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### Resolution Example and Exercises

Solutions to Selected Problems

** Example:**

Consider the following axioms:

- All hounds howl at night.
- Anyone who has any cats will not have any mice.
- Light sleepers do not have anything which howls at night.
- John has either a cat or a hound.
- (Conclusion) If John is a light sleeper, then John does not

have any mice.

The conclusion can be proved using Resolution as shown below. The first

step is to write each axiom as a well-formed formula in first-order

predicate calculus. The clauses written for the above axioms are shown

below, using LS(x) for `light sleeper’.

The next step is to transform each wff into Prenex Normal Form, skolemize,

and rewrite as clauses in conjunctive normal form; these transformations

are shown below.

The set of CNF clauses for this problem is thus as follows:

Now we proceed to prove the conclusion by resolution using the above clauses.

Each result clause is numbered; the numbers of its parent clauses are shown

to its left.

[1.,4.(b):] 6.

[2,5.(c):]

7.

[7,5.(b):]

8.

[6,8:]

9.

[4.(a),9:]

10.

[3,10:]

11.

[4.(a),11:]

12.

[5.(a),12:]

13.

** Exercises:**

** 1.** Unify (if possible) the following pairs of predicates and give the

resulting substitutions. is a constant.

a.

b.

c.

** 2.** Consider the following axioms:

- Every child loves Santa.
- Everyone who loves Santa loves any reindeer.
- Rudolph is a reindeer, and Rudolph has a red nose.
- Anything which has a red nose is weird or is a clown.
- No reindeer is a clown.
- Scrooge does not love anything which is weird.
- (Conclusion) Scrooge is not a child.

Represent these axioms in predicate calculus; skolemize as necessary

and convert each formula to clause form. (Note: `has a red nose’ can

be a single predicate. Remember to negate the conclusion.) Prove the

unsatisfiability of the set of clauses by resolution.

** 3.** Consider the following axioms:

- Anyone who buys carrots by the bushel owns either a rabbit or a

grocery store. - Every dog chases some rabbit.
- Mary buys carrots by the bushel.
- Anyone who owns a rabbit hates anything that chases any rabbit.
- John owns a dog.
- Someone who hates something owned by another person will not date

that person. - (Conclusion) If Mary does not own a grocery store, she will not date

John.

Represent these clauses in predicate calculus, using only those predicates

which are necessary. For example, you need not represent `person’, and

phrases such as `who buys carrots by the bushel’ may be represented by

a single predicate. Negate the conclusion and convert to clause form,

skolemizing as necessary. Prove the unsatisfiability of the resulting

set of clauses by resolution.

** 4.** Consider the following axioms:

- Every Austinite who is not conservative loves some armadillo.
- Anyone who wears maroon-and-white shirts is an Aggie.
- Every Aggie loves every dog.
- Nobody who loves every dog loves any armadillo.
- Clem is an Austinite, and Clem wears maroon-and-white shirts.
- (Conclusion) Is there a conservative Austinite?

** 5.** Consider the following axioms:

- Anyone whom Mary loves is a football star.
- Any student who does not pass does not play.
- John is a student.
- Any student who does not study does not pass.
- Anyone who does not play is not a football star.
- (Conclusion) If John does not study, then Mary does not love John.

** 6.** Consider the following axioms:

- Every coyote chases some roadrunner.
- Every roadrunner who says “beep-beep” is smart.
- No coyote catches any smart roadrunner.
- Any coyote who chases some roadrunner but does not

catch it is frustrated. - (Conclusion) If all roadrunners say “beep-beep”, then all coyotes

are frustrated.

** 7.** Consider the following axioms:

- Anyone who rides any Harley is a rough character.
- Every biker rides [something that is] either a Harley or a BMW.
- Anyone who rides any BMW is a yuppie.
- Every yuppie is a lawyer.
- Any nice girl does not date anyone who is a rough character.
- Mary is a nice girl, and John is a biker.
- (Conclusion) If John is not a lawyer, then Mary does not date John.

** 8.** Consider the following axioms:

- Every child loves anyone who gives the child any present.
- Every child will be given some present by Santa if Santa

can travel on Christmas eve. - It is foggy on Christmas eve.
- Anytime it is foggy, anyone can travel if he has some

source of light. - Any reindeer with a red nose is a source of light.
- (Conclusion) If Santa has some reindeer with a red nose, then

every child loves Santa.

** 9.** Consider the following axioms:

- Every investor bought [something that is] stocks or bonds.
- If the Dow-Jones Average crashes, then all stocks that are

not gold stocks fall. - If the T-Bill interest rate rises, then all bonds fall.
- Every investor who bought something that falls is not happy.
- (Conclusion) If the Dow-Jones Average crashes and the T-Bill

interest rate rises, then any investor who is happy bought some gold stock.

** 10.** Consider the following axioms:

- Every child loves every candy.
- Anyone who loves some candy is not a nutrition fanatic.
- Anyone who eats any pumpkin is a nutrition fanatic.
- Anyone who buys any pumpkin either carves it or eats it.
- John buys a pumpkin.
- Lifesavers is a candy.
- (Conclusion) If John is a child, then John carves some pumpkin.

