Knights, Knaves, and Logical Reasoning Mechanising the Laws of Thought
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1 Knights, Knaves, and Logical Reasoning Mechanising the Laws of Thought Who? From? Fabio Papacchini and Francis Southern The University of Manchester When? 3rd December, 2014
2 Introduction Thinking Formalising Modelling Computing
3 A Puzzle You have decided to take a trip to strange far-off island where the native people have the unusual tradition of dividing themselves into two castes: knights and knaves. Knights are forbidden from ever telling a lie, and knaves are forbidden from ever telling the truth. Both male and female natives are described as knights and knaves.
4 A Puzzle You have decided to take a trip to strange far-off island where the native people have the unusual tradition of dividing themselves into two castes: knights and knaves. Knights are forbidden from ever telling a lie, and knaves are forbidden from ever telling the truth. Both male and female natives are described as knights and knaves. Now, upon your arrival you are greeted by two natives of the island (called A and B) and, eager to learn more about their customs, you ask Are you knights or knaves?
5 A Puzzle You have decided to take a trip to strange far-off island where the native people have the unusual tradition of dividing themselves into two castes: knights and knaves. Knights are forbidden from ever telling a lie, and knaves are forbidden from ever telling the truth. Both male and female natives are described as knights and knaves. Now, upon your arrival you are greeted by two natives of the island (called A and B) and, eager to learn more about their customs, you ask Are you knights or knaves? The first native A replies At least one of us is a knave. What are you to understand from this?
6 Mathematical(?) logic
7 Sudoku
8 Sudoku If there isn t a 7 in this row, and there isn t a 7 in this column, and there isn t a 7 in this square, then you can put a 7 in this box.
9 Sudoku If there isn t a 7 in this row, and there isn t a 7 in this column, and there isn t a 7 in this square, then you can put a 7 in this box. This box must contain a 7 or a 9.
10 Propositions An expression which is either true or false.
11 Propositions An expression which is either true or false. Proposition test: Is it true that...?
12 Propositions An expression which is either true or false. Proposition test: Is it true that...? = 5
13 Propositions An expression which is either true or false. Proposition test: Is it true that...? = 5 Manchester
14 Propositions An expression which is either true or false. Proposition test: Is it true that...? = 5 Manchester Grass is green
15 Propositions An expression which is either true or false. Proposition test: Is it true that...? = 5 Manchester Grass is green We re in Manchester
16 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?
17 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
18 Not p p F T T F
19 Not p p F T T F It s not raining.
20 Not p p F T T F It s not raining. Grass is not green.
21 And & p q p&q F F F F T F T F F T T T
22 And & p q p&q F F F F T F T F F T T T Grass is green and it s raining.
23 And & p q p&q F F F F T F T F F T T T Grass is green and it s raining. We re in Manchester and we re in France.
24 Or p q p q F F F F T T T F T T T T
25 Or p q p q F F F F T T T F T T T T Take an aspirin or lie down.
26 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.
27 Implication - If, then p q p q F F T F T T T F F T T T
28 Implication - If, then p q p q F F T F T T T F F T T T If you get 90% on this assignment, then you ll pass the course.
29 Implication - If, then p q p q F F T F T T T F F T T T If you get 90% on this assignment, then you ll pass the course. If you re late, then you ll give me a fiver.
30 Biimplication - If and only if p q p q F F T F T F T F F T T T
31 Biimplication - If and only if 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.
32 Biimplication - If and only if 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.
33 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
34 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
35 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
36 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
37 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
38 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
39 The Trick k A means A is a knight.
40 The Trick k A means A is a knight. k A means A is a knave.
41 The Trick k A means A is a knight. k A means A is a knave. A said X is the same as k A X.
42 The Trick k A means A is a knight. k A means A is a knave. A said X is the same as k A X. Draw a truth table for this and find a satisfying assignment. (A row where the final column contains T.)
43 A Solution At least one of us is a knave.
44 A Solution At least one of us is a knave. k A means A is a knight. k A means A is a knave.
45 A Solution At least one of us is a knave. k A means A is a knight. k A means A is a knave. 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
46 A Solution At least one of us is a knave. k A means A is a knight. k A means A is a knave. 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
47 A Solution At least one of us is a knave. k A means A is a knight. k A means A is a knave. 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
48 A Solution At least one of us is a knave. k A means A is a knight. k A means A is a knave. 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
49 A Solution At least one of us is a knave. k A means A is a knight. k A means A is a knave. 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
50 Automating the Process Truth table mechanical time consuming (2 n rows!) tedious
51 Automating the Process Truth table mechanical time consuming (2 n rows!) tedious Let a computer do it for you!
52 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
53 From Solving to Modelling Computers solve the puzzle (part of the fun is gone)
54 From Solving to Modelling Computers solve the puzzle (part of the fun is gone) Our contribution is still fundamental! finding the right procedure (hopefully a fast one) changing focus: Solving Modelling A says At least one of us is a knave. k A k A k B
55 From Solving to Modelling Computers solve the puzzle (part of the fun is gone) Our contribution is still fundamental! finding the right procedure (hopefully a fast one) changing focus: Solving Modelling A 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!
56 Modelling a Sudoku What propositions do we need?
57 Modelling a Sudoku What propositions do we need? Number n is in row i and column j
58 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 4 number 2 is in row 6 and column 7
59 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 4 number 2 is in row 6 and column propositions!
60 Modelling a Sudoku (cont d) What to model at least one number per cell at most one number per cell
61 Modelling a Sudoku (cont d) What to model at least one number per cell at most one number per cell no number can be repeated in a row
62 Modelling a Sudoku (cont d) What to model at least one number per cell at most one number per cell no number can be repeated in a row no number can be repeated in a column
63 Modelling a Sudoku (cont d) What to model at least one number per cell at most one number per cell no number can be repeated in a row no number can be repeated in a column no number can be repeated in a region
64 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
65 Where Could Have Been Used Software bug in Therac-25 a radiation therapy machine led to the death of six patients.
66 Where Could Have Been Used (cont d) Ariane 5 rocket failure due to a software bug, cost $370 million.
67 Automated Reasoning Competitions The CADE ATP System Competition (CASC) OWL Reasoning Competition (ORE) SAT-Race
68 Automated Reasoning Competitions The CADE ATP System Competition (CASC) OWL Reasoning Competition (ORE) SAT-Race You can bet on the winner!
69 Do You Want to Know More? Look at the references on the handout!
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