Informatics 1 - Computation & Logic: Tutorial 1

Transcription

1 Informatics 1 - Computation & Logic: Tutorial 1 Propositional Logic: An Introduction Week 3: 5 9 October 2015 Please attempt the entire worksheet in advance of the tutorial, and bring with ou all work, including (if a computer is involved) printouts of code and test results. Tutorials cannot function properl unless ou do the work in advance. You ma work with others, but ou must understand the work; ou can t phone a friend during the eam. Assessment is formative, meaning that marks from coursework do not contribute to the final mark. But coursework is not optional. If ou do not do the coursework ou are unlikel to pass the eams. Attendance at tutorials is obligator; please let our tutor know if ou cannot attend. 1

2 Eercise There are 8 regions in the diagram. How man subsets of this set of 8 regions are there? Given an subset of the eight regions can ou write a comple proposition to which it corresponds (using and, or, and not as connectives)? 17 You ma want to come back to this question after ou have completed the rest of the tutorial. 2

3 Eercise 1.2 current net R = R or A A = G or (R and not A) G = R and A Slide 25 (lecture 1) shows an implementation of the traffic light controller. We could have designed our logic differentl. For eample, letting A = G or (R and not A). Draw the circuit for this implementation. Is this a correct implementation of the controller? Eplain our answer. 3

4 Eercise 1.3 A _ B A! B A B A ^ B Each of the tables above represents the truth table of a binar boolean operation. Label each table with a boolean epression for which it is the truth table (five tables are alread labelled begin b checking whether these are correct). How man of the binar operations actuall depend on both variables? How man depend on onl one variable? How man depend on no variables? 4

5 Eercise 1.4 R RA R RA A G A G 4. As discussed in the lecture, the diagram represents the beginnings of a refinement of our description of the traffic light controller. We model a sensor that detects a car read to pass the light. For each state of the lights, (R, RA, G, A) we have two states, one (with a double circle) where there is a car, and the other, without a car, as before. Draw arrows to indicate state changes that still obe the correct sequence for the lights, but also respect the following two rules. 1. A car can onl pass the light if it is green. 2. The light onl changes from red to red-amber when a car is detected Optional: You ma also design the logic for the controller. Use a new boolean variable C to represent the presence of a car, and give equations for R A and G. Should we also give an equation for C? 29 5

6 5. Here are some propositional smbols, together with the English sentences the represent: A Sam Mendes directed Skfall B Leonardo DiCaprio starred in Skfall C Leonardo DiCaprio starred in Django Unchained D Quentin Tarantino is a director E Leonardo DiCaprio is an actress F Judi Dench is an actress G Judi Dench acted in Skfall H Leonardo DiCaprio was married to Judi Dench J Django Unchained was released in 2012 K Skfall is set in 16th centur Scotland L Leonardo DiCaprio is a woman M Judi Dench used to be married N Leonardo DiCaprio is an actor Ever epression of propositional logic is either true or false, and no epression can be both true and false. Based on the relationship between propositional smbols and English sentences above, our own general knowledge, and, if need be, the Internet Movie Database, decide whether each proposition is true or false. 6. Assume the propositional smbols in Question 5. Assume also that: (a) the smbol represents the negation operator not ; (b) the smbol represents the conjunction connective and ; (c) the smbol represents the disjunction connective or ; (d) the smbol represents the implication connective; and (e) the smbol represents the equivalence connective. Some of the following are well-formed epressions of propositional logic and the others are smbol soup. Decide which is which. (a) A C (b) (F D) (c) (N B) (d) (G L) E (e) A (C H) (f) (K B) (g) F D (h) H (A C) 6

7 7. Translate the following epressions of propositional logic into reasonabl natural English, assuming the ke in Question 5: (a) E B (b) J K (c) E L (d) (C L) N 7

8 8. Translate the following English sentences into propositional logic, using the appropriate propositional smbols from Question 5: (a) Leonardo DiCaprio and Judi Dench both starred in Skfall (b) If Leonardo DiCaprio was Judi Dench s husband, then Judi Dench was married and Leonardo DiCaprio is not an actress (c) Skfall did not star either Judi Dench or Leonardo DiCaprio (d) If Leonardo DiCaprio is a woman, then he isn t an actor and isn t married to Judi Dench 9. The truth or falsit of a comple epression of propositional logic is a function of the truth/falsit of the propositional smbols it is consists of. Based on the answers ou gave in Question 5, and our knowledge of the truth tables for negation, conjunction, disjunction, implication and equivalence, work out whether the following epressions are true or false: (a) E B (b) (C L) N (c) (H M) (M H) (d) (K L) N 10. Consider the epression (H M). Work out whether this epression is true or false. Can ou eplain wh this epression is true/false, considering its English translation and the true/false values of the literals? This tutorial eercise sheet was written b Mark McConville, revised b Paolo Besana, Thomas French, Areti Manataki, and Michael Fourman. Send comments to michael.fourman@ed.ac.uk 8

9 Truth tables of the basic operators A B A B T T T T F F F T F F F F A B A B T T T T F T F T T F F F A B A B T T T T F F F T T F F T We can also epress the truth tables for binar operations in a different stle, which makes the smmetries more immediatel apparent