Loading in 2 Seconds...

University of Aberdeen, Computing Science CS2013 Mathematics for Computing Science Adam Wyner

Loading in 2 Seconds...

- By
**afya** - Follow User

- 114 Views
- Uploaded on

Download Presentation
## University of Aberdeen, Computing Science CS2013 Mathematics for Computing Science Adam Wyner

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

**University of Aberdeen, Computing ScienceCS2013Mathematics**for Computing ScienceAdam Wyner Slides adapted from Michael P. Frank’s course based on the textDiscrete Mathematics & Its Applications(5th Edition)by Kenneth H. Rosen**Propositional Logic**Rosen 5th ed., §§1.1-1.2 (but much extended) ~85 slides, ~2 lectures**Foundations of Logic**Mathematical Logic is a tool for working with compound statements. It includes: • A formal language for expressing them. • A methodology for reasoning about their truth or falsity. • It is the foundation for expressing formal proofs in all branches of mathematics. Frank / van Deemter / Wyner**Two Logical Systems:**• Propositional logic • Predicate logic (extends 1. ) Many other logical `calculi` exist, but they tend to resemble these two Frank / van Deemter / Wyner**Propositional Logic (§1.1)**Propositional Logic is the logic of compound statements built from simpler statements using so-called Booleanconnectives. Some applications in computer science: • Design of digital electronic circuits. • Expressing conditions in programs. • Queries to databases & search engines. George Boole(1815-1864) Chrysippus of Soli(ca. 281 B.C. – 205 B.C.) Frank / van Deemter / Wyner**Propositions in natural language**In propositional logic, a proposition is simply: • a statement (i.e., a declarative sentence) • with some definite meaning • having a truth value that’s either true (T) or false (F). Only values statements can have. • Never both, or somewhere “in between”. However, you might not know the truth value Frank / van Deemter / Wyner**Examples of NL Propositions**• “It is raining.”(In a given situation.) • “Beijing is the capital of China, and 1 + 2 = 2” But, the following are NOT propositions: • “Who’s there?”(interrogative: no truth value) • “x := x+1”(imperative: no truth value) • “1 + 2”(term: no truth value) Frank / van Deemter / Wyner**Propositions in Propositional Logic**• Atoms: p, q, r, …(Corresponds with simple English sentences, e.g.‘I had salad for lunch’) • Complex propositions : built up from atoms using operators: pq(Corresponds with compound English sentences, e.g., “I had salad for lunch andI had steak for dinner.”) Frank / van Deemter / Wyner**Defining Propositions**• Logic defines notions of atomic and complex propositions and what complex propositions “mean”. • We explain by example, giving precise definitions. Frank / van Deemter / Wyner**Operators / Connectives**An operator or connectivecombines with noperand expressions into a larger expression. • Unary operators take 1 operand; • Binary operators take 2 operands. • Propositional or Boolean operators operate on propositions instead of on numbers (+,-). Frank / van Deemter / Wyner**Common Boolean Operators**Frank / van Deemter / Wyner**The Negation Operator**The unary negation operator“¬” (NOT) combines with one prop, transforming the prop into its negation. E.g. If p = “I have brown hair.” then ¬p = “I do not have brown hair.” The truth table for NOT: T :≡ True; F :≡ False “:≡” means “is defined as” Operandcolumn Resultcolumn Frank / van Deemter / Wyner**Truth-functionality**• Truth table expresses truth/falsity of ¬pin terms of truth/falsity of p • This not possible for the operator ‘tomorrow’, or `probably’: • ‘Tomorrow p’ is true iff p is ….’?? • ‘Probably p’ is true iff p is ….’?? Frank / van Deemter / Wyner**Truth-functionality**• Truth table expresses truth/falsity of ¬pin terms of truth/falsity of p. • Each horizontal line of the table expresses some alternative context. • Truth-functional operator: an operator that is a function from the truth values of the component expressions to a truth value. • NOT is truth functional. Yesterday is not. • Propositional logic is only about truth-functional operators. • We can compute the values of the complex expressions. Frank / van Deemter / Wyner**Comment on Truth**• Not concerned with the "meaning" of a proposition other than whether it is true or false. Not about "how" we know it is true or false, but supposing it is, what else do we know. Abstraction. • The "truth" of a proposition determined "by inspection" – The book is on the table. The "real" world. • The "truth" determined by "stipulation"- suppose The book is on the table is true. Not the "real" world. • Mostly we are stipulate truth of a proposition. The truth value of