Translating English Sentences • Steps to convert an English sentence to a statement in propositional logic ○ Identify atomic propositions and represent using propositional variables. ○ Determine appropriate logical connectives • Example: “If I go to Harry’s or to the country, I will not go shopping.” ○ p: “I go to Harry’s.” ○ q: “I go to the country.” ○ r: “I will go shopping.” ○ p∨q→¬r • Example: “You can get an extra piece of pie if you have completed your homework or if you are extremely hungry” ○ p: You have completed your homework ○ q: You are extremely hungry ○ r: You can get an extra piece of pie ○ p∨q→r System Specifications • System and Software engineers take requirements in English and express them in a precise specification language based on logic. • Example: “The automated reply cannot be sent when the file system is full” ○ p: “The automated reply can be sent” ○ q: “The file system is full.” ○ q→¬ p Logic Puzzles • An island has two kinds of inhabitants, knights, who always tell the truth, and knaves, who always lie. • You go to the island and meet A and B. ○ A says “B is a knight.” ○ B says “The two of us are of opposite types.” • What are the types of A and B? ○ Let p and q be the statements that A is a knight and B is a knight, respectively. ○ So, then Øp represents the proposition that A is a knave and Øq that B is a knave. ○ If A is a knight, then p is true. Since knights tell the truth, q must also be true. Then (p ∧Øq)∨(Øp∧q) would have to be true, but it is not. So, A is not a knight and therefore Øp must be true. ○ If A is a knave, then B must not be a knight since knaves always lie. So, then both Øp and Øq hold since both are knaves. Logic Circuits • Electronic circuits; each input/output signal can be viewed as a 0 or 1. ○ 0 represents False ○ 1 represents True • Complicated circuits are constructed from three basic circuits called gates. ○ The inverter (NOT gate) takes an input bit and produces the negation of that bit. ○ The OR gate takes two input bits and produces the value equivalent to the disjunction of the two bits. ○ The AND gate takes two input bits and produces the value equivalent to the conjunction of the two bits. • More complicated digital circuits can be constructed by combining these basic circuits to produce the desired output given the input signals by building a circuit for each piece of the output expression and then combining them.
