CSE 311 Lecture 05: Canonical Forms and Predicate Logic
Topics
- Boolean algebra
- A review of Lecture 04 with another end-to-end example.
- Canonical forms
- Standard forms for a Boolean expression.
- Predicate logic
- Extending propositional logic with predicates and quantifiers.
Boolean algebra
A review of Lecture 04 with another end-to-end example.
Boolean algebra is a notation for combinational logic
Think of it as a notation for propositional logic used in circuit design.
Boolean algebra consists of the following elements and operations:
- a set of elements $B = \{0,1\}$,
- binary operations $\{ +, \cdot \}$,
- a unary operation $\{\,^\prime\}$.
These correspond to the truth values $\{\F,\T\}$, and the logical connectives $\vee,\wedge,\neg$.
Boolean operations satisfy the following axioms for any $a,b,c\in B$:
- Closure
- $a + b \in B$
- $a \cdot b \in B$
- Commutativity
- $a + b = b + a$
- $a \cdot b = b \cdot a$
- Associativity
- $a + (b + c) = (a + b) + c$
- $ a \cdot (b \cdot c) = (a \cdot b) \cdot c$
- Distributivity
- $a + (b \cdot c) = (a + b) \cdot (a + c)$
- $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$
- Identity
- $a + 0 = a$
- $a \cdot 1 = a$
- Complementarity
- $a + a’ = 1$
- $a \cdot a’ = 0$
- Null
- $a + 1 = 1$
- $a \cdot 0 = 0$
- Idempotency
- $a + a = a$
- $a \cdot a = a$
- Involution
- $(a’)’ = a$
Example: from spec to an adder circuit
Binary addition is a basic operation implemented by every computer.
It works just like decimal addition: we add numbers digit by digit, from least to most significant, keeping track of the current sum and carry.
| carry | 0010 |
| input A | 101 |
| input B | + 001 |
| sum | 110 |
We can implement $n$-bit addition by chaining together $n$ 1-bit adders:
Let’s implement the 1-bit adder circuit!
Example: from spec to logic via a truth table
- Inputs: $A, B, C_{in}$ (input bits and carry-in)
- Outputs: $S, C_{out}$ (sum and carry out)
| $A$ | $B$ | $C_{in}$ | $C_{out}$ | $S$ |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 |
$S =\,$$A’\cdot B’\cdot C_{in}$$\,+\,A’\cdot B\cdot C_{in}’$$\,+\,A\cdot B’\cdot C_{in}’$$\,+\,A\cdot B\cdot C_{in}$
$C_{out} =\,$$A’\cdot B\cdot C_{in}$$\,+\,A\cdot B’\cdot C_{in}$$\,+\,A\cdot B\cdot C_{in}’$$\,+\,A\cdot B\cdot C_{in}$
Example: apply theorems to simplify the logic
| $C_{out}$ | $=$ | $A’\cdot B\cdot C_{in} + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in}’ + A\cdot B\cdot C_{in} $ | |
| $=$ | $A’\cdot B\cdot C_{in} + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in}’ + A\cdot B\cdot C_{in} + A\cdot B\cdot C_{in} $ | Idempotence | |
| $=$ | $A\cdot B\cdot C_{in} + A’\cdot B\cdot C_{in} + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in}’ + A\cdot B\cdot C_{in} $ | Commutativity | |
| $=$ | $B\cdot C_{in}\cdot A + B\cdot C_{in}\cdot A' + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in}’ + A\cdot B\cdot C_{in} $ | Commutativity | |
| $=$ | $B\cdot C_{in}\cdot (A + A') + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in}’ + A\cdot B\cdot C_{in} $ | Distributivity | |
| $=$ | $B\cdot C_{in}\cdot 1 + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in}’ + A\cdot B\cdot C_{in} $ | Complementarity | |
| $=$ | $B\cdot C_{in} + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in}’+ A\cdot B\cdot C_{in} $ | Identity | |
| $=$ | $B\cdot C_{in} + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in}’+ A\cdot B\cdot C_{in} + A\cdot B\cdot C_{in} $ | Idempotence | |
| $=$ | $B\cdot C_{in} + A\cdot B\cdot C_{in} + A\cdot B’\cdot C_{in} + A\cdot B\cdot C_{in} + A\cdot B\cdot C_{in}’ $ | Commutativity | |
| $=$ | $B\cdot C_{in} + A\cdot C_{in} \cdot B + A\cdot C_{in} \cdot B’ + A\cdot B\cdot C_{in} + A\cdot B\cdot C_{in}’ $ | Commutativity | |
| $=$ | $B\cdot C_{in} + A\cdot C_{in} \cdot (B + B’) + A\cdot B\cdot (C_{in} + C_{in}’) $ | Distributivity | |
| $=$ | $B\cdot C_{in} + A\cdot C_{in} \cdot 1 + A\cdot B\cdot 1 $ | Complementarity | |
| $=$ | $B\cdot C_{in} + A\cdot C_{in} + A\cdot B $ | Identity |
Example: map the logic to the available gates
$C_{out} = B\cdot C_{in} + A\cdot C_{in} + A\cdot B$
$C_{out}$ mapped to AND, OR, and NOT gates.
