CS203 Discrete Mathematical Structures Logic (2)

Predicate Logic - everybody loves somebody Proposition, YES or NO? 3 + 2 = 5 X + 2 = 5 X + 2 = 5 for any choice of X in {1, 2, 3} X + 2 = 5 for some X in {1, 2, 3} YES NO YES YES

… Predicates Ahmed eats pizza at least once a week. Define: EP(x) = “x eats pizza at least once a week.” Universe of Discourse - x is a student in CS203 A predicate is a function that takes some variable(s) as arguments and returns True or False. Note that EP(x) is not a proposition, EP(Hassan) is. …

Predicates A predicate is a property or description of subjects in the universe of discourse. The following predicates are all italicized : Johnny is tall. The bridge is structurally sound. 17 is a prime number. Java: predicates are boolean-valued method calls- someLinkedList.isEmpty() isPrime(17)

Quantifiers There are two quantifiers Existential Quantifier “” reads “there exists” Universal Quantifier “” reads “for all” Each is placed in front of a propositional function and binds it to obtain a proposition with semantic value. Mnemonics:  -- reverse E signifies “there Exists”  -- upside-down A signifies “for All”

Predicates - the universal quantifier Suppose P(x) is a predicate on some universe of discourse. Ex. B(x) = “x is carrying a backpack,” x is set of CS203 students. The universal quantifier of P(x) is the proposition: “P(x) is true for all x in the universe of discourse.” We write it x P(x), and say “for all x, P(x)” x P(x) is TRUE if P(x) is true for every single x. x P(x) is FALSE if there is an x for which P(x) is false. x B(x)?

Predicates - the existential quantifier Suppose P(x) is a predicate on some universe of discourse. Ex. C(x) = “x has a cat,” x is set of CS203 students. The existential quantifier of P(x) is the proposition: “P(x) is true for some x in the universe of discourse.” We write it x P(x), and say “for some x, P(x)” x P(x) is TRUE if there is an x for which P(x) is true. x P(x) is FALSE if P(x) is false for every single x. x C(x)?

Predicates - the existential quantifier Universe of discourse is people in this room. B(x) = “x is wearing sneakers.” L(x) = “x is at least 16 years old.” Y(x)= “x is less than 24 years old.” Are either of these propositions true? x B(x) x (Y(x)  L(x)) A: only a is true B: only b is true C: both are true D: neither is true

Predicates - more examples Universe of discourse is all creatures. L(x) = “x is a lion.” F(x) = “x is fierce.” C(x) = “x drinks coffee.” All lions are fierce. Some lions don’t drink coffee. Some fierce creatures don’t drink coffee. x (L(x)  F(x)) x (L(x)  C(x)) x (F(x)  C(x))

Predicates - more examples B(x) = “x is a hummingbird.” L(x) = “x is a large bird.” H(x) = “x lives on honey.” R(x) = “x is richly colored.” All hummingbirds are richly colored. No large birds live on honey. Birds that do not live on honey are dully colored. Universe of discourse is all creatures. x (B(x)  R(x)) x (L(x)  H(x)) x (H(x)  R(x))

Fundamentals of Logic The Use of Quantifiers Ex universe: real numbers

Predicates - the universal quantifier Universe of discourse is people in this Hall. B(x) = “x is wearing sneakers.” L(x) = “x is at least 18 years old.” Y(x)= “x is less than 24 years old.” Are either of these propositions true? x (Y(x)  B(x)) x (Y(x)  L(x)) A: only a is true B: only b is true C: both are true D: neither is true

Predicates - quantifier negation x (L(x)  H(x)) Not all large birds live on honey. x P(x) means “P(x) is true for every x.” What about x P(x) ? Not [“P(x) is true for every x.”] “There is an x for which P(x) is not true.” x P(x) So, x P(x) is the same as x P(x). x (L(x)  H(x))

Predicates - quantifier negation x (L(x)  H(x)) No large birds live on honey. x P(x) means “P(x) is true for some x.” What about x P(x) ? Not [“P(x) is true for some x.”] “P(x) is not true for all x.” x P(x) So, x P(x) is the same as x P(x). x (L(x)  H(x))

Predicates - quantifier negation So, x P(x) is the same as x P(x). So, x P(x) is the same as x P(x). General rule: to negate a quantifier, move negation to the right, changing quantifiers as you go.

