Predicate Logic 06-06-2016 · 2016. 7. 13. · Predicate Logic Jason Filippou CMSC250 @ UMCP...

Predicate Logic

Jason Filippou

CMSC250 @ UMCP

06-06-2016

Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 1 / 42

Outline

1 Propositional logic falls short

2 Predicate LogicSyntaxSemanticsProof theory


Propositional logic falls short




Modelling worlds

The goal of logic has, is, and will be to model domain knowledgeabout the world, and make certain inferences, based on a certaintheory of proof.

So, for every scenario, we have an agreement on what our world is.

E.g CSIC, CS department, State of Maryland

Consider how the world affects the truth value of certainpropositional logic statements!

freshman ∨ sophomore ∨ junior ∨ seniormotorcycle ∧ red light⇒ wait for green



Semantic simplicity of propositional symbols

Suppose we already have the propositional symbol charlie.

How do we express the fact that Charlie is a unicorn?

1 Insert a new symbol, charlie the unicorn, retract (?) symbolcharlie.

2 Insert rule charlie ∧ horned charlie⇒ charlie the unicorn and thesymbol horned charlie, use modus ponens.

What about the pink and gray unicorns?




Suppose we already have the propositional symbol charlie.

How do we express the fact that Charlie is a unicorn?1 Insert a new symbol, charlie the unicorn, retract (?) symbol

charlie.2 Insert rule charlie ∧ horned charlie⇒ charlie the unicorn and the

symbol horned charlie, use modus ponens.

What about the pink and gray unicorns?




Manually curated knowledge is time-consuming, error-prone, andsometimes contradicting.

Stable modeling example (whiteboard).

Beats the point of inference rules: Why did we even come up withthe automated construction of new knowledge if we end up puttingstuff in ourselves?




Modeling properties of an element of our world is virtuallyimpossible in propositional logic.

For every object in our world, we need to replicate every property!(whiteboard)

How do we relate objects to one another? E.g siblings,coworkers,...



The need for more symbols

How can I write a statement that says “Every CS250 student willsit for a midterm”?

Need a symbol to express the notion of “every item x that satisfiessome property P”...

How about “There’s at least two people in this classroom whoshare a birth month?”



Propositional Logic is not enough...

Expressiveness - tractability trade-off.

Tracta-what?

Propositional logic is the most basic kind of logic.

Excellent for:

Modeling hardware (boolean gates).The study of computational complexity (SAT problem).

Not-so-excellent for:

Translating language into computer-readable format.Building deductive databases.Efficient inference on large domains.

The next-step: First-Order logic!

Only we won’t do full FOL §


Predicate Logic

Predicate Logic


Predicate Logic

What is predicate logic?

An extension of propositional logic we have come up with.

A subset of FOL suitable for introducing formal proofs.

“The logic of quantified statements” is another suitablecharacterization (Epp).


Predicate Logic

A hierarchy of logics

Propositional Logic

First- Order Logic

Second-Order Logic

Type Theory

Infinitary Logic


Predicate Logic


Propositional Logic

First- Order LogicPredicates, quantifiers,

functors, backward / forward chaning,undecidability of inference

Second-Order Logic

Type Theory

Infinitary Logic


Predicate Logic


Only aspectsof FOLincluded in“Predicate Logic”.

Propositional Logic

First- Order LogicPredicates, quantifiers,

functors, backward / forward chaning,undecidability of inference

Second-Order Logic

Type Theory

Infinitary Logic


Predicate Logic Syntax

Syntax



Variables and Constants

Our syntax has some crucial additions over Propositional Logic.

Variables (denoted lowercase) and their (sometimes implicit)domains. E.g:

E.g x ∈ R (Dom(c) = R)E.g c, with Dom(c) = {green, red, blue}

Constants (denoted uppercase): Unique identifiers of objects inour database (similar to Propositional Logic’s “propositionalsymbols”).

Sun,Earth,Benedict Cumberbatch, Jason Filippou



Predicates

Predicate Symbols: typically used to denote properties ofobjects, like adverbs or adjectives in language.

Written with uppercase first letter: P,Q, Father,Rainy

Predicates (denoted uppercase): consist of a predicate symbolfollowed by at least one constant and variable as an “argument”within parentheses. E.g:

Odd(x), Even(y), Father(q, r), with Dom(x) = Dom(y) = N,Dom(q) = {s | s is a MD resident under 18} andDom(r) = {s | s is a male PA resident over 22},King(Charlie, Bananas), Enrolled(x,CMSC 250), withDom(x) = CS UMD Undergraduates.