$S = A’\cdot B’\cdot C_{in} + A’\cdot B\cdot C_{in}’ + A\cdot B’\cdot C_{in}’ + A\cdot B\cdot C_{in}$
$S$ mapped to an XOR gate: $S \equiv A \oplus B \oplus C_{in}$.
Example: the recipe for translating a spec to a circuit
To translate a specification to a circuit:
- Write the truth table (and, optionally, the program) for the spec.
- Write the Boolean expression for the output bits.
- Minimize the Boolean expressions for the output bits.
- Map the minimized expressions to the available logic gates.
①
| $A$ | $B$ | $C$ | $F$ |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |
②③
| $F$ | $=$ | $A'\cdot B \cdot C' + A'\cdot B \cdot C + A\cdot B' \cdot C + A\cdot B \cdot C $ | |
| $=$ | $A'\cdot B \cdot C' + A'\cdot B \cdot C + A \cdot C\cdot B' + A\cdot C \cdot B $ | Commutativity | |
| $=$ | $A'\cdot B\cdot(C' + C) + A\cdot C\cdot(B' + B) $ | Distributivity | |
| $=$ | $A'\cdot B\cdot(C + C') + A\cdot C\cdot(B + B') $ | Commutativity | |
| $=$ | $A'\cdot B\cdot 1 + A\cdot C\cdot 1 $ | Complementarity | |
| $=$ | $A'\cdot B + A\cdot C $ | Identity |
④
Canonical forms
Standard forms for a Boolean expression.
Why do we need canonical forms?
- A truth table is the unique signature of a Boolean function.
- It captures the semantics (meaning) of the function.
- The same truth table can have many realizations in Boolean algebra.
- One function can have many different syntactic representations.
- Depends on how good we are at Boolean simplification.
- Canonical forms are standard form for a Boolean expression.
- We all come up with the same expression.
- Also used internally by theorem provers.
- We will cover two useful canonical forms.
- Sum-of-products form.
- Product-of-sums form.
Sum-of-products canonical form
- Also known as …
- Disjunctive Normal Form (DNF)
- Minterm Expansion
- To convert a truth table to sum-of-products:
- ① Read the rows with true (1) output.
- ② Convert to Boolean algebra.
- ③ Add the minterms together.
| $A$ | $B$ | $C$ | $F$ | ① | ② |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | ||
| 0 | 0 | 1 | 0 | ||
| 0 | 1 | 0 | 1 | 010 | $A’BC’$ |
| 0 | 1 | 1 | 1 | 011 | $A’BC$ |
| 1 | 0 | 0 | 0 | ||
| 1 | 0 | 1 | 1 | 101 | $AB’C$ |
| 1 | 1 | 0 | 0 | ||
| 1 | 1 | 1 | 1 | 111 | $ABC$ |
③ $F = A’BC’ + A’BC + AB’C + ABC$
Sum-of-products canonical form: properties
- Product term (or minterm)
- Conjunction of literals, which are variables or their negations.
- Represents an input combination for which output is true.
