Predicate Logic 06-06-2016 · 2016. 7. 13. · Predicate Logic Jason Filippou CMSC250 @ UMCP...
Transcript of 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
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 2 / 42
Propositional logic falls short
Propositional logic falls short
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 3 / 42
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
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 4 / 42
Propositional logic falls short
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?
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 5 / 42
Propositional logic falls short
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 (?) 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?
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 5 / 42
Propositional logic falls short
Semantic simplicity of propositional symbols
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?
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 6 / 42
Propositional logic falls short
Semantic simplicity of propositional symbols
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,...
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 7 / 42
Propositional logic falls short
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?”
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 8 / 42
Propositional logic falls short
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 §
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 9 / 42
Propositional logic falls short
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 §
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 9 / 42
Propositional logic falls short
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 §
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 9 / 42
Predicate Logic
Predicate Logic
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 10 / 42
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).
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 11 / 42
Predicate Logic
A hierarchy of logics
Propositional Logic
First- Order Logic
Second-Order Logic
Type Theory
Infinitary Logic
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 12 / 42
Predicate Logic
A hierarchy of logics
Propositional Logic
First- Order LogicPredicates, quantifiers,
functors, backward / forward chaning,undecidability of inference
Second-Order Logic
Type Theory
Infinitary Logic
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 13 / 42
Predicate Logic
A hierarchy of logics
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
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Predicate Logic Syntax
Syntax
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Predicate Logic 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
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 16 / 42
Predicate Logic Syntax
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.
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Predicate Logic Syntax
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?
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 18 / 42
Predicate Logic Syntax
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?
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 18 / 42
Predicate Logic Syntax
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))
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 19 / 42
Predicate Logic Syntax
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.
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Predicate Logic Semantics
Semantics
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Predicate Logic 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 Logic Semantics
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.
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Predicate Logic Semantics
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 24 / 42
Predicate Logic Semantics
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 24 / 42
Predicate Logic Semantics
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?
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 25 / 42
Predicate Logic Semantics
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?
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 25 / 42
Predicate Logic Semantics
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 26 / 42
Predicate Logic Semantics
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 26 / 42
Predicate Logic Semantics
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 26 / 42
Predicate Logic Semantics
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”.
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Predicate Logic Semantics
Modeling Example: Family tree
Jill Phil
Steven Keegan
Hailey Bailey
Cathy
SargeMarge WesleyLesley
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Predicate Logic Semantics
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
SargeMarge WesleyLesley
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 29 / 42
Predicate Logic Semantics
Example: Family tree
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)
Female(Cathy) Male(Steven)
Male(Keegan)
Jill Phil
Steven Keegan
Hailey Bailey
Cathy
SargeMarge WesleyLesley
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 30 / 42
Predicate Logic Semantics
Example: Family tree
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)
Female(Cathy) Male(Steven)
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
SargeMarge WesleyLesley
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 31 / 42
Predicate Logic Semantics
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 32 / 42
Predicate Logic Semantics
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?
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Predicate Logic Proof theory
Proof theory
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Predicate Logic 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).
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 35 / 42
Predicate Logic Proof theory
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 36 / 42
Predicate Logic Proof theory
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 36 / 42
Predicate Logic Proof theory
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 36 / 42
Predicate Logic Proof theory
Universal modus tollens
Universal Modus Tollens
(∀x)P (x)⇒ Q(x)
∼Q(A) for some A ∈ Dom(x)
∴ ∼P (A)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 37 / 42
Predicate Logic Proof theory
Quantified converse error
Quantified Converse Error
(∀x)P (x)⇒ Q(x)
Q(A) for some A ∈ Dom(x)
∴ P (A)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 38 / 42
Predicate Logic Proof theory
Quantified inverse error
Quantified inverse error
(∀x)P (x)⇒ Q(x)
∼P (A) for some A ∈ Dom(x)
∴ ∼Q(A)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 39 / 42
Predicate Logic Proof theory
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.
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Predicate Logic Proof theory
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.
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 41 / 42
Predicate Logic Proof theory
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 42 / 42
Predicate Logic Proof theory
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)
Jason Filippou (CMSC250 @ UMCP) Predicates 06-06-2016 42 / 42