sept 18, 2003© vadim bulitko : cmput 272, fall 2003, uofa1 cmput 272 formal systems logic in cs i....

Sept 18, 2003 © Vadim Bulitko : CMPUT 272, Fall 2003, UofA 1

CMPUT 272CMPUT 272Formal Systems Logic in Formal Systems Logic in

CSCS

CMPUT 272CMPUT 272Formal Systems Logic in Formal Systems Logic in

CSCS

I. E. LeonardUniversity of Alberta

http://www.cs.ualberta.ca/~isaac/cmput272/f03


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Chapter 2.1Formal FOPC introClean-up

Chapter 2.2

Chapter 2.3

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AnnouncementsAnnouncementsAnnouncementsAnnouncements

Assignment #1 is due Thursday, September 25


The PlanThe PlanThe PlanThe Plan

Last time considered an informal introduction to predicate logic

We will introduce predicate logic formally today and dwell on some subtle issues


Predicate Logic Predicate Logic (simplified)(simplified)

Predicate Logic Predicate Logic (simplified)(simplified)

Boolean variablesP, Q, R, … can be T/FE.g., P=“5 is a prime number”

Connectives:&, v, , , ~

Domain variablesA,B,C,… draw values from a domain setsE.g., j is a variable over integer numbers

Predicates:P(X1,…,XN),… map from domain inputs to T/FE.g., P(j) = “j is a prime number”

Quantifiers:For all , Exists E.g., [j prime(j)] = “any number is a prime number”

Statements/formulae:Atomic, unit, negated, conjunctive, …, universally/existentially quantifiedE.g., [j [even(j) & ~equal(j,2) ~prime(j) ]] = “no even numbers are prime except 2”


NoteNoteNoteNote

Notice that we did away with 0-arity predicates such as P(), etc.

Any Boolean variable is a 0-arity predicate so we will just drop the parentheses…


Questions?Questions?Questions?Questions?


Def : Well-formed Def : Well-formed FormulaeFormulae

Def : Well-formed Def : Well-formed FormulaeFormulae

A well-formed formula/sentence (wff) is a construction defined recursively:

Any Boolean variable is a wffAny predicate is a wffFor any wff X, ~(X) is a wffTwo wffs parenthesized and connected by v,&,, form a wff. For any wff X and a domain variable k:

k (X) is a wffk (X) is a wffHere X is within the scope of k (or k)


Find wffs…Find wffs…Find wffs…Find wffs…

x [man(X) mortal(x)]x [sandwich(x) better-than(x,Nothing)]x [duck(x) flies(x)]~[x [man(x) & lives-forever(x)]]x [man(x) ~lives-forever(x)]x [person(x) [Knave(x) v Knight(x)]]x [person(x) [Knave(x) v Knight(x)] & ~[Knave(x) & Knight(x)]]


Def : InterpretationsDef : InterpretationsDef : InterpretationsDef : Interpretations

Interpretation for a wff :Assigns T/F values to all Boolean variablesDefines a single domain set for all domain variablesDefines all predicates

Example: consider wff = (j prime(j)) & P

A possible interpretation:Boolean variable P=TDomain variable j runs over integer numbersPredicate prime(x) holds iff x is a prime number


Interpretation ExampleInterpretation ExampleInterpretation ExampleInterpretation Example

Consider wff:x y Likes(x,y)

Does it hold?Depends on the interpretation!

Interpretation: Domain set for x,y is {Angela,Belinda,Jean}Define predicate Likes(liker,liked) as:

Likes(Angela,Belinda)Likes(Belinda,Angela)Likes(Jean,Belinda)Likes(Jean,Angela)

Does it hold in this interpretation?Yes


Model / CountermodelModel / CountermodelModel / CountermodelModel / Countermodel

Formula holds given an interpretation I:

is satisfied by IThen is satisfiableThen I is a model for

Formula doesn’t hold given I: is not satisfied (or falsified) by IThen I is a countermodel for


Tautologies & Tautologies & ContradictionsContradictionsTautologies & Tautologies & ContradictionsContradictions

Formula holds under any interpretation I

is valid

is a tautology

Formula doesn’t hold under any I is a contradiction

is unsatisfiable


ExampleExampleExampleExample

Define Predicate P(X) as “person X is having a drink”

Considerxy [P(x) P(y)]“There exists a person such that when he is having a drink then everyone is having a drink”

True or false?TRUE – it is a tautology!


