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Ontologies

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

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Ontologies

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  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)

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