** 11.** Consider the following axioms:

- Every tree that is an oak contains some grackle.
- If anyone walks under any tree that contains any grackle,

then he hates every grackle. - For every building, there is some tree that is beside it.
- Taylor Hall is a building.
- Every CS student visits Taylor Hall.
- If anyone visits any building, then he walks under every tree

that is beside that building. - (Conclusion) If some CS student does not hate some grackle,

then there is some tree beside Taylor Hall that is not an oak.

** 12.** Consider the following axioms:

- Every child sees some witch.
- No witch has both a black cat and a pointed hat.
- Every witch is good or bad.
- Every child who sees any good witch gets candy.
- Every witch that is bad has a black cat.
- (Conclusion) If every witch that is seen by any child has a pointed hat,

then every child gets candy.

** 13.** Consider the following axioms:

- Every boy or girl is a child.
- Every child gets a doll or a train or a lump of coal.
- No boy gets any doll.
- No child who is good gets any lump of coal.
- (Conclusion) If no child gets a train, then no boy is good.

** 14.** Consider the following axioms:

- Every child who finds some [thing that is an] egg or chocolate

bunny is happy. - Every child who is helped finds some egg.
- Every child who is not young or who tries hard finds some

chocolate bunny. - (Conclusion) If every young child tries hard or is helped, then

every child is happy.

** 15.** Consider the following axioms:

- Anything that is played by any student is tennis, soccer, or chess.
- Anything that is chess is not vigorous.
- Anyone who is healthy plays something that is vigorous.
- Anyone who plays any chess does not play any soccer.
- (Conclusion) If every student is healthy, then every student

who plays any chess plays some tennis.

** 16.** Consider the following axioms:

- Every student who makes good grades is brilliant or studies.
- Every student who is a CS major has some roommate.

[Make “roommate” a two-place predicate.] - Every student who has any roommate who likes to party

goes to Sixth Street. - Anyone who goes to Sixth Street does not study.
- (Conclusion) If every roommate of every CS major likes to party,

then every student who is a CS major and makes good grades is brilliant.

** 17.** Consider the following axioms:

- Everyone who aces any final exam studies or is brilliant or is lucky.
- Everyone who makes an A aces some final exam.
- No CS major is lucky.
- Anyone who drinks beer does not study.
- (Conclusion) If every CS major makes an A, then every CS major

who drinks beer is brilliant.

** 18.** Consider the following axioms:

- Anyone who loves any lottery is a gambler.
- Everyone who favors the lottery proposition loves some lottery.
- Everyone favors the lottery proposition or opposes the lottery

proposition. - If every Baptist votes and opposes the lottery proposition, then

the lottery proposition does not win. - Every Baptist who is faithful is not a gambler.
- (Conclusion) If every Baptist votes and the lottery proposition wins,

then some Baptist is not faithful.

** 19.** Consider the following axioms. Hint: the predicates WHITE,

SNOWY, HAPPY, and GETS should each have a “time” argument.

- Anyone who owns any sled is happy when it is snowy.
- When it is white, it is snowy.
- If Santa is happy at Christmas, then every child who is good

gets some toy at Christmas. - Any child who gets a toy at any time is happy at that time.
- Santa owns a sled.
- (Conclusion) If it is white at Christmas and every child who is

not good owns some sled, then every child is happy at Christmas.

** 20.** Consider the following axioms.

- Anyone who is on Sixth Street and is not a police officer has some

costume. - No CS student is a police officer.
- Every costume that is good is a robot costume.
- For anyone, if they are on Sixth Street and are happy, then every

costume they have is good or they are drunk. - (Conclusion) If no CS student is drunk and every CS student on

Sixth Street is happy, then every CS student on Sixth Street has some

costume that is a robot costume.

** 21.** Consider the following axioms.

- For every mall, there is some Santa who is at the mall.
- Every child who visits anywhere talks with every Santa who is at

the place visited. [Don’t make a predicate for “the place visited”;

it should just be a variable.] - Every child who is a city child visits some mall.
- Every child who is good or who talks with some Santa gets some toy.
- (Conclusion) If every child who is not a city child is good,

then every child gets some toy.

** 22.** Consider the following axioms.

- Everyone who feels warm either is drunk, or every costume they

have is warm. - Every costume that is warm is furry.
- Every AI student is a CS student.
- Every AI student has some robot costume.
- No robot costume is furry.
- (Conclusion) If every CS student feels warm, then every AI

student is drunk.

** 23.** Consider the following axioms.

- Every bird sleeps in some tree.
- Every loon is a bird, and every loon is aquatic.
- Every tree in which any aquatic bird sleeps is beside some lake.
- Anything that sleeps in anything that is beside any lake eats fish.
- (Conclusion) Every loon eats fish.

** 24.** Consider the following axioms.

- Everyone is like him/her self.
- If someone is a member of a club, then they want to be a member

and the club will accept them. - Groucho does not want to be a member of any club

that will accept anyone like him. - (Conclusion) Groucho is not a member of any club.

** 25.**“Schubert’s Steamroller”, for the ambitious. (See C. Walther,

AAAI-84, p. 330.)

