CSCE 420 Lecture 20

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Homework 4 Due next week

First-Order Logic

Showing Categories, Defintions, and Part-Whole Relationships:

∀x dog(x) → mammal(x)
∀y banana(y) → fruit(y)
∀x finger(x) → ∃y hand(y) ∧ partof(x,y)
∃c,d computer(c) ∧ hard_drive(d) ∧ ¬working(d) ∧ partof(d,c)
∃e case(e,c) ∧ in(d,e)

Other example spatial relationships:

• on
• over
• around
• above
• touching/contact
• left-of
• north-of (geographical)

Axioms:

∀x,y,z in(x,y) ∧ in(y,z) → in(x,z)
∀x,y in(x,y) → ¬in(y,x)

Level of Detail is qualitative, but in some cases (e.g. coordinate system) it is also quantitative.

• qualitative: is a particular building north of another building?

"There is a gallon of milk in Bill's shopping cart."

∃m milk(m) ∧ ∃s cart(s) ∧ use(s,shopping) ∧ belongs^?(s, bill) ∧ in(m,s) ∧ volume(m) = gallons(1)

∀ x,y volume(x) = gallons(y) → volume(x) = quarts(4*y)

gallons and quarts are special functions that map numbers into quantity scales (units)

Event Calculus

What is an event? A thing that has some temporal extent

Assume an infinite continuum (totally ordered) of time points (i.e. a timeline).

• Events occupy closed intervals on the timeline (e.g. [a,b])

In FOL, we use happens(e,i) that maps an event to an interval.

thanksgiving_of_2011 = [2011-11-23T00:00:00-06:00, 2011-11-23T23:59:59-06:00]
∃x game(x) ∧ type_of_game(x,football) ∧ duration(x) = hours(3) ∧ location(x,college_station) ∧
    ∃i happens(x,i) ∧ during(i, thanksgiving_of_2011)

Interval Logic:

begin(i) ⟼ time_point
end(i) ⟼ time_point

∀i,j during(i,j) ↔ begin(i) ≥ begin(j) ∧ end(i) ≤ end(j)
∀i,j before(i,j) ↔ end(i) ≤ begin(j)
∀i,j after(i,j) ↔ before(j,i)
∀i,j meets(i,j) ↔ end(i) = begin(j) ∨ end(j) = begin(i)
∀i,j overlap(i,j) ↔ begin(j) ≤ end(i) ∧ begin(j) ≥ begin(i)

Fluents: dynamic conditions that can change over time:

# Are the lights on in HRBB 113 right now?
# T(f,t): Throughout an interval
T(on(lights(113_HRBB)), now)

Other Fluent Predicates:

• initiates
• teminates
• clipped
• restored

John Graduated in 2013

graduated(john, year(2013)) → ∃g graduation(g) ∧ school(g) = TAMU ∧
    ∃i happens(g,i) ∧ interval(i) ∧ during(i, year(2013)) ∧ participant(g,john) ∧
    ∃d,g receives(john,d,g) ∧ degree(d, BS) ∧ field(d,CS)

Cool logic stuff:

• Cycorp
• Semantic Web