Semester 3 / Artificial Intelligence /Knowledge Representation

Knowledge Representation

3 min read Updated Fri Apr 24 2026 07:36:29 GMT+0000 (Coordinated Universal Time)
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Defines how information about the world is stored and structured in knowledge base. Focuses on what knowledge should be represented and how.

Knowledge representation can either be done using propositional logic or first-order logic. Here, first-order logic is assumed.

Substance

Substance is continuous, divisible.

Can either be:

  • Intrinsic: color, density (retain under division).
  • Extrinsic: weight, shape (lost when divided).

Object

Defines a physical entity with properties and relationships. Discrete entities.

Physical Composition

Objects can be parts of other objects using the relation:

  • Transitive
    PartOf(x,y)∧PartOf(y,z)  ⟹  PartOf(x,z)\text{PartOf}(x, y) \land \text{PartOf}(y, z) \implies \text{PartOf}(x, z)
  • Reflexive
    PartOf(x,x)\text{PartOf}(x, x)

Composite Objects

Objects made of multiple objects.

Categories

Group of objects. Grouped for reasoning and prediction. Agents use categories to:

  • Infer unseen object properties.
  • Predict behavior from category membership.

Relations between a group and its members:

  • Membership
    B∈Basketballs\text{B} \in \text{Basketballs}
  • Subclass
    Basketballs⊂Balls\text{Basketballs} \subset \text{Balls}
  • Property inheritance
    (x∈Basketballs)  ⟹  Spherical(x)(x \in \text{Basketballs}) \implies \text{Spherical}(x)

Relations between 2 groups:

  • Disjoint: no shared members
    Disjoint(Animals, Vegetables)\text{Disjoint({Animals, Vegetables})}
  • Exhaustive decomposition: must belong to one
    ExhaustiveDecomposition(Americans, Canadians, Mexicans, NorthAmericans)\text{ExhaustiveDecomposition({Americans, Canadians, Mexicans}, NorthAmericans)}
  • Partition: both disjoint and exhaustive
    Partition(Males, Females, Animals)\text{Partition({Males, Females}, Animals)}

Measurements

Objects can have measurable properties. Objects can be ordered by such measures even when non-numeric.

Quantitative

Numeric objective measurements such as length, weight.

Qualitative

Non-numeric subjective measurements such as beauty, spiciness.

Time and Event Calculus

Basic Terms

  • Action: logical term representing activity (e.g., Turn(Right))\text{Turn}(\text{Right}))
  • Fluent: predicate/function varying over time
  • Atemporal predicates: permanent truths (e.g., Gold(G1)\text{Gold}(\text{G1}))

Event Calculus

Used to model actions over time intervals.

Predicates

PredicateMeaning
T(f,t)\text{T}(f, t)Fluent ff true at time tt
Happens(e,i)\text{Happens}(e, i)Event ee occurs over interval ii
Initiates(e,f,t)\text{Initiates}(e, f, t)ee starts ff at time tt
Terminates(e,f,t)\text{Terminates}(e, f, t)ee stops ff at time tt
Clipped(f,i)\text{Clipped}(f, i)ff ceases to hold within ii
Restored(f,i)\text{Restored}(f, i)ff becomes true within ii

Axioms

  1. Happens(e,(t1,t2))∧Initiates(e,f,t1)∧¬Clipped(f,(t1,t))∧t1<t⇒T(f,t)\text{Happens}(e, (t_1,t_2)) \land \text{Initiates}(e, f, t_1) \land \text{¬Clipped}(f, (t_1,t)) ∧ t_1<t ⇒ T(f, t)
  2. Happens(e,(t1,t2))∧Terminates(e,f,t1)∧¬Restored(f,(t1,t))∧t1<t⇒¬T(f,t)\text{Happens}(e, (t_1,t_2)) \land \text{Terminates}(e, f, t_1) ∧ ¬\text{Restored}(f, (t_1,t)) ∧ t_1<t ⇒ ¬T(f, t)

Processes

Aka. liquid events. Continuous events where any subinterval is also valid:

Suppose e∈Processese \in \text{Processes} and tnt_n denotes a timestamp (tn<tn+1t_n \lt t_{n+1}):

Happens(e,(t1,t4))  ⟹  Happens(e,(t2,t3))\text{Happens}(e, (t_1,t_4)) \implies \text{Happens}(e, (t_2,t_3))

Intervals

Zero duration intervals are moments. Non-zero duration intervals are extended intervals.

Allen’s Relations

RelationDefinition
Meet(i,j)\text{Meet}(i,j)End(i)=Begin(j)\text{End}(i)=\text{Begin}(j)
Before(i,j)\text{Before}(i,j)End(i)<Begin(j)\text{End}(i)<\text{Begin}(j)
During(i,j)\text{During}(i,j)Begin(j)<Begin(i)<End(i)<End(j)\text{Begin}(j)<\text{Begin}(i)<\text{End}(i)<\text{End}(j)
Overlap(i,j)\text{Overlap}(i,j)Begin(i)<Begin(j)<End(i)<End(j)\text{Begin}(i)<\text{Begin}(j)<\text{End}(i)<\text{End}(j)
Begins(i,j)\text{Begins}(i,j)Begin(i)=Begin(j)\text{Begin}(i)=\text{Begin}(j)
Finishes(i,j)\text{Finishes}(i,j)End(i)=End(j)\text{End}(i)=\text{End}(j)
Equals(i,j)\text{Equals}(i,j)Begin(i)=Begin(j)∧End(i)=End(j)\text{Begin}(i)=\text{Begin}(j) ∧ \text{End}(i)=\text{End}(j)

Mental Events and Mental Objects

Agents can reason about their own and others’ beliefs.

Knowledge bases now include mental objects (beliefs, intentions, goals).

Relations expressing mental states:

  • Believes(Lois,Flies(Superman))\text{Believes}(\text{Lois}, \text{Flies}(\text{Superman}))
  • Knows(Lois,CanFly(Superman))\text{Knows}(\text{Lois}, \text{CanFly}(\text{Superman}))

Referential Opacity

Substitution inside belief statements may fail:

(Superman=Clark)∧Knows(Lois,CanFly(Superman))(\text{Superman} = \text{Clark}) ∧ \text{Knows}(\text{Lois}, \text{CanFly}(\text{Superman})) ≠Knows(Lois,CanFly(Clark))\neq \text{Knows}(\text{Lois}, \text{CanFly}(\text{Clark}))