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UTA027: Artificial Intelligence
TIET Patiala

A01: Predicate Calculus

Instructors:

1. Raghav B. Venkataramaiyer (bv.raghav)
2. Stuti Chug (stuti.chug)

Task

This exercise requires a student to translate situations (decision logic) described in natural english language to be translated to predicate calculus. Here are a few examples. There are a total of 75 instances for each student. The submission is in the form of a git repository the commit access to which shall be rescinded after the deadline.

Important Dates

S.No.	Desc	Timestamp
1.	Submission Starts	07-01-2025 0800 IST
2.	Submission Ends	20-01-2025 0500 IST

Getting started

This assignment should already be available on your userspace as a git repository. Navigate to the git repository in your userspace;

1. Navigate to generateAssignment.ipynb, (in your userspace).
2. Click “Open in Colab.”
   
3. Upon opening in Colab; the runtime should connect all by itself and the files pane should be populated. Click on folder/ files icon to view.
   
4. Fill you rollno in the widget; and Ctrl+F9 (or else, via Menu  ⧽  Runtime  ⧽  Run all). This should result in a CSV file named a01-<YOUR_ROLL_NO>.csv in the files pane. Click on three-dots to download it.
   
5. Edit the csv with your favourite spreadsheet editor (Google Sheets, MS Excel, LibreOffice, VIm, Emacs or otherwise) to fill in your answers.
6. Save or download the file as CSV, and commit back into your repository.
7. A submission may be updated multiple times before the deadline. So, one may save the progress online. By default a student assignment is a private repo; so unless exposed intentionally by the student, your submission is secure.

Some Tricks

In your lab session, your instructor may be able to guide to use save some time by:

1. Getting started with the Github Classrooms;
2. Save the file to google drive directly instead of downloading;
3. Editing within Github Codespace itself;
4. Using automated string substitution(s), e.g.
   FA X tomorrow?(X) AND rains?(X) IMPL bring?(me,umbrella)
   instead of
   ∀ X tomorrow?(X) ∧ rains?(X) → bring?(me,umbrella)
   and this translation is done automatically.

Examples

A situation description in natural language, may be expressed formally using predicate logic. The following ten cases serve as exemplars.

=== "Case 1"

*If it rains tomorrow, I'll bring an umbrella.*

`∀ X tomorrow?(X) ∧ rains?(X) → bring?(me,umbrella)`

Here, `X` is a variable chosen for day, and we know
that there are two predicates for the day, that
define it; namely its `tomorrow?`, and it `rains?`
on the day.  This forms our premise (or antecedent).
The consequent is that I (represented by a constant
`me` because `i` might be too generic) would bear an
umbrella.

=== "Case 2"

*If you heat water to 100°C, it boils.*

`∀ T geq?(temp(water, T),100) → boils?(water,T)`

From our understanding of the world, heating of
water is a process, which bears its signature as
water temperature.  Hence the author chose a
function `temp` to extract the substance property at
time `T`.  Function Expression `temp(water,T)`
represents that water temperature is being measured,
probably periodically through some sensor.  `geq?`
is a predicate defined for comparison, and returns
“truthy” if the first argument is greater than or
equal to the second.  `boils?` is a (predicate)
property that will be visible on water at time `T`
as a consequence.

=== "Case 3"

*If I had more time, I would learn to play piano.*

`∀ X greater?(free_time(me),X) → learning_piano?(me)`

More time is a relative concept; hence the reference
point here is taken as a variable `X`, to define the
premise.  And the nomenclature `learning_piano?` for
the predicate has been designed to reflect continuity.

Please note here that in the formulation here the
english grammatical tense makes no difference,
e.g. “If I am having more time, I am learning piano”
might be poor, perhaps incorrect, english but that’s
the rough meaning of this well-formed expression
(WFE) in predicate calculus (PC).

=== "Case 4"

*If she arrives early, we can start the meeting sooner.*

`∀ X less?(arr_time(she),X) → less?(start_time(meeting),X)`

Again here, understanding the world of discourse
would help frame a (WFE-PC)

Time is taken as variable and subtly embedded as
“return value” of a “function”.

=== "Case 5"

*If you mix blue and yellow, you get green.*

`∀ X blue?(X) ∧ ∀ Y yellow?(Y) → green?(mix(X,Y))`

`X` and `Y` have been chosen as variable colours,
because blue (or yellow or green) is not just one
colour, they are a range of colours. eg. both sky
and sea are blue, but not same.  And if you mix
either of them with *a yellow*, you get *a green*.

=== "Case 6"

*If I were you, I'd take that job offer.*

`as_if?(me,you) → ∀ X (job_offer?(X) ∧
accept?(me,X))`

=== "Case 7"

*If the store is open, could you buy some milk?*

`open?(store) → ∀ X may_buy_from?(you,store,milk)`

The question like “could you…?” is implicit in all
of *PC*, because the expression evaluates to true
only if certain conditions are met.  So, in *PC*,
this english sentence is equivalent to saying “If
the store is open, by me some milk.”

=== "Case 8"

*If my dog hears a noise, he always barks.*

`∀ X noise?(X) ∧ ∀ Y my_dog?(Y) ∧ hears?(X,Y) →
barks?(Y)`  
or,  
`∀ X noise?(X) ∧ ∀ Y dog?(Y) ∧ pet?(me, Y) ∧
hears?(X,Y) → barks?(Y)`

Please note the equivalence here: `my_dog?(Y) ≡
dog?(Y) ∧ pet?(me, Y)`

=== "Case 9"

*If we leave now, we'll catch the train.*

`eq?(dep_time(us),now()) → catch?(us,train)`  
or,  
`∀T now?(T) ∧ eq?(dep_time(us),T) → catch?(us,train)`

`now()` has essentially been represented, more
formally in the latter version, as a “unary
function” because it’s value changes depending upon
the time of invocation.

=== "Case 10"

*If you need help, just ask me.*

`need?(you,help) → ask?(you,me,help)`

*[PC]: Predicate Calculus *[WFE]: Well-Formed Expression