- Each variable appears exactly once, true or negated (but not both).
| $A$ | $B$ | $C$ | $F$ | minterms |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | |
| 0 | 0 | 1 | 0 | |
| 0 | 1 | 0 | 1 | $A’BC’$ |
| 0 | 1 | 1 | 1 | $A’BC$ |
| 1 | 0 | 0 | 0 | |
| 1 | 0 | 1 | 1 | $AB’C$ |
| 1 | 1 | 0 | 0 | |
| 1 | 1 | 1 | 1 | $ABC$ |
$F$ in canonical form
$F = A’BC’ + A’BC + AB’C + ABC$
canonical form $\neq$ minimal form
| $F$ | $=$ | $A'BC' + A'BC + AB'C + ABC $ | |
| $=$ | $A'B(C + C') + AC(B + B') $ | ||
| $=$ | $A'B + AC $ |
Product-of-sums canonical form
- Also known as …
- Conjunctive Normal Form (CNF)
- Maxterm Expansion
- To convert a truth table to product-of-sums:
- ① Read the rows with false (0) output.
- ② Negate all bits.
- ③ Convert to Boolean algebra.
- ④ Multiply the maxterms together.
| $A$ | $B$ | $C$ | $F$ | ① | ② | ③ |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 000 | 111 | $A+B+C$ |
| 0 | 0 | 1 | 0 | 001 | 110 | $A+B+C’$ |
| 0 | 1 | 0 | 1 | |||
| 0 | 1 | 1 | 1 | |||
| 1 | 0 | 0 | 0 | 100 | 011 | $A’+B+C$ |
| 1 | 0 | 1 | 1 | |||
| 1 | 1 | 0 | 0 | 110 | 001 | $A’+B’+C$ |
| 1 | 1 | 1 | 1 |
④
Product-of-sums canonical form: why does it work?
- What we know …
- $(F’)’ = F$ by Involution.
- How to get a minterm expansion for $F’$.
| $A$ | $B$ | $C$ | $F$ |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |
$F’ = A’B’C’ + A’B’C + AB’C’ + ABC’$
- Taking the complement of both sides
- $(F’)’ = (A’B’C’ + A’B’C + AB’C’ + ABC’)’$
- Using Involution and DeMorgan Laws
Product-of-sums canonical form: properties
- Sum term (or maxterm)
- Disjunction of literals, which are variables or their negations.
- Represents an input combination for which output is false.
- Each variable appears exactly once, true or negated (but not both).
| $A$ | $B$ | $C$ | $F$ | maxterms |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | $A+B+C$ |
| 0 | 0 | 1 | 0 | $A+B+C’$ |
| 0 | 1 | 0 | 1 | |
| 0 | 1 | 1 | 1 | |
| 1 | 0 | 0 | 0 | $A’+B+C$ |
| 1 | 0 | 1 | 1 | |
| 1 | 1 | 0 | 0 | $A’+B’+C$ |
| 1 | 1 | 1 | 1 |
$F$ in canonical form
$F = (A+B+C)(A+B+C’)(A’+B+C)(A’+B’+C)$
canonical form $\neq$ minimal form
| $F$ | $=$ | $(A+B+C)(A+B+C')(A'+B+C)(A'+B'+C) $ | |
| $=$ | $(A+B+CC')(A'+C+BB') $ | ||
| $=$ | $(A+B)(A'+C) $ |
Predicate logic
Extending propositional logic with predicates and quantifiers.
Predicate logic versus propositional logic
- Propositional logic
- “If Garfield is an orange cat and likes lasagna, then he has black stripes.”
- Predicate logic
- “All positive integers $x,y,z$ satisfy $x^3 + y^3 \neq z^3$.”
Propositional logic lets us express complex propositions in terms of their constituent parts (atomic propositions) joined by connectives. Predicate logic lets us express how propositions depend on the objects they mention.
Key notions in predicate logic
- Syntax
- Predicate logic extends propositional logic with two key constructs: predicates and quantifiers ($\exists,\forall$).
- Semantics
- We define the meaning of formulas in predicate logic with respect to a domain of discourse.
Predicates
- Predicate is a function that returns a truth value.
- Cat(x) $::=$ “x is a cat”
- Prime(x) $::=$ “x is prime”
- HasTaken(x, y) $::=$ “student x has taken course y”
- LessThan(x, y) $::=$ “x < y”
- Sum(x, y, z) $::=$ “x + y = z”
- GreaterThan5(x) $::=$ “x > 5”
- HasNChars(s, n) $::=$ “string s has length n”
- Predicates can have varying arity (numbers of arguments).
