) 1 a P (a) 2 yp (y) -I, 1 3 P (a) yp (y) -I, x(p (x) yp (y)) -I, 3 e) Give a natural deduction roof of z(( xp (x) Q(x)) Q(z) P (z)) 4
5 1 a ( xp (x) Q(x)) Q(a) 2 ( xp (x) Q(x)) -E, 1 3 Q(a) -E, 1 4 P (a) 5 P (a) Q(a) -E, 2 6 Q(a) -E, 5, 4 7 F -E, 3, 6 8 P (a) PC, xp (x) > Q(x)) Q(a) P (a) -E, z( xp (x) > Q(x)) Q(z) P (z)) -I, 9 (Parts b, d and e were art of the 2017 exam) 5
6 Aendix 1 Natural Deduction Rules Proositional Calculus ( I) ( E) [] [] ( I) ( E) r r r [] ( I) ( E) [] ( I) F ( E) F [ ] (PC) F (T ) T Predicate Calculus ( I) P (a) (a arbitrary) x P (x) ( E) x P (x) P (a) ( I) P (a) xp (x) [P (a)] ( E) xp (x) (a is not free in ) (a arbitrary) 6
7 Aendix 2 Truth Table Values T T T T T F T T F T F F F F F T T F T T F F F F F T T T 7
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