P is true/false. Frank / van Deemter / Wyner**The Conjunction Operator**The binary conjunction operator“” (AND) combines two propositions to form their logical conjunction. E.g. If p=“I will have salad for lunch.” and q=“I will have steak for dinner.”, then pq=“I will have salad for lunch andI will have steak for dinner.” Frank / van Deemter / Wyner**Conjunction Truth Table**Operand columns • Note that aconjunctionp1p2 … pnof n propositionswill have 2n rowsin its truth table. • Also: ¬ and operations together can express any Boolean truth table! more later Frank / van Deemter / Wyner**The Disjunction Operator**The binary disjunction operator“” (OR) combines two propositions to form their logical disjunction. p=“My car has a bad engine.” q=“My car has a bad carburator.” pq=“Either my car has a bad engine, ormy car has a bad carburetor.” Frank / van Deemter / Wyner**Disjunction Truth Table**• Note that pq meansthat p is true, or q istrue, or both are true! • So, this operation isalso called inclusive or,because it includes thepossibility that both p and q are true. • “¬” and “” together are also universal. Notedifferencefrom AND Frank / van Deemter / Wyner**Nested Propositional Expressions**• Use parentheses to group sub-expressions:“I just saw my old friend, and either he’s grown or I’veshrunk.” = f (g s) (f g) s would mean something different f g s would be ambiguous • By convention, “¬” takes precedence over both “” and “”. ¬s fmeans (¬s) f rather than ¬ (s f) Frank / van Deemter / Wyner**Logic as shorthand for NL**Let p=“It rained last night”, q=“The sprinklers came on last night,”r=“The lawn was wet this morning.” ¬p = r ¬p = ¬ r p q = It didn't rain last night. The lawn was wet this morning, andit didn’t rain last night. Either the lawn wasn't wet this morning, or it rained last night, or the sprinklers came on last night. Frank / van Deemter / Wyner**Some important ideas:**• Distinguishing between different kinds of formulas • Seeing that some formulas that look different may express the same information • First: different kinds of formulas Frank / van Deemter / Wyner**Tautologies**A tautology is a compound proposition that is trueno matter what the truth values of its atomic propositions are! Ex.p p[What is its truth table?] Frank / van Deemter / Wyner**Tautologies**• When every row of the truth table gives T. • Example: p p T T FT F T TF Frank / van Deemter / Wyner**Contradictions**A contradiction is a compound proposition that is false no matter what! Ex.p p [Truth table?] Frank / van Deemter / Wyner**Contradictions**• When every row of the truth table gives F • Example: p p T F FT F F TF Frank / van Deemter / Wyner**Contingencies**All other props. are contingencies: Some rows give T, others give F Now: formulas that have the same meaning Frank / van Deemter / Wyner**Propositional Equivalence**Two syntactically (i.e., textually) different compound propositions may be semantically identical (i.e., have the same meaning). Here semantically identical means just that they have the same truth table for input truth values of the propositions. We call them logically equivalent. Notation: … … Frank / van Deemter / Wyner**Logical Equivalence**Compound proposition p is logically equivalent to compound proposition q, written pq, IFFp and q contain the same truth values in all rows of their truth tables We will also say: they express the same truth function (= the same function from values for atoms to values for the whole formula). Frank / van Deemter / Wyner**Proving Equivalencevia Truth Tables**Ex. Prove that pq (p q). F T T T F T T F F T T F T F T T F F F T Shows that OR is equivalent to a combination of NOT and AND. Frank / van Deemter / Wyner**Before introducing more connectives**• … let us step back and ask a few questions about truth tables Frank / van Deemter / Wyner**Questions for you to think about**• What does each line of the table "mean"? • Consider a conjunction p1p2 p3How many rows are there in its truth table? • Consider a conjunctionp1p2 … pn of n propositions.How many rows are there in its truth table? • Explain why ¬ and together are sufficient to express any Boolean truth table Frank / van Deemter / Wyner**Questions for you to think about**• Consider a conjunction p1p2 p3How many rows are there in its truth table? 