Ontologies

1. Ontologies • a system of concepts and relationships among them to reason about the world • commit to the existence of certain types of things, like mammals, or bodies, or gallons, or elections, or waves, or explosions • define categories or “types” • add axioms to define in terms of other concepts • add axioms to ascribe properties • causal rules: r,x Pit(s)&Adj(r,s)Breezy(r) • diagnostic rules: r Breezy(r)s Adj(r,s)&Pit(s) • choice of granularity depends on application, inferences required, scope of related concepts, context (in this cave, at this time...) read Sec 8.3-8.4, pp. 320-328 (10.1-10.2), 334-340, and 10.5 skip Situation Calculus, Beliefs, and Sec 10.6-10.7 for now - we will cover them later

2. Kinship • primitives: spouse(x,y), child(x,y), male(x), female(x) • x,y husband(x,y)  male(x) & spouse(x,y) • m mother(m)  female(m) & c child(c,m) • a,b child(a,b)  parent(b,a) • a,b grandparent(a,b)  c parent(a,c) & parent(c,b) • a,b grandfather(a,b)  grandparent(a,b) & male(a) • x,y sibling(x,y)  p parent(x,p) & parent(y,p) & xy

3. Taxonomies (Hierarchies) thing livingThing animal plant mammal bird fish dog human horse • x dog(x)mammal(x) • x human(x)mammal(x) • x mammal(x)animal(x) • x bird(x)animal(x) • x mammal(x)warm_blooded(x) & live_birth(x) & has_fur(x) & breast_feed_young(x) • x bird(x)lay_eggs(x) • x dog(x)numLegs(x)=4 • x human(x)numLegs(x)=2 • dog(snoopy) & human(charlie_brown) & canary(woodstock) • inheritance of properties via modus ponens

4. Sets • sets are formed from other sets by adjoining an item (empty set is basis) • s set(s)s= v x,s2 set(s2)&s={x|s2} • x,s {x|s}= • x,s xss={x|s} • a,b Subset(a,b)  (x xaxb) • a,b,x xIntersection(a,b)  (xa & xb)

5. Numbers • Natural numbers – Peano axioms • NatNum(0) • n NatNum(n)NatNum(succ(0)) • n 0succ(n) • n,m nm  succ(n)succ(m) • generates 0, S(0), S(S(0)), S(S(S(0)))... in any model • define addition (as function ‘+’): • n,m NatNum(n)&NatNum(m) +(succ(n),m)=succ(+(m,n)) • n NatNum(n)+(n,0)=n • define comparisons: • n,m n<mp NatNum(p) & p0 & m=+(n,p) • How to define...? • negatives, rationals, irrationals, reals, transfinite(,0), imaginary... • prove theorems in number theory, groups, fields, polynomials...

6. Lists • Lists are formed from nil (the empty list), or by concatenating items to other lists, function “[|]” • list(nil), x,y list(x)list([x|y]) • a,x,y a=x  member(a,y)member(a,[x|y]) • a,x,y first([x|y])=x & rest([x|y])=y • length(nil)=0; x,y length([x|y])=1+length(y)

7. Quantities • mass nouns vs. count nouns • merology, “some stuff” • “give me 3 apples and a (teaspoon of) sugar” • can’t say “a butter” or “a water” • units of measure: functions • bowl(p) & volume(p,gallons(1)) • what is the denotation of the second term? • can p hold more than 2 quarts? • x gallons(x)=quarts(4*x) • cost(book,dollars(18.50)) • dollars(18.50)=euros(12.38)

8. Space • predicates, often transitive, anti-reflexive • in, above, on, touching, support, • functions: surface, interior, frontOf, topOf • topological properties: connected, has_hole, 3D • partOf – physical subset • x,y human(x)&body(x,y)->a,b,c,d,e head(a)&arm(b)&arm(c)&leg(d)&leg(e) &partOf(a,y)&partOf(b,y)... • partOf(Germany,Europe) • if a person is in a car, is his hand in the car? • Shapes • a triangle has 3 sides; an isosceles triangle has 2 equal-length sides; an equilateral triangle has 3 sides of equal length • ...but how is this different than a square? or an asterisk? depends on intersection of edges; need to introduce points? • define blocks forming “arches”, or objects “arranged in a circle”