ContingenciesContingenciesContingenciesContingencies

holds under any interpretation tautology does NOT hold under any interpretation contradictionWhat if holds under some interpretations but not others?

Then is a contingency


Classify…Classify…Classify…Classify…

P & QP & ~PP v ~Px P(x)x ((P(x) Q(x)) & P(x)) Q(x)x (P(x) v ~P(x)) (P(x) & ~P(x))


Bindings & ScopeBindings & ScopeBindings & ScopeBindings & Scope

A variable can be free:Sharp(X)

…or bound to a quantifierx P(x)

If variable X is bound to a quantifier then there is a binding between themThen X is said to be in the quantifier’s scopeHow about:

x y P(x) & Q(x,y) v R(y)x y (equal(x,y) v P(z) & (y B(X,Y)))

CMPUT 272 is case-sensitive! X is not the same as x


Free variables Free variables Free variables Free variables

Given an interpretation we can evaluate a formula, e.g.:

x y Likes(x,y)

Consider the following formula:y Likes(x,y)

Suppose you are given the interpretation I:Domain set for x,y is {Angela,Belinda,Jean}Define predicate Likes(liker,liked) as:

Likes(Angela,Belinda)Likes(Jean,Belinda)Likes(Jean,Angela)

Is y Likes(x,y) satisfied by I ?Cannot tell – x is a free variable…


SubstitutionsSubstitutionsSubstitutionsSubstitutions

A wff that can be evaluated to T/F given an interpretation is called a statementFormulae with free variables may not be statements

That is so because an interpretation doesn’t touch free domain variables

A substitution “complements” an interpretation by assigning (substituting) all free domain variables


Example Example Example Example

Revisit the formula:y Likes(x,y)

Suppose you are given the interpretation I:Domain set for x,y is {Angela,Belinda,Jean}Define predicate Likes(liker,liked) as:

Likes(Angela,Belinda)Likes(Jean,Belinda)Likes(Jean,Angela)

Additionally you have a substitution S:Free domain variable x is set to Angela

Is y Likes(x,y) satisfied by I and S?Yes…


Updated DefinitionUpdated DefinitionUpdated DefinitionUpdated Definition

Formula holds given an interpretation I and a substitution S:

is satisfied by I and SThen is satisfiableThen I and S form a model for

Formula doesn’t hold given I and S : is not satisfied (or falsified) by I and SThen I and S form a countermodel for



Formula holds under every interpretation I and substitution S

is valid

is a tautology

Formula doesn’t hold under any I and substitution S

is a contradiction

is unsatisfiable



holds under some interpretations and substitutions but not others?

Then is a contingency


Note 1Note 1Note 1Note 1

Many formulae with free variables cannot be tautologies or contradictions

Their value depends on the substitution

However, some can:Likes(x,y) v ~Likes(x,y)… is a tautology despite two free variables


Implicit QuantifiersImplicit QuantifiersImplicit QuantifiersImplicit Quantifiers

If x > 2 then x2 > 4

Textbook: real numbers x, if x>2 then x2 > 4

More formally:x [x>2 y square(x,y) & y>4]where predicates square(a,b) and c>d are defined over the set of real numbers

Implicit Implicit quantifierquantifier

ss


ss


ss


ss


Domain FunctionsDomain FunctionsDomain FunctionsDomain Functions

Consider the last statement again: real numbers x, if x>2 then x2 > 4Is it a wff? No! Why?

Well x2 is not a predicate and is not a variableIt is a domain function (square)!