Arity of a predicate: The number of its arguments.

We constrain predicates to have arity at least 1, otherwise (a) Theymake no sense and (b) They are undistinguishable from constants.



Quantifiers

The symbols “exists”: ∃ and “forall”: ∀, followed by at least onevariable and one predicate.

(∃ x)(Prime(x))(∀x)(Politician(x)⇒ Liar(x))

Parentheses can be used to define the scope of a quantifier. Whenthe scope is obvious, they can be ommitted (e.g above).

Can I have more than 1 predicate on the right of a quantifier?



Quantified Statements

Absolutely! We will call those quantified statements, and they aresomewhat equivalent to propositional logic’s “compoundstatements”

1 Existential statements follow ∃.2 Universal statements follow ∀.3 Mixed statements follow both:

(∀x)(Person(x) ⇒ (∃z)(Loves(z, x))).(∀p1, p2 ∈ R2)(∃p3 ∈ R2)dist(p3, p1) = dist(p3, p2)(∀q)(Prime(q) ⇒ (∃p)(Prime(p) ∧ p > q))

We can also have regular, non-quantified statements that involveconstants instead of variables (ground statements), orstatements that involve both:

Hates(Jason,Artichokes) ∧Hates(Jason,Brussel Sprouts),Form Triangle(P1, P2, P3) ∨ Colinear(P1, P2, P3)(∀x)(Lives(x,North America)⇔Lives(x,Canada) ∨ Lives(x, USA) ∨ Lives(x,Mexico))



Bound / Free variables in statements

Bound variable: A variable that is quantified. E.g:

(∃z) Unicorn(z) ∧Nuts(z)(∀x, y ∈ N)Divides(x, y)⇔ (∃z ∈ N)y = x ∗ z

Free variable: A variable that isn’t bound. E.g:

(∀x)P (x, y)(∃z)(R(z, s) ∨Q(z))⇒ F (z)Q(x)⇒ (∀x)Q(x)

Use parentheses when necessary!

Sentence: A quantified statement with only bound variables.


Predicate Logic Semantics

Semantics



Knowledge Bases / Grounding

Knowledge base (KB): A set of ground (variable-free) predicates1

that represent what we know about the world.

Grounding: The substitution of variables in quantified statementswith constants corresponding to predicate arguments.

Groundings can be true or false with respect to the KB.

Closed world assumption: Anything not mentioned in ourknowledge base is assumed to be false!

1We technically call those ground atoms.Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 22 / 42


Predicate truth

Semantics in Predicate Logic will be tied to the notion of the truthof a - possibly quantified - statement.

Ground statement: Similar to propositional logic. Consists of anon-quantified statement that can be either true or false givenour knowledge base.Sentence: Have to introduce the notions of universal andexistential instantiation / generalization.



Universal instantiation

Rule of Universal Instantiation

(∀x ∈ D)P (x)∴ P (A) for any particular A ∈ D

Examples:

(∀x ∈ R) x2 ≥ 0

(∀p ∈ UMD Undergrads) Smart(p)



Existential instantiation

Rule of Existential instantiation

(∃x ∈ D)P (x)∴ P (A) for a specific A ∈ D

Examples:

(∃z ∈ Classroom) Name(z, Jason)

(∃c ∈ USA) NameContains(y, “Truth”) ∧NameContains(y, “Consequences”)

Can I have more > 1 “A”’s?



Universal generalization

Rule of Universal Generalization

P (A) for some A ∈ D selected arbitrarily.∴ (∀x)P (x)

A is then often called the generic particular.Compare and contrast:

Let A ∈ N Let A ∈ Neven

. . . . . .P (A) P (A)

∴ (∀n ∈ N)P (n) ∴ (∀n ∈ N)P (n)






A is then often called the generic particular.

Compare and contrast:


. . . . . .P (A) P (A)

∴ (∀n ∈ N)P (n) ∴ (∀n ∈ N)P (n)






A is then often called the generic particular.Compare and contrast:


. . . . . .P (A) P (A)

∴ (∀n ∈ N)P (n) ∴ (∀n ∈ N)P (n)



Existential generalization

Rule of existential generalization

P (A) for any A ∈ D.∴ (∃x)P (x)

Good practice: Pay attention to the usages of “any” and “some”.