8p1p2 p31 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 Two truth values (0,1) and three propositions: 23 = 8. Frank / van Deemter / Wyner**Questions for you to think about**2. Consider p1p2 … pnHow many rows are there in its truth table?2*2*2*… *2 (n factors)Hence 2n (This grows exponentially!) Frank / van Deemter / Wyner**Questions for you to think about**3. Explain why ¬ and together are sufficient to express anyother complex expression in propositional logic. Frank / van Deemter / Wyner**Questions for you to think about**• Explain why ¬ and together are sufficient to express any other complex expression in propositional logic. • Obviously, if we add new connectives(like ) we can write new formulas. • CLAIM: these formulas would always be equivalent with ones that only use ¬ and (This is what we need to prove). Frank / van Deemter / Wyner**Relating AND and OR**• Saying this in a different way: if we add new connectives, we can write new formulas, but these formulas will always only express truth functions that can already be expressed by formulas that only use ¬ and . • That is, they will be equivalent. • Example of writing a disjunction in another form (equivalence shown before): p q ¬(¬p ¬q) Frank / van Deemter / Wyner**Mystery Operator**PQRFormula (containing P,Q,R) 1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 Does there exist such a Formula? Suppose, given the truth values of P, Q, and R, we construct a Formula with the given resulting truth value. This is our 'mystery' operator . Can it be written equivalently with NOT and AND. Frank / van Deemter / Wyner**T-values in Conjunction**3. Explain why ¬ and together are sufficient to express any Boolean truth table • Suppose precisely two rows give T.For example, the rows where • P=T, Q=T, R=F. This is P Q ¬R • P=T, Q=F, R=T. This is P ¬Q R Frank / van Deemter / Wyner**Table as a disjunction of T-rows**Suppose precisely two rows give T.For example, the rows where • P=T, Q=T, R=F. This is P Q ¬R • P=T, Q=F, R=T. This is P ¬Q R • We’ve proven our claim if we can express the disjunction of these two rows: (P Q ¬R) (P ¬Q R) Frank / van Deemter / Wyner**Disjoining rows of the table**• We’ve arrived if we can express their disjunction: (P Q ¬R) (P ¬Q R) • But we’ve seen that disjunction can be expressed using and ¬: AB ¬(¬A ¬B) • So: (P Q ¬R) (P ¬Q R) ¬(¬ (P Q ¬R) (P ¬Q R)) • We’ve only used and . Frank / van Deemter / Wyner**Check**(P Q ¬R) (P ¬Q R) ¬(¬ (P Q ¬R) (P ¬Q R)) 1 1 0 01 0 1 01 0 1 0 1 1 1 0 01 1 1 1 01 0 1 1 1 1 10 1 1 01 0 0 1 0 1 1 1 10 0 1 1 01 0 0 1 00 01 1 1 10 1 1 1 1 1 0 0 01 0 0 1 10 1 1 1 0 0 10 0 1 10 0 0 0 1 1 0 0 10 1 1 1 10 0 0 0 1 0 01 0 0 01 0 1 0 1 0 1 0 01 1 1 0 01 0 1 0 1 0 10 0 0 01 0 0 0 1 0 1 0 10 1 1 0 01 0 0 0 0 0 01 0 0 10 0 1 0 1 0 0 0 01 1 1 0 10 0 1 0 0 0 10 0 0 10 0 0 0 1 0 0 0 10 1 1 0 10 0 0 Notice that we only state where the new mystery connective is true as it is false elsewhere. Frank / van Deemter / Wyner**About this proof …**• We’ve made our task a bit easier, assuming that there were only 2 rows resulting in T • But the case with 1 or 3 or 4 or …. rows is analogous (and there are always only finitely many rows.) • So, the proof can be made precise Frank / van Deemter / Wyner**Having proved this …**• We can express every possible truth-functional operator in propositional logic in terms of AND and NOT • This is sometimes called functional completeness. Also universality. • Reduce other operators to other more basic operators. • Very useful in computing to reduce complexity of formulas (Normal forms) Frank / van Deemter / Wyner**Let’s introduce some additional connectives**• A variant of disjunction • The conditional • The biconditional Frank / van Deemter / Wyner**The Exclusive Or Operator**The binary exclusive-or operator“” (XOR) combines two propositions to form their logical “exclusive or”. p = “I will earn an A in this course,” q =“I will drop this course,” p q = “I will either earn an A in this course, or I will drop it (but not both!)” Frank / van Deemter / Wyner**Exclusive-Or Truth Table**• Note that pq meansthat p is true, or q istrue, but not both! • This operation iscalled exclusive or,because it excludes thepossibility that both p and q are true. • “¬” and “” together are not universal. Notedifferencefrom OR. Frank / van Deemter / Wyner**Natural Language is Ambiguous**Note that English“or” can be ambiguous regarding the “both” case! Need context to disambiguate the meaning! For this class, assume “or” means inclusive. Frank / van Deemter / Wyner**Test your understanding of the two types of disjunction**• Suppose p q is true.Does it follow that pqis true? • Suppose pqis true.Does it follow that p q is true? Frank / van Deemter / Wyner**Test your understanding of the two types of disjunction**• Suppose p q is true.Does it follow that pqis true?No: consider p TRUE, q TRUE • Suppose pqis true. Does it follow that p q is true? Yes. Check each of the two assignments that make pq true:a) p TRUE, q FALSE (makes p q true) b) p FALSE, q TRUE (makes p q true) Frank / van Deemter / Wyner

Download Presentation

Connecting to Server..