Wolves, foxes, birds, caterpillars, and snails are animals, and there

are some of each of them. Also there are some grains, and grains are

plants. Every animal either likes to eat all plants or all animals much

smaller than itself that like to eat some plants. Caterpillars and snails

are much smaller than birds, which are much smaller than foxes, which

in turn are much smaller than wolves. Wolves do not like to eat foxes

or grains, while birds like to eat caterpillars but not snails. Caterpillars

and snails like to eat some plants. Therefore there is an animal that

likes to eat a grain-eating animal.

Solutions to Selected Problems

## [NEW] The Basics of the Excel IF Function | if will – NATAVIGUIDES

The IF function is useful for making quick decisions or performing simple comparisons in Excel. The solution lies in understanding how to use the IF function in Excel. It’s an easy but versatile logical function for handling many scenarios, ranging from detailed calculations to even assigning allowances. (Tutorial includes Excel worksheet with formula examples.)

What is Excel’s IF Function?

The **IF function** is one of the simplest and most useful Excel functions. It can fill cell fields for you based on evaluating a condition such as a cell’s content. You can find it in the **Logical** category.

The wizard-like dialog allows you to fill 3 **Function Arguments** or data elements. This is the easiest way to learn Excel formulas because you can see if it returns your expected result. For example, at the bottom left of the dialog, a line reads “**Formula result =**.”

Pin

Blank IF function dialog with empty Formula result

The IF Function Arguments

FieldDefinitionLogical_testA test on a cell value that is either TRUE or FALSE.Value_if_trueThe value Excel will put in a cell if the test is true.Value_if_falseThe value Excel will put in a cell if the test fails.

Despite not having Microsoft Excel, my parents routinely employed this type of IF logic when calculating my allowance. Their version read:

**IF** you empty the garbage **AND** mow the lawn **AND** wash the dishes **AND** walk the dog, you **get your full allowance**. And since I grew up in New England, this logic would change with the seasons to account for things like leaves and snow.

Setting Up the IF Function

Although Excel can’t issue an allowance, it can calculate the amount using a **logic test** based on whether a cell met a formula condition.

For example, I could create a spreadsheet with the tasks needed to get an allowance. If the task was completed (TRUE situation), a value would be applied toward the allowance. If the task wasn’t completed (FALSE situation), nothing would be added.

These examples are noted by labels [1] and [2] in the screen snap below.

Pin

Logical test examples

Using the example above, you might express the logic in the following way:

**IF** cell **B2 equals “Y” **, then use the Rate value from cell C2 ($3.00) in D2.

**IF** cell **B2 does not equal “Y”**, then place 0 in cell D2.

As you can see in this example, the IF logical condition is either TRUE or FALSE. And it pays to take out the garbage.

To enter your IF Function Arguments,

- Click the spreadsheet cell where you wish to use the Excel formula.
- From the Formulas tab, click Insert function…
- In the Insert Function dialog text box, type “if”.
- Make sure your cursor is in the Logical_test text box.
- Click the spreadsheet cell you wish to evaluate. Excel will fill in the cell reference such as “B2”.
- Add the equals sign = and your desired value in quotes. For example =”Y”.
- In the Value_if_true field, type the value you would like entered in your cell if B2 equals “Y”. In our example, I’ll click cell C3.
- In the Value_if_false: field, enter the value the cell should have if B2 does not have a “Y”. I’ll enter 0. I could leave it blank, but the cell would show “FALSE”.
- Review the dialog to see if the
**Formula result=**value (label 1 below) is what you expect. If not, check to see if any errors show to the right of the fields (label [2] below).

Pin

Example of a completed IF formula and result

- Click
**OK**. - Copy the formula to the other cells in your column.

✪** Tip**: Even though the **Value_if_false** field is optional, it’s best to provide a value. Otherwise, Excel will use FALSE in the cell value.

Excel IF Example 2

The above spreadsheet might have been Version 1 for my parents. A new incentive program would appear based on some parent/child negotiations and competitive neighborhood rates. I probably would’ve requested pay for partial tasks. No doubt, my parents would counter with a penalty clause if something was less than half done.

Excel is flexible when it comes to IF statements and can evaluate more than a simple “Y” or “N.” For example if we convert our previous **Done?** column to a **% Done** column with a number, we can accommodate these new requirements such as:

`=IF(B2>0.5,B2*C2,-C2)`

The new formula returns the allowance based on the % Done in Column B. If the task completion number is greater than .5, a prorated amount was applied to the allowance. If the task completion rate was .5 or below, a negative amount was applied to the allowance. Loosely translated, a “half-assed” performance costs money. You could also apply colors using conditional formatting.

Pin

Using a numeric value instead of Y/N values

Excel’s IF function is a versatile and useful function. Once you get the hang of it, you’ll start using it in more scenarios. The two examples presented here were foundational. But you can use IF functions to handle other transactions such as applying sales tax, shipping charges, fixing Excel DIV 0 errors, or even nested IF functions with Boolean logic. And if you have kids, let them build the Excel spreadsheet and give them a bonus for using the IF function.

Related Excel Resource

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