Knights, Knaves, and Logical Reasoning
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1 Knights, Knaves, and Logical Reasoning Mechanising the Laws of Thought Fabio Papacchini 1 8 March Special thanks to Francis Southern F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
2 Introduction Thinking Formalising Modelling Computing F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
3 Thinking F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
4 A Puzzle You are on a strange island where people are divided into Knights always saying the truth Knaves always saying lies You meet two natives of the island Alice and Bob, and ask them Are you knights or knaves? Alice answers At least one of us is a knave What are Alice and Bob? F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
5 Alice: At least one of us is a knave Alice Bob Alice Bob Alice Bob Alice Bob F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
6 Alice: At least one of us is a knave Alice Bob Alice Bob Alice Bob Alice Bob F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
7 Alice: At least one of us is a knave Alice Bob Alice Bob Alice Bob Alice Bob F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
8 Formalising F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
9 Formalising Correct Reasoning A: Socrates is a man B: All men are mortal C: All men are Socrates C: Socrates is mortal F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
10 Formalising Correct Reasoning A: Socrates is a man B: All men are mortal C: All men are Socrates C: Socrates is mortal Woody Allen - Life and Death Aristotle F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
11 Formalising Correct Reasoning A: Socrates is a man B: All men are mortal C: All men are Socrates C: Socrates is mortal Woody Allen - Life and Death Aristotle Linguistic, philosophical, or mathematical approaches to formalisation Today: Propositional Logic F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
12 Propositions An expression which is either true or false. F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
13 Propositions An expression which is either true or false. Proposition test: Is it true that...? = 5 Manchester Grass is green We re in Manchester What s your name? It s raining F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
14 Not, And & Not p p F T T F It s not raining Grass is not green. F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
15 Not, And & Not p p F T T F And p q p & q F F F F T F T F F T T T It s not raining Grass is not green. Grass is green and it s raining. We re in Manchester and we re in France. F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
16 Or, Implication (If, then) Or p q p q F F F F T T T F T T T T Take an aspirin or lie down. You can have milk or sugar in your tea. F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
17 Or, Implication (If, then) Or p q p q F F F F T T T F T T T T Implication p q p q F F T F T T T F F T T T Take an aspirin or lie down. You can have milk or sugar in your tea. If you get 90% on this assignment, then you ll pass the course. If you re late, then you ll give me a fiver. F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
18 Biimplication (If and only if) Biimplication p q p q F F T F T F T F F T T T I ll buy you a new wallet if (and only if) you need one. He studies if (and only if) he can. F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
19 An Example: (p & q) r p q r (p & q) (p & q) r F F F F F T F T F F T T T F F T F T T T F T T T F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
20 An Example: (p & q) r p q r (p & q) (p & q) r F F F F F F T F F T F F F T T F T F F F T F T F T T F T T T F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
21 An Example: (p & q) r p q r (p & q) (p & q) r F F F F F F T F F T F F F T T F T F F F T F T F T T F T T T T T F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
22 An Example: (p & q) r p q r (p & q) (p & q) r F F F F T F F T F T F T F F T F T T F T T F F F T T F T F T T T F T T T T T F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
23 An Example: (p & q) r p q r (p & q) (p & q) r F F F F T F F T F T F T F F T F T T F T T F F F T T F T F T T T F T F T T T T F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
24 An Example: (p & q) r p q r (p & q) (p & q) r F F F F T F F T F T F T F F T F T T F T T F F F T T F T F T T T F T F T T T T T F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
25 Modelling F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
26 The Trick k A = Alice is a knight k A = Alice is a knave Alice says X is the same as k A X. F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
27 The Trick k A = Alice is a knight k A = Alice is a knave Alice says X is the same as k A X. Alice says at least one of us is a knave I m a knave or Bob is a knave k A k B k A ( k A k B ) F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
28 The Trick k A = Alice is a knight k A = Alice is a knave Alice says X is the same as k A X. Alice says at least one of us is a knave I m a knave or Bob is a knave k A k B k A ( k A k B ) k A k B k A k B k A k B k A ( k A k B ) F F T T T F T T F T T F F T T T T F F F F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
29 The Trick k A = Alice is a knight k A = Alice is a knave Alice says X is the same as k A X. Alice says at least one of us is a knave I m a knave or Bob is a knave k A k B k A ( k A k B ) k A k B k A k B k A k B k A ( k A k B ) F F T T T F F T T F T T F F T T T T F F F F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