Fundamentals of Logic Ex. p(x): x is odd. q(x): x2-1 is even. Negate (If x is odd, then x2-1 is even.) There exists an integer x such that x is odd and x2-1 is odd. (a false statement, the original is true)

Predicates - quantifier negation No large birds live on honey. x (L(x)  H(x))  x (L(x)  H(x)) Negation rule  x (L(x)  H(x)) DeMorgan’s  x (L(x)  H(x)) Subst for  What’s wrong with this proof?

Fundamentals of Logic multiple variables

Fundamentals of Logic BUT Ex. p(x,y): x+y=17. : For every integer x, there exists an integer y such that x+y=17. (TRUE) : There exists an integer y so that for all integer x, x+y=17. (FALSE) Therefore,

Order matters Set the universe of discourse to be all natural numbers {0, 1, 2, 3, … }. Let R (x,y ) = “x < y”. Q1: What does x y R (x,y ) mean? Q2: What does y x R (x,y ) mean?

Order matters R (x,y ) = “x < y” A1: x y R (x,y ): “All numbers x admit a bigger number y ” A2: y x R (x,y ): “Some number y is bigger than all x” Q: What’s the true value of each expression?

More Practice for Predicate Logic Nermin likes all movies that Rehab likes (and possibly more). x [Movie(x)  Likes(Rehab, x)  Likes(Nermin, x)] There is exactly one AU professor who won a Nobel prize x[AU_Prof(x)  Wins(x, NobelPrize)]  y,z[y  z  AU_Prof(y)  AU_Prof(z)  Wins(y, NobelPrize)  Wins(z, NobelPrize)]

Review of Boolean algebra Not is a horizontal bar above the number 0 = 1 1 = 0 Or is a plus 0+0 = 0 0+1 = 1 1+0 = 1 1+1 = 1 And is multiplication 0*0 = 0 0*1 = 0 1*0 = 0 1*1 = 1

NOT Y = ~X Y = !X Y = not X Y = X’ not(Y,X) Logic Gate: A A’ or -A (also called an inverter) A A’ or -A Truth Table: Single-throw Double-pole Switch: A A -A 1 A’ or -A Y = ~X Y = !X Y = not X Y = X’ not(Y,X)

AND A Logic Gate: A*B Truth Table: B A B A*B 1 A B Series Circuit: A*B

AND X & Y (Verilog and ABEL) X and Y (VHDL) X Y X * Y XY (textbook) and(Z,X,Y) (Verilog) V U

OR A Logic Gate: A+B Truth Table: B A B A+B 1 A Parallel Circuit: B 1 A Parallel Circuit: B A+B

OR X | Y (Verilog) X # Y (ABEL) X or Y (VHDL) X + Y (textbook) X V Y X U Y or(Z,X,Y) (Verilog)

NAND Gate NAND X Y Z 0 0 1 0 1 1 X 1 0 1 1 1 0 Z Y Z = ~(X & Y) 0 0 1 0 1 1 1 0 1 1 1 0 X Z Y Z = ~(X & Y) nand(Z,X,Y)

NOR Gate NOR X Y Z 0 0 1 0 1 0 X 1 0 0 Z 1 1 0 Y Z = ~(X | Y) 0 0 1 0 1 0 1 0 0 1 1 0 X Z Y Z = ~(X | Y) nor(Z,X,Y)

Exclusive-OR Gate XOR X Y Z X Z 0 0 0 Y 0 1 1 1 0 1 1 1 0 Z = X ^ Y 0 0 0 Y 0 1 1 Z = X ^ Y xor(Z,X,Y) 1 0 1 1 1 0

XOR X ^ Y (Verilog) X $ Y (ABEL) X @ Y xor(Z,X,Y) (Verilog)

Logic Circuits ≡ Boolean Expressions All logic circuits are equivalent to Boolean expressions and any boolean expression can be rendered as a logic circuit. AND-OR logic circuits are equivalent to sum-of-products form. Consider the following circuits: A y=aB+Bc abc B C A B C Y aBc y Ab y=abc+aBc+Ab

question x+y (x+y)y y Find the output of the following circuit Answer: (x+y)y Or (xy)y x+y (x+y)y y __

question Write the circuits for the following Boolean algebraic expressions (x+y)x _______ x+y (x+y)x x+y

Draw a circuit diagram for  = (xy' + x'y)z.

Let’s compare the resulting circuits Here are two different but equivalent circuits. In general the one with fewer gates is “better”: It costs less to build It requires less power But we had to do some work to find the second form