9. spatial relations • qualitative representation • a,b above(a,b)¬above(b,a) • a,b,c above(a,b)&above(b,c)above(a,c) • a,b above(a,b)below(b,a) • a,b on(a,b)above(a,b)&touching(a,b) • quantitative representation • a,b,p,q,px,py,qx,qy above(a,b) in(p,a)& coords(p,px,py) &in(q,b) &coords(q,qx,qy) pyqy &u,v,w in(u,a)&in(v,b)&in(w,a)...  uxvx & vxwx • must all of p be within vertical column above q? • depends on sizes, angle, distance? • ontology forces you to specify what you really mean • point of reference: geocentric, egocentric, landmarks • behind(bicycle,church) – relative to church/street/earth, me? • “going S. on Wellborn, turn left at the bell tower” – relative to my direction of motion • what is the difference between: “the helicopter is over the town” and the helicopter flew over the town” and “a blanket of snow covers over the town” (and “the test is over”)? • how can a fly be on the wall?

10. Fluids • Hayes (Naive Physics Manifesto, 1986) • a glass of water is not quite like a lake (with streams in and out) • 15 states of water: water can be... • static like a pond (bulk water) • bounded 3D region of space, except possibly top • flowing like a river • there is a region of space with two 2D portals on surface through which water is flowing, flux(A)=-flux(B) • filling or draining (bounded, volume incr/decr) • constantly re-filling like a lake or cup overflowing • wetting of a surface (2-dimensional) • spreading out on a 2D surface • mist in air • ontology: define types of water based on boundaries, dimensionality, change over time (movement, flux in/out) • (see link on web to look at axioms)

11. Time • frequently, we want to refer to when something was/will be true, or when an event occurred • fluents – facts whose truth value depends on time • could add extra arg to predicates for “time index” • e.g. open(NYSE,2:15pm), alive(Clinton,2009) • book also uses T(In(Shankar,NewYork),Today) • important: • no matter how you think of it, the time index is always an interval (at some level of granularity); • ontology assumes properties hold static within intervals • what are these objects (in our universe)? • moments: points on timeline, assume totally ordered • intervals: closed, bounded by start/end • predicates: start(int,tp), end(int,tp), in(tp,int) • denotations of ‘now’, ‘yesterday’, ‘lunchtime’, ‘1980’ • duration(yesterday)=hours(24)

12. Events • ‘structured’ objects in our universe • have an interval, among other properties • birthday_party(p) & host(p,john) & in_honor_of(p,sue) & location(p,house(john)) & SubEvent(p,Saturday) • map event to its interval using E:<evt X int> • ... & E(p,i) & during(i,Saturday) & start(i)=4:00 & duration(i)=hours(2) • can relate intervals to other intervals via predicates • before, during, meets, after, overlaps • “It rained just before the graduation ceremony yesterday afternoon.” • r,g raining_event(e) & graduation(g) & during(g,yesterday) & meets(e,g)

13. Interval Logic • primitives: • meet(i,j)  end(i)=start(j) • before(i,j)  end(i)<start(j) • after(i,j)  start(i)<end(j) <-> before(j,i) • during(i,j)  start(i)start(j) & end(i)end(j) • overlap(i,j) k during(k,i) & during(k,j) • inferences: • before(A,B) & before(B,C)  before(A,B) • during(A,B) & before(B,C)  before(A,C) • after(A,B) & meets(B,C)  ?

14. Processes • events that occur over an interval, as opposed to discrete points of time • Shankar flew to NewYork yesterday. • Shankar was flying to NewYork during lunch.

15. Event Calculus • these are “instantaneous events” that change the state • events happen at time points, fluents hold between events • happens(e,t) • initiates(e,f,t), terminates(e,f,t) • T(f,t1) <-> e,t2 Happens(e,t2) & Initiates(e,f,t2) & t2<t1 & Clipped(f,t1,t2) • fluent is “true throughout” interval • example: initiates(flip_switch,dark,12:00)