Proofs Propositions and Calculuses

Transcription

1 Lecture 2 CS 1813 Discrete Mathematics Proofs Propositions and Calculuses 1

2 City of Königsberg (Kaliningrad) 2

3 Bridges of Königsberg Problem Find a route that crosses each bridge exactly once Must the route have a matching exit to each land mass for each entry? For a particular land mass, can the total number of entries and exits be and odd number? Can there be an odd number of bridges to a land mass with an even entry/exit count? So, there must be either no land masses with an odd number of bridges, or exactly two. Right? How many land masses have an odd number of bridges? TILT 3 o, 1997 Adapted from Singh, Fermat s Enigma, Walker & C

4 Tracing a Square and Its Diagonals Square + Diagonals Problem Start at any corner Trace some line to another corner Then trace from that corner to another Keep going until all six lines are traced Don t trace any line more than once If it s possible to do such a tracing, how many corners will have an odd number of lines emanating from them? Euler solved the bridges of None. Or two, right? Königsberg problem in the 1700s. The field of graph theory includes problems of this kind. TILT 4

5 Logic a tool for careful reasoning Software and hardware development Formal logic can lead to defect-free products Used in chip design (since Pentium disaster) Used in safety-critical software Used in network-secure software Positive influence on all software development Foundation for programming languages Lambda calculus semantic base for all PLs Type checking and type inference basis in logic Computation models limits of computabilty Artificial intelligence, database systems Grounded in formal logic 5

6 Starting Here Propositional Logic Proposition atomic entity, true or false Capital letters denote propositional variables Standing for specific propositions P, Q, specific propositions - value: true or false Lowercase letters are metavariables Denoting propositional formulas a, b, value: formula, such as P, (P Q), Calculus a method of reasoning by computation of symbols Propositional Calculus Scheme for calculating with logic formulas Three different Propositional Calculuses Semantics-based reasoning truth tables Syntax-based reasoning inference rules Equational reasoning Boolean algebra What does this mean? all three consistent 6

7 Logical And Logical Or Logical Operators P Q, (P Q) R, P Q, P (Q R), Logical Not P, (( P) Q), Logical Implication P Q, (P Q) Q, Logical Equivalence P Q, (P Q) (Q P) Exclusive Or P Q, (P Q) R, Logical operators provide ways to combine propositions for calculation 7

8 Truth Table Logical And P Q P Q Pronounced if and only if iff both operands are Define P = x > 0, Q = x < 10 P Q is iff x is between 0 and 10 8

9 Truth Table Logical Or P Q P Q iff either operand is Define P = x < 0, Q = x > 10 P Q is iff x is outside 0 to 10 This is the inclusive or Usually and/or in English 9

10 Truth Table Logical Not P P iff operand is Define P = x < 0, Q = x > 10 P is iff x is non-negative (P Q) is iff x is between 0 and 10 10

11 Truth Table Logical Implication P Q P Q iff 1 st operand is and 2 nd Define P = x > 10, Q = x > 0 Consider x = 15, x = 5, and x = -5 P Q is, regardless of x s value That is, third line in table does not occur Q P is when x is between 0 and 10 So, line 3 in table sometimes applies to this formula This confirms intuition about inference We expect to infer Q, knowing P (P Q is ) But not vice versa (cannot always infer P, knowing Q) 11

12 Truth Table Logical Equivalence P Q P Q iff both operands have the same value P Q has same value as (P Q) (Q P) 12

13 Truth Table Exclusive Or P Q P Q iff operands have different values Define P = x > 0, Q = y > 0 Quadrant 2 x < 0, y > 0 Quadrant 3 x < 0, y < 0 P Q is iff (x, y) is in 2 nd or 4 th quadrant P Q also has same value as (P Q) y Quadrant 1 x > 0, y > 0 x Quadrant 4 x > 0, y < 0 13

14 Well-Formed Formulas WFFs the syntax of logic formulas Classes of WFFs atomic formulas Any propositional variable ( p) if p is a WFF (p q) if p and q are WFFs (p q) if p and q are WFFs (p q) if p and q are WFFs (p q) if p and q are WFFs (p q) if p and q are WFFs formulas with connectives (not atomic) This syntax defines WFFs Inductive definition P, Q, Examples ( P), ( Q), ( ( P)), (P Q), (( P) (P Q)), (P Q), ((P Q) ( Q)), (P Q), ((P Q) (P Q)), (P Q), ((P Q) (Q P)), (P Q), (( P) Q), In case of missing parens: Precedence: Association: P (Q R) 14

15 Confirming WFFness by Analyzing Constituents WFF? ((P Q) (( P) Q)) Constituents (must be WFFS) (P Q) (( P) Q) Constituents (of constituents) P Q ( P) Q Constituents (of constituents of constituents) P Yes, it s a WFF constituents match WFF patterns all the way down to atomic formulas 15

16 Constituent Analysis Can Confirm non-wffness, Too WFF? (( P ( ( Q))) (( P) Q)) Constituents (must be WFFS) (P ( ( Q))) (( P) Q) Constituents (of constituents) P ( ( Q)) ( P) Q Constituents (of constituents of constituents) ( Q) Whoops! Q not a WFF No WFF begins with P TILT 16

17 Semantic Reasoning with Truth Tables Proposition (WFF): ((P Q) (( P) Q)) P Q (P Q) ( P) (( P) Q) ((P Q) (( P) Q)) If prop is when all variables are: P, Q ((P Q) (( P) Q)) Formal statement about meaning Proved via truth table double turnstile Some : prop is Satisfiable If they were all : Tautology All : Contradiction (not satisfiable) 17

18 Another Truth Table Proposition (WFF): ((( P) Q) (( P) Q)) P Q ( P) (( P) Q) (( P) Q) ( ((( P) Q) (( P) Q)) Prop is when all variables are P, Q (( P) Q) (( P) Q)) 18

19 Why Reasoning with Truth Tables Is Infeasible Works fine when there are 2 variables {T,F} {T,F} = set of potential values of variables 2 2 lines in truth table Three variables starts to get tedious {T,F} {T,F} {T,F} = set of potential values lines in truth table Twenty variables definitely out of hand lines (2 20 ) You want to look at a million lines? If you did, how would you avoid making errors? Hundreds of variables not in a million years 19

20 End of Lecture 20