We have not formally introduced domain functions

But we will use them anyway for the sake of:

SimplicityConsistency with the text book


Conversion to Conversion to PredicatesPredicates

Conversion to Conversion to PredicatesPredicates

The use of domain functions is justified:Because they can be expressed through domain predicates

Example:let function f map numbers to numbers:

f(n)=m

Then whenever we see f(x) in our wffWe will change it (with some care) to:

y Pf(x,y)

Where Pf(n,m) holds iff f(n)=m


Evaluating FormulaeEvaluating FormulaeEvaluating FormulaeEvaluating Formulae

Boolean variablesInterpretation directly

Predicate with variables : Likes(x)Use the assignment of x and the semantics of Likes()

Universally quantified formula : x P(x)Evaluates to true iff P(x) holds on all possible values of x

Existentially quantified formulae : x P(x)Evaluates to true iff P(x) holds on at least one possible value of x


ExamplesExamplesExamplesExamples

Consider formula S:x (x2 > x)

Assume the standard algebraic interpretationSuppose the domain set for variable x is {2,3,4}Does S hold or not?

Yes : 4>2, 9>3, 16>4

Suppose the domain set for variable x is the set of real numbersDoes S hold or not?

No : ~(0 > 0)


ExamplesExamplesExamplesExamplesConsider formula S:

x (x2 > x)

Assume the standard algebraic interpretation

Suppose the domain set for variable x is {2,3,4}Does S hold or not?

Yes : 4>2

Suppose the domain set for variable x is the segment [0,1]Does S hold or not?

No : all numbers between 0 and 1 have their squares not greater than the number itself


SummarySummarySummarySummaryGIVEN a particular interpretation

To show that a -quantified formula holds:Exhaust all possibilities – show that all examples satisfy the formula

To show that a -quantified formula does NOT hold:

Find a falsifying example

To show that a -quantified formula holds:Find a satisfying example

To show that a -quantified formula does NOT hold:

Exhaust all possibilities and show that all examples falsify the formula


SummarySummarySummarySummary

How about classifying a formula:Tautology / Contradiction / Contingency

Contingencies are, perhaps, the easiest:

Need to find a model and a counter-model

Tautologies / contradictions:Need to show that all possible interpretations satisfy/falsify the formulaMore difficult


Restricted Restricted --QuantifiersQuantifiers

Restricted Restricted --QuantifiersQuantifiers

Informal: Any number greater than 2 is positive

Formal: x [x>2 x>0]

Formal shorthand (“syntactic sugar”): x>2 [x>0]

Common mistake:Use of & instead of x [x>2 & x>0]“Any number is greater than 2 and greater 0”How about 1?


Restricted Restricted -Quantifiers-QuantifiersRestricted Restricted -Quantifiers-Quantifiers

Informal: There is a positive solution to x2=1

Formal: x [x>0 & x2=1]

Formal shorthand (“syntactic sugar”): x>0 [x2=1]

Common mistake:Use of instead of &x [x>0 x2=1]“There exists a number such that if it is positive then it is a solution to x2=1”This formula is satisfied by any interpretation that contains 0 among the numbers


Restricted Restricted -Quantifiers-QuantifiersRestricted Restricted -Quantifiers-Quantifiers

Another example of…Common mistake: use of instead of &

“There exists an immortal man”

Correct:x [man(x) & ~mortal(x)]

Incorrect:x [man(x) ~mortal(x)]“There exists something such that if it is a man then it is immortal”

The latter formula is satisfied by any interpretation that allows x to run over non-man objects

E.g., x can be a duck satisfying the formula


LimitationsLimitationsLimitationsLimitations

It is Ok to use the shorthand with:Algebraic predicates: e.g., x>0Set membership: e.g., xD

Why?Because it is clear that your quantifiers is applied to the variable which follows it immediately:

x>0 [x2=1]

It is not Ok to use such shorthands with other predicates:

E.g., P(x) [x2=1]


Special cases : finite Special cases : finite domainsdomains

Special cases : finite Special cases : finite domainsdomains

Suppose you have a formula over a single variable x

If the domain of variable x is finite:{x1,…,xN}

Then x P(x) is equivalent to:P(x1) & … & P(xN)