Modeling Example: Family tree

Jill Phil

Steven Keegan

Hailey Bailey

Cathy

SargeMarge WesleyLesley



Example: Family tree

Female(Marge) Male(Sarge)

Female(Lesley) Male(Phil)

Female(Jill) Male(Wesley)

Female(Hailey) Male(Bailey)

Female(Cathy) Male(Steven)

Male(Keegan)

Jill Phil

Steven Keegan

Hailey Bailey

Cathy





Female(Marge) Male(Sarge) Couple(Marge, Sarge)

Female(Lesley) Male(Phil) Couple(Lesley,Wesley)

Female(Jill) Male(Wesley) Couple(Jill, Phil)

Female(Hailey) Male(Bailey) Couple(Hailey,Bailey)


Male(Keegan)

Jill Phil

Steven Keegan

Hailey Bailey

Cathy





Female(Marge) Male(Sarge) Couple(Marge, Sarge)

Female(Lesley) Male(Phil) Couple(Lesley,Wesley)

Female(Jill) Male(Wesley) Couple(Jill, Phil)

Female(Hailey) Male(Bailey) Couple(Hailey,Bailey)


Mother(Marge, Jill) Male(Keegan)

Mother(Lesley, Phil)

Mother(Lesley,Hailey)

Mother(Jill, Steven)

Mother(Jill,Keegan)

Mother(Hailey, Cathy)

Jill Phil

Steven Keegan

Hailey Bailey

Cathy




Negated quantifiers

For variable set x and quantified statement P (x), the following hold:

(@x)P (x) ≡ (∀x) ∼P (x)

(∃x)P (x) ≡ ∼(∀x)∼P (x)

This applies recursively to nested quantifiers! (whiteboard examples)



Vacuous truth of quantified statements

Critique the following quantified statement:

(∀x)(Marker(x) ∧ Location(x,WhiteBoard)⇒ Blue(x))

What truth value would you attach to it?


Predicate Logic Proof theory

Proof theory



Major and Minor premises

Major premise: A universally quantified implication, i.e of form(∀x) P (x)⇒ Q(x)

Minor premise: The association of an object with the domain ofthe quantified variable, i.e P (A) for some A.

Conclusion: Q(A).



Universal modus ponens

The quantified version of modus ponens.

Universal Modus Ponens

(∀x)P (x)⇒ Q(x)

P (A) for some A ∈ Dom(x)

∴ Q(A)

Theorem

Universal Modus Ponens is a valid rule of inference.

Proof.

Diagramatic (whiteboard)



Universal modus tollens

Universal Modus Tollens

(∀x)P (x)⇒ Q(x)

∼Q(A) for some A ∈ Dom(x)

∴ ∼P (A)



Quantified converse error

Quantified Converse Error

(∀x)P (x)⇒ Q(x)

Q(A) for some A ∈ Dom(x)

∴ P (A)



Quantified inverse error

Quantified inverse error

(∀x)P (x)⇒ Q(x)

∼P (A) for some A ∈ Dom(x)

∴ ∼Q(A)



Complex inferences

Assume a simplified version of 250, called Mini250.

2 midterms, 1 final.

We want to author rules that dictate when a student passes acourse, and formally prove that a student called Trisha will, infact, pass 250.



Complex inferences

We will use the following predicates:

Predicate Meaning

Midterm(n, s, g) The grade of student s in midterm number n(1 or 2) was g (A, B or C).

Final(s, g) The grade of student s in midterm in the finalwas g (A, B, or C).

Present(s) Student s was consistently present in lecture.

Studies(s, l) Student s studies in mode l (Lazily, Well orHard)

Passes(s, g) Student s passed the course with grade g.

Fails(s) Student s failed the course.



Complex inferences

Let’s translate the following statements into Predicate Logic:

1 Every student who studies hard will get As in both midterms andat least a B in the final.

2 Any student who is consistently present in lecture will score atleast a B in both midterms and at least a C in the final.

3 One will pass the course if, and only if, one scores at least a Cin the final, and a C or B in either one of the two midterms.

4 Any student who studies well or hard will score at least a B inboth midterms and the final.

5 One cannot pass and fail the course at the same time.

Now, assume that Trisha is a student who studies well and isconsistently present in lecture. Prove that Trisha will pass the course(with any grade)


Predicate Logic 06-06-2016 · 2016. 7. 13. · Predicate Logic Jason Filippou CMSC250 @ UMCP...

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