Attempto Controlled English Language, Tools and Applications
Exercises 3

Reasoning in ACE

By reasoning we can find out whether a given ACE text contains a contradiction, whether a given ACE text is entailed by another ACE text, whether two ACE texts are semanticaly equivalent (i.e. the first entails the second and the second entails the first), etc. Because ACE is a formal language, this kind of reasoning can be automated, but in the following exercises we try to reason manually.

In the following, assume that ACE is mapped to first-order logic with the standard semantics, e.g. including the open world assumption which states that everything that is not specified is simply unknown and not necessarily false.

Assume also that there is no background knowledge. The texts speak for themselves, there is no text in the background. The only background knowledge is knowledge about numbers, e.g. that 5 is smaller than 7, and linguistic knowledge about the words "somebody" and "something", i.e.

∀ X object(X, atomic, _, person, _, _, _, _) → object(X, dom, _, person, _, _, _, _)
∀ X object(X, atomic, _, object, _, _, _, _) → object(X, dom, _, object, _, _, _, _)

Discuss the key concepts

Entailment
Redundancy
Informativity
Contradiction
Consistentcy
Satisfiability
Validity

Order the following texts by entailment

John is a mortal.
Every man is a human and every human is a mortal. John is a man.
Every man is a mortal. John is a man.
Somebody is a mortal.

2, 3, 1, 4

Does the entailment hold?

(no) "Every time-machine is a machine." → "There is a time-machine."
(no) "Every man owns a car." → "There is a man."
(no) "Every man owns a car." → "A man owns a car."
(yes) "John is a man. Every man owns a car." → "A man owns a car."
(yes) "John is a man. Every man owns a car." → "John owns a car."
(no) "No man owns a car." → "A man does not own a car."
(yes) "John is a man. No man owns a car." → "A man does not own a car."
(yes) "John is a man. No man owns a car." → "John does not own a car."
(yes) "John is a man. Every man owns more than 2 cars." → "John owns at least 3 cars."
(no) "John is a man. Every man owns more than 2 cars." → "John owns less than 3 cars."
(no) "John is a man. Every man owns less than 2 cars." → "John owns a car."
(no) "No man owns something that is not a car." → "Every man owns a car."
(yes) "No man is a woman." → "No woman is a man."

Write an ACE sentence which follows from the given text

Every man is a human.
Every man is a human. John is a man.
Every man is a human. John is a human.
No man is a woman. John is a man.
No man is a woman. Mary is a woman.
Every man owns a car. John is a man.
No man owns something that is not a car. John is a man.
No man owns something that is not a car. John who is a man owns something.

"Every man is a human." entails e.g.

"Every man is somebody."
"Everybody who is not a human is not a man."

Which of the following ACE texts are semantically equivalent?

i.e. number the texts so that the texts with the same meaning have also the same number.

Implication and coordination

(1) Every dog is an animal and is a pet.
(1) Every dog is an animal and every dog is a pet.
(2) Every programmer likes Java or likes Perl.
(3) Every programmer likes Java and every programmer likes Perl.
(4) Every programmer likes Java or every programmer likes Perl.
(2) Every programmer likes something that is Java or that is Perl.

Implication and negation

(1) Every kinase is a polymerase.
(2) Every polymerase is a kinase.
(3) No kinase is a polymerase.
(3) Every kinase is not a polymerase.
(4) Every kinase is a non-polymerase.
(3) No polymerase is a kinase.
(5) Everything that is not a kinase is a polymerase.
(5) Everything that is not a polymerase is a kinase.
(2) Everything that is not a kinase is not a polymerase.
(1) Everything that is not a polymerase is not a kinase.
(3) Nothing that is a kinase is a polymerase.
(3) Nothing that is a polymerase is a kinase.
(2) Nothing that is not a kinase is a polymerase.
(1) Nothing that is not a polymerase is a kinase.

Leaving aside the sentence with "non-polymerase" which cannot be linked to "polymerase" unless we have some background knowledge, there are basically 8 cases, and 4 equivalence classes.

The cases are the following (in propositional logic for conciseness reasons):

A → B
B → A
A → ¬ B
B → ¬ A
¬ A → B
¬ B → A
¬ A → ¬ B
¬ B → ¬ A

The representatives of the equivalent classes are:

A → B
B → A
A → ¬ B
¬ A → B

Implication, negation and generalized quantifiers

(1) No customer owns more than 5 cars.
(1) Every customer owns less than 6 cars.

More complex sentences

(1) Everybody that owns something that is not a bike is not a student.
(1) No student owns something that is not a bike.
(1) Everything that a student owns is a bike.
(2) No man sees somebody that is not somebody that is a singer and that is a violinist.
(2) Everybody that a man sees is a singer and is a violinist.
(2) Everybody that is seen by a man is a singer and is a violinist.
(2) If a man sees somebody then he/she is a singer and is a violinist.
(ERROR) If a man sees somebody then he/she is a singer. He/she is a violinist.
(2) Everybody that a man sees is a singer and everybody that a man sees is a violinist.

Inconsistency

Write a short ACE text which is inconsistent.
Continue the ACE text "John likes Mary." so that it becomes inconsistent.
Continue the ACE text "No dog likes a cat. Fido is a dog that loves a cat." with an every-sentence so that the text becomes inconsistent.
Continue the ACE text "No dog likes a cat. Fido is a dog that loves a cat." with an every-sentence which doesn't contain any common nouns so that the text becomes inconsistent.
Continue the ACE text "Everybody owns at most 2 cars." so that it becomes inconsistent.
The text "Everybody who John likes is a woman. Mary is not a woman. John likes Mary." is inconsistent. How can this text be turned into a consistent text. (There are various ways, list all of them.)

If Fido loves a cat then Fido likes the cat.
If a dog loves a cat then the dog likes the cat.
If something X loves something Y then X likes Y.

Alternatively, one can use every-sentences, e.g.
Every dog that loves a cat likes the cat.

Do those statements hold?

Removing all sentences that contain anaphoric references from a consistent ACE text can make the text inconsistent.
Adding/removing a sentence to/from a/an consistent/inconsitent ACE text always/sometimes/never makes it consistent/inconsistent. (i.e. 2 * 2 * 3 * 2 = 24 cases to discuss)

Norbert E. Fuchs, Kaarel Kaljurand {fuchs,kalju}@ifi.uzh.ch

Attempto Controlled English Language, Tools and Applications Exercises 3