Logical Commonsense

Bijan Parsia

Commonsense knowledge

• We all know quite a bit
• If the power is out, there will be no hot shower, nor tea unless the kettle just popped which means the power outage is recent and the shower will be hot if one hurries.
• I can't see my foot, but it's still there.
• I can't take a train to Atlanta from Manchester.
• I've started balding. It will only get worse as I get older, unless someone finds a cure.
  
• Much of what we know is wrong, incomplete, or tentative
• If I leave now, I can catch the train.
• Hair plugs or a hair piece will fool people.
• I am incredibly witty when drunk.
• You can't extrapolate from 547 to 665,000.
  
• Explicating what we know is surprisingly complex
  

Four Sorts of Problem

• Representing...
• Toy Problems
• Yale Shooting Problem, Suitcase Problem, "Logic puzzles"
• Microworlds
• Model first, then axiomatize
• Artifical: Blocks world, Roller coaster, Zoo
• Single domain: Time, Space, Action, Solid Objects
• Mid-sized problems
• Egg Cracking Problem, Suprise Birthday Problem
• Deep, multi-domain, realistic, no prior model
• Large to very large chunks
• Cyc, HPKB, ConceptNet
• 100ks of axioms
• Broad rather than deep

A Mathy Example

• Basic ordering
• (∀X,Y)(X<Y ⇒ ¬(Y<X))
• (∀X,Y,Z)(X<Y ∧ Y<Z ⇒ X<Z)
• (∀X,Y)(X<Y ∨ Y<X ∨ X=Y)
• We can relax totality to get a partial order
• Intervals (several approaches)
• Take a primative:
• meets(I, J)
• A certain set of points from a totally ordered space
• invl(I)⇔(∀X,Y,Z)(X,Y∈I∧X<Z<Y z∈I)
• lb(X,I) ⇔ (∀Y∈I)(Y>=X)
• ub(X,I) ⇔ (∀Y∈I)(Y<=X)
• X=st(I)⇔lb(X,I)∧(∀Y)(lb(Y,I)Y<=X)
• X=end(I)⇔ub(X,I)∧(∀Y)(ub(Y,I)Y>=X)

Interval relations (1)

meets(I, J)	meets(I,J) (∀X∈I,Y∈J)(X<Y) ∧ ¬(∃X∀W∈I,Z∈J)      ⇒(W<X<Z) end(I)=start(J)
before(I,J)	(∃K)(meets(I,K) ∧ meets(K,J)) (∃X∀W∈I,Z∈J)(W<X<Z) end(I)<start(J)

Interval relations (2)

overlaps(I,J)	(∃A-E)(meets(A,I)∧meets(I,D)∧ meets(D,E)∧meets(A,B)∧meets(B,J)∧meets(J,E)) (∃X∈I∀Y∈J)(X<Y)∧ (∃Y∈J∀X∈I)(X<Y) ∧ (∃X)(X∈I∧X∈J) start(I)<start(J)<end(I)<end(J)
starts(I,J) during(I,J) finishes(I,J) equal(I,J)

A Toy Everyday Example

• "Imagine a suitcase with two locks and a spring loaded mechanism which will open the suitcase when both locks are in the up position. Apparently, because of the spring loaded mechanism of the locks, then this action may also cause, as an indirect effect, the suitcase to open." --Suitcase Problem
• Simple, yes?
• ∀s(up(l1,S)∧up(l2,S)⇒open(S))
• (S is a situation not a suitcase! There is only one suitcase.)
• up(l1,S)∧¬open(S)⇒¬up(L2,S)
• Imagine a 2 lock, spring loaded suitcase, where if one lock turned up while the other is up and the suitcase close, the second lock will turn down.
• Logically equivalent axioms!
•  Involves causation.

Situation Calculus: Basics

• A theory of time (sort of), action, and happenings
• Objects, actions, situations
• Situations can be seen as states or as histories
• Actions
• Special function do(action, sit1) to sit2
• Special predicate Poss(action, sit)
• "Fluents"
• holds(f, s) where f is a formula  true at s
• Also use an extra situation argument for terseness
• up(l,s) is just holds(up(l), s)
• Basic axioms (abbreviated, including a missing second order one!)
• s ≠ do(a,s)
• do(a,s)=do(a',s')⇒(a=a' ∧ s=s')

Effects and causation

• A caused predicate
• Caused(P, true, S)⇒holds(P, S)
• Caused(P, false, S)⇒¬holds(P, S)
• true ≠ false ∧ (∀V)(V = true ∨ V = false)
•  (Direct) Effects
• Poss(flip(X), S)⇒
•   up(X,S)⇒Caused(up(X),false,do(flip(X),S))
• Poss(flip(X), S)⇒
•   ¬up(X,S)⇒Caused(up(X),true,do(flip(X),S))
• Lock theory
• Poss(flip(X), S)⇒lock(X)
• l1≠l2 ∧ lock(l1) ∧ lock(l2)
• up(l1,S)∧up(l2,S) ⇒ Caused(open, true, S)

Using FOL for Commonsense

• Powerful
• Function symbols alone let you do a lot
• Inequality is helpful
• Reasoners (human and automated) can work things out
•  Can be tedious
• Really must state quite a lot
• Tons missing in the above example solution to a toy problem
• Even in the more elaborated paper!
• Not powerful enough, always
• Sit-calc needs a second order axiom (like arithmetic)
• Can fake it often, though

A Seemingly Simple Example

• "Alice and Bob want to surprise their sister Carol with a joint present for her birthday, two weeks from now. They therefore go into a closed room to decide on the present and to plan how they will buy it." --Surprise Birthday Problem
But what does it involve?
• Calender and clock time and situations
• Actions, actors, and (multi-actor) plans
• Scheduling and coordination
• Beliefs, knowledge, likelihood, and a bit of pyschology (for being surprised)
• Cash, presents, gift giving
• Some sort of container/physical location theory
• ...

Elaboration tolorence (1)

• Evaluating a representation of a commonsense problem:
• Is it correct:
• Does it get the right answers as theorems?
• Does it avoid wrong answers?
• How strong answers?
• Is it complete:
• Does it get all the answers?
• Is it intelligible?
• Is it practical?
• Is it robust?
• For the last, the key question is how well does it tolorate additions and complications.
  

Elaboration tolorence (2)

• We need for the plan to not work:
• If Carol is also in the room.
• If the door is open and Carol is in the next room.
• If one of them tells Carol.
• If they do not consult together.
• If they can't agree on a present.
• If they wait until after Carol's birthday.
• It will probably not work if Carol sees that they have the present before they give it to her.
• On the other hand, it probably will work if Carol sees the present in the store.
• The plan will work if, instead,
  • They discuss the plan during a walk outside.
  • They pass a hidden written message.
  • They go together to buy the present, or one of them goes singly to buy the present.

Elaboration tolorence (3)

• Elaborations it's ok to miss:
• Carol spontaneously combusts before her birthday.
• Carol is a space alien with ESP.
• Carol finds out but space aliens erase her memory.
• If Carol has a certain neurological condition she'll be surprised since she is suprised by anything.
• Relations between surprise and brain states.
• If the present is toxic, then she'll be suprised
• Carol hates suprise parties, thus she'll be upset if she is surprised and miffed if she isn't.
• Scheduling can be done via a simple CSP.
• Three sorts of outlier elaboration:
• Bizarrities (but these are just things commonsense holds unlikely or false)
• Scientific things
• Out-of-scope (these are mid-sized problems)