30 The Trick k A = Alice is a knight k A = Alice is a knave Alice says X is the same as k A X. Alice says at least one of us is a knave I m a knave or Bob is a knave k A k B k A ( k A k B ) k A k B k A k B k A k B k A ( k A k B ) F F T T T F F T T F T F T F F T T T T F F F F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
31 The Trick k A = Alice is a knight k A = Alice is a knave Alice says X is the same as k A X. Alice says at least one of us is a knave I m a knave or Bob is a knave k A k B k A ( k A k B ) k A k B k A k B k A k B k A ( k A k B ) F F T T T F F T T F T F T F F T T T T T F F F F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
32 The Trick k A = Alice is a knight k A = Alice is a knave Alice says X is the same as k A X. Alice says at least one of us is a knave I m a knave or Bob is a knave k A k B k A ( k A k B ) k A k B k A k B k A k B k A ( k A k B ) F F T T T F F T T F T F T F F T T T T T F F F F F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
33 From Solving to Modelling Alice Alice: At least one of us is a knave Bob Alice Bob Alice Bob Alice Bob F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
34 From Solving to Modelling Alice: At least one of us is a knave k A = Alice is a knight The trick: Alice says X is the same as k A X At least one of us is a knave = k A k B Alice says At least one of us is a knave = k A ( k A k B ) F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
35 From Solving to Modelling Alice: At least one of us is a knave k A = Alice is a knight The trick: Alice says X is the same as k A X At least one of us is a knave = k A k B Alice says At least one of us is a knave = k A ( k A k B ) Can be (really) hard, but you only have to do it once! F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
36 Modelling a Sudoku What propositions do we need? Number n is in row i and column j F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
37 Modelling a Sudoku What propositions do we need? Number n is in row i and column j number 7 is in row 1 and column propositions! p (1,1,1), p (1,1,2),..., p (9,9,9) rows in a truth table atoms in the visible universe F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
38 Modelling a Sudoku What propositions do we need? Number n is in row i and column j number 7 is in row 1 and column propositions! p (1,1,1), p (1,1,2),..., p (9,9,9) rows in a truth table atoms in the visible universe What do we have to model? at least one number per cell (p (1,1,1) p (2,1,1)... p (9,1,1) ) at most one number per cell (p (1,1,1) p (2,1,1),... ) F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
39 Modelling a Sudoku What propositions do we need? Number n is in row i and column j number 7 is in row 1 and column propositions! p (1,1,1), p (1,1,2),..., p (9,9,9) rows in a truth table atoms in the visible universe What do we have to model? at least one number per cell (p (1,1,1) p (2,1,1)... p (9,1,1) ) at most one number per cell (p (1,1,1) p (2,1,1),... ) no number can be repeated in a row F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
40 Modelling a Sudoku What propositions do we need? Number n is in row i and column j number 7 is in row 1 and column propositions! p (1,1,1), p (1,1,2),..., p (9,9,9) rows in a truth table atoms in the visible universe What do we have to model? at least one number per cell (p (1,1,1) p (2,1,1)... p (9,1,1) ) at most one number per cell (p (1,1,1) p (2,1,1),... ) no number can be repeated in a row/column F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
41 Modelling a Sudoku What propositions do we need? Number n is in row i and column j number 7 is in row 1 and column propositions! p (1,1,1), p (1,1,2),..., p (9,9,9) rows in a truth table atoms in the visible universe What do we have to model? at least one number per cell (p (1,1,1) p (2,1,1)... p (9,1,1) ) at most one number per cell (p (1,1,1) p (2,1,1),... ) no number can be repeated in a row/column/region F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
42 Computing F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
43 Automating the Process Truth table mechanical time consuming (2 n rows!) tedious F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
44 Automating the Process Truth table mechanical time consuming (2 n rows!) tedious Let a computer do it for you! ideal for mechanical tasks only needs an input formula much faster than us the output is easily customisable F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
45 Automated Reasoning Much more than solving puzzles! software and hardware verification Intel and Microsoft information management biomedical ontologies, Semantic Web, databases combinatorial reasoning constraint satisfaction, planning, scheduling Internet security theorem proving in mathematics F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
46 Where Could Have Been Used Ariane 5 rocket failure due to a software bug, cost $370 million. F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
47 Where Has Been Used To find and fix a bug in a widely used sorting algorithm! F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
48 Where Has Been Used To find and fix a bug in a widely used sorting algorithm! Even Amazon and Facebook use automated reasoning techniques! F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
49 Automated Reasoning Competitions The CADE ATP System Competition (CASC) OWL Reasoning Competition (ORE) SAT-Race F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
50 Do You Want to Know More? Feel free to ask questions or look at the references on the handout! F. Papacchini Knights, Knaves, and Logical Reasoning 8 March / 23
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