Domain of discourse
To give meaning to predicates in a formula, we define a set of objects that those predicates can take as input.
This set of objects is called the domain of discourse for a formula.
- For each of the following, what might the domain be?
- “x is a cat”, “x barks”, “x ruined my couch”
- “mammals” or “sentient beings” or “cats and dogs” or …
- “x is prime”, “x = 0”, “x < 0”, “x is a power of two”
- “numbers” or “integers” or “integers greater than 5” or …
- “student x has taken course y” “x is a pre-req for z”
- “students and courses” or “university entities” or …
Quantifiers
- Quantifiers let us talk about all or some objects in the domain.
- $\forall x. P(x)$
- $P(x)$ is true for every $x$ in the domain.
- Read as “for all $x$, $P(x)$”.
- Called the universal quantifier.
- $\exists x. P(x)$
- There is an $x$ in the domain for which $P(x)$ is true.
- Read as “there exists $x$, $P(x)$”.
- Called the existential quantifier.
Universal quantifier $\forall$
- $\forall x. P(x)$
- $P(x)$ is true for every $x$ in the domain.
- Examples: are these true?
- $\forall x. \text{Odd}(x)$
- $\forall x. \text{LessThan5}(x)$
Depends on the domain.
| $\{-3, 3\}$ | Integers | Odd Integers | |
|---|---|---|---|
| $\forall x. \text{Odd}(x)$ | True | False | True |
| $\forall x. \text{LessThan5}(x)$ | True | False | False |
You can think of $\forall x. P(x)$ as conjunction over all objects in the domain.
- $\forall x. \text{Odd}(x)$
- over $\{-3, 3\}$ is the conjunction $\text{Odd}(-3)\wedge\text{Odd}(3)$
- over integers is the infinite conjunction $\ldots\wedge\text{Odd}(-1)\wedge\text{Odd}(0)\wedge\text{Odd}(1)\wedge\ldots$
Existential quantifier $\exists$
- $\exists x. P(x)$
- There is an $x$ in the domain for which $P(x)$ is true.
- Examples: are these true?
- $\exists x. \text{Odd}(x)$
- $\exists x. \text{LessThan5}(x)$
Depends on the domain.
| $\{-3, 3\}$ | Integers | Positive Multiples of 5 | |
|---|---|---|---|
| $\exists x. \text{Odd}(x)$ | True | True | True |
| $\exists x. \text{LessThan5}(x)$ | True | True | False |
You can think of $\exists x. P(x)$ as disjunction over all objects in the domain.
- $\exists x. \text{Odd}(x)$
- over $\{-3, 3\}$ is the disjunction $\text{Odd}(-3)\vee\text{Odd}(3)$
- over integers is the infinite disjunction $\ldots\vee\text{Odd}(-1)\vee\text{Odd}(0)\vee\text{Odd}(1)\vee\ldots$
Statements with quantifiers
Just like with propositional logic, we need to define variables (this time predicates). And we must also now define a domain of discourse.
What is the truth value of these statements?
| $\exists x. \text{Even}(x)$ | T |
| $\forall x. \text{Odd}(x)$ | F |
| $\forall x. \text{Even}(x)\vee\text{Odd}(x)$ | T |
| $\exists x. \text{Even}(x)\wedge\text{Odd}(x)$ | F |
| $\forall x. \text{Greater}(x+1,x)$ | T |
| $\exists x. \text{Even}(x)\wedge\text{Prime}(x)$ | T |
- Domain of discourse
- Positive integers
- Predicate definitions
- Even(x) $:=$ “x is even”
- Odd(x) $:=$ “x is odd”
- Prime(x) $:=$ “x is prime”
- Greater(x, y) $:=$ “x > y”
- Equal(x, y) $:=$ “x = y”
- Sum(x, y, z) $:=$ “z = x + y”
Predicate logic to English
- Translate the following statements to English
- $\forall x. \exists y. \text{Greater}(y, x)$
- For every positive integer $x$, there is a positive integer $y$, such that $y > x$.