And x P(x) is equivalent to:P(x1) v … v P(xN)


Satisfying Satisfying // statementsstatements

Satisfying Satisfying // statementsstatements

To show that x P(x) we need to:Prove that every x satisfies P(x)

To show that x P(x) we need to:Prove that there exists at least one x that satisfies P(x)


De Morgan again…De Morgan again…De Morgan again…De Morgan again…

To show that ~(x P(x)) we need to:Prove that not every x satisfies P(x)In other words, we need to find at least one x that does not satisfy P(X)But that is the same as proving that x ~P(x)

Therefore:~(x P(x)) x (~P(x))

are logically equivalentDe Morgan’s law for quantifiers


And again…And again…And again…And again…

To show that ~(x P(x)) we need to:Prove that there does not exists a single x satisfies P(x)In other words, we need to show that all x do not satisfy P(X)But that is the same as proving that x ~P(x)

Therefore:~(x P(x)) x (~P(x))

are logically equivalentDe Morgan’s law for quantifiers



Statement:“Everyone snoozes in CMPUT 272”

Negation:“Not every one snoozes in CMPUT272”“There is someone who doesn’t snooze in CMPUT 272”

A common mistake:“Everyone does NOT snooze in CMPUT 272”



Statement:“Professors don’t make much money”

Negation:“There is a professor who makes much money”

A common mistake:“Professors make much money”



Statement:There is a secret agent who appeals to all women

Negation:For every secret agent there is a woman that the agent doesn’t appeal to

A common mistake:There is a secret agent who doesn’t appeal to all women


In a nutshell…In a nutshell…In a nutshell…In a nutshell…

When negating a quantified formula:Quantifier1 x1… QuantifierN xN P(x1,..,xN)

Do this:Change all to Change all to Negate the innermost formula P(…)

Be careful with scoping and connectives!


Vacuous Truth of UQSVacuous Truth of UQSVacuous Truth of UQSVacuous Truth of UQS

Suppose I say:“Presently, all men on the moon are happy”

Is it true or false?Think of it this way:

x [OnTheMoon(x) Happy(x)]

So, is it true or false?The internal implication is always vacuously true for there is presently no man on the MoonThus, the entire statement holds for any x

The entire statement holds


Logical ImplicationLogical ImplicationLogical ImplicationLogical Implication

In propositional logic formula A logically implies formula B iff:

Every interpretation that satisfies A also satisfies B

In predicate logic?The same!


Validity of ArgumentsValidity of ArgumentsValidity of ArgumentsValidity of Arguments

Hence validity of arguments is defined in the same way

The difference is: in predicate logic it is not always possible to go through all interpretations to prove that A logically implies B

Why?The number of interpretations can be infinite


Inference RulesInference RulesInference RulesInference Rules

Thus, proving arguments with inference rules becomes the method of choice

We can also derive new inference rules for our toolbox


Universal InstantiationUniversal InstantiationUniversal InstantiationUniversal Instantiation

Consider a universally quantified statement:

xD [P(x)]

Suppose you have a particular x0D

What can you say about P(x0)?It holds!

Why? Because predicate P(…) holds for all members of set DExample: “mortal Socrates” argument


Universal Modus Universal Modus PonensPonens

Universal Modus Universal Modus PonensPonens

Propositional modus ponens PQPThus, Q

Universal modus ponensx [P(x)Q(x)]P(a)Thus, Q(a)


Diagrams for Validity (p. Diagrams for Validity (p. 104)104)

Diagrams for Validity (p. Diagrams for Validity (p. 104)104)

Diagrams can sometimes be used to:support a validity of an argumentor, show that an argument is invalid

Diagrams are not a formal proof!Use them to illustrate your reasoning ONLYExamples:

Valid : Modus PonensInvalid : abduction

mortalmortal

human

SocrateSocratess

SocrateSocratess

pneumoniapneumonia

fever

patientpatientpatientpatient

sept 18, 2003© vadim bulitko : cmput 272, fall 2003, uofa1 cmput 272 formal systems logic in cs i....

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