- $\forall x. \exists y. \text{Greater}(x, y)$
- For every positive integer $x$, there is a positive integer $y$, such that $x > y$.
- $\forall x. \exists y. \text{Greater}(y, x)\wedge\text{Prime}(y)$
- For every positive integer $x$, there is a positive integer $y$, such that $y > x$ and $y$ is prime.
- $\forall x. \text{Prime}(x)\rightarrow(\text{Equal}(x,2)\vee\text{Odd}(x))$
- For every positive integer $x$, if $x$ is prime then $x=2$ or $x$ is odd.
- $\exists x. \exists y. \text{Sum}(x,2,y)\wedge\text{Prime}(x)\wedge\text{Prime}(y)$
- There exist positive integers $x$ and $y$ such that $x+2=y$ and $x$ and $y$ are prime.
- Domain of discourse
- Positive integers
- Predicate definitions
- Even(x) $:=$ “x is even”
- Odd(x) $:=$ “x is odd”
- Prime(x) $:=$ “x is prime”
- Greater(x, y) $:=$ “x > y”
- Equal(x, y) $:=$ “x = y”
- Sum(x, y, z) $:=$ “z = x + y”
Predicate logic to English: natural translations
- Translate the following statements to English
- $\forall x. \exists y. \text{Greater}(y, x)$
- There is no greatest positive integer.
- $\forall x. \exists y. \text{Greater}(x, y)$
- There is no least positive integer.
- $\forall x. \exists y. \text{Greater}(y, x)\wedge\text{Prime}(y)$
- For every positive integer there is a larger number that is prime.
- $\forall x. \text{Prime}(x)\rightarrow(\text{Equal}(x,2)\vee\text{Odd}(x))$
- Every prime number is 2 or odd.
- $\exists x. \exists y. \text{Sum}(x,2,y)\wedge\text{Prime}(x)\wedge\text{Prime}(y)$
- There exist prime numbers that differ by two.
- Domain of discourse
- Positive integers
- Predicate definitions
- Even(x) $:=$ “x is even”
- Odd(x) $:=$ “x is odd”
- Prime(x) $:=$ “x is prime”
- Greater(x, y) $:=$ “x > y”
- Equal(x, y) $:=$ “x = y”
- Sum(x, y, z) $:=$ “z = x + y”
English to predicate logic
- “Orange cats like lasagna.”
- $\forall x. ((\text{Orange}(x) \wedge \text{Cat}(x)) \rightarrow \text{LikesLasagna}(x))$
- “Some orange cats don’t like lasagna.”
- $\exists x. ((\text{Orange}(x) \wedge \text{Cat}(x)) \wedge \neg \text{LikesLasagna}(x))$
- Domain of discourse
- Mammals
- Predicate definitions
- Cat(x) $:=$ “x is a cat”
- Orange(x) $:=$ “x is orange”
- LikesLasagna(x) $:=$ “x likes lasagna”
English to predicate logic: translation hints
- “Orange cats like lasagna.”
- $\forall x. ((\text{Orange}(x) \wedge \text{Cat}(x)) \rightarrow \text{LikesLasagna}(x))$
- When there’s no leading quantification, it means “for all”.
- When restricting to a smaller domain in a “for all”, use implication.
- “Some orange cats don’t like lasagna.”
- $\exists x. ((\text{Orange}(x) \wedge \text{Cat}(x)) \wedge \neg \text{LikesLasagna}(x))$
- “Some” means “there exists”.
- When restricting to a smaller domain in an “exists”, use conjunction.
- When putting predicates together, like orange cats, use conjunction.
- Domain of discourse
- Mammals
- Predicate definitions
- Cat(x) $:=$ “x is a cat”
- Orange(x) $:=$ “x is orange”
- LikesLasagna(x) $:=$ “x likes lasagna”
Summary
- Canonical forms are standard form for a Boolean expression.
- Sum-of-products form.
- Product-of-sums form.
- Predicate logic adds predicates and quantifiers to propositional logic.
- Predicate is a function that returns a truth value.
- Quantifiers let us talk about all ($\forall$) or some ($\exists$) objects in the domain.
- The domain of discourse is the set of objects over which the predicates and quantifiers in a formula are evaluated.