Knowledge Representation

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Knowledge Representation

• AIMA 2nd Ed. Chapter 10

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Ontological Engineering

• Ontological Engineering representing abstract
  concepts of the world. E.g. Actions, Time,
  Physical Objects, Beliefs.
• The general framework of concepts upper
  ontology.
• See fig. 10.1

3
Categories and Objects

• Categories organization of objects.
• Important because much reasoning takes place at
  the level of categories.
• Serve to simplify and organize knowledge base
  through inheritance.
• Subclass relations organize categories into
  taxonomy.
• FOL can be used to state facts
• An object is a member of category
• A category is a subclass of another category
• All members of a category have some properties
• Members of a category can be recognized by some
  properties
• A category as a whole has some properties

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Categories and Objects

• Two or more categories are disjoint if they have
  no members in common.
• E.g. males and females are disjoint, but we will
  not know that an animal that is not a male must
  be a female, unless males and females constitute
  an exhaustive decomposition.
• A disjoint exhaustive decomposition is known as
  partition.

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Physical Composition

• Use general PartOf relation to say that one thing
  is part of another. Objects can be grouped into
  part of hierarchies.
• Categories of composite objects are often
  characterized by structural relations among
  parts.
• Use BunchOf to define composite objects with
  definite parts but no particular structure.
  BunchOf is different from Set that closer related
  to categories.

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Measurements

• Objects have properties such as height, mass,
  const, etc.
• The value assigned to those properties are called
  measure.
• Measures can be represented by combining Units
  Function with a number.
• Measures are not necessarily particular numbers
  but more importantly, measures can be ordered.

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Actions, Situations and Events

• Situations are logical terms consisting of the
  initial situation (S0) and all situations that
  are generated by applying an action to a
  situation. Function Result(a,s) names the
  situation that results when action a is executed
  in situation s.
• Fluents are functions and predicates that vary
  from one situation to the next. E.g. location of
  the agent, aliveness of the wumpuss.
• E.g. Holding(G1, S4) says the agent is holding
  the gold G1 in the situation S4.
• Approach to represent the dynamic environment
  using above concepts is called situation
  calculus.
• Situation calculus also allows logical terms
• temporal or eternal predicates.
• actions

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Situation calculus agent

• A situation calculus agent has two kind of tasks
• Projection task the agent should be able to
  deduce the outcome of a given sequence of
  actions
• Planning task the agent should be able to find a
  sequence that achieves a desired effect with a
  suitable constructive inference algorithm.

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Situation calculus

• Simplest version of situation calculus includes
  two axioms to represent each action
• Possibility axiom that says when its possible to
  execute the action and
• Effect axiom that says what happen when a
  possible action is executed.
• Possibility axiom Preconditions ? Poss(a,s).
• Effect axiom Poss(a,s) ? Changes that result
  from taking action

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Frame problem

• Problem effect axioms say what changes, but do
  not says what stays the same.
• Representing all things that stay the same is
  called the frame problem.
• This problem must be solved efficiently because
  almost everything in the real world stays the
  same almost all the time. Each action affects
  only tiny fraction of all fluents.

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Representational frame problem

• One approach to solve frame problem writing
  explicit frame axioms that do say what stays the
  same.
• E.g. agents movements leave other objects
  stationary unless they are held.
• If there are F fluent predicates and A actions we
  will need O(AF) frame axioms.
• On the other hand, if each action has at most E
  effects, where E is typically much less than F,
  then the knowledge base can be much smaller of
  the size O(AE).
• The latter case is representational frame problem.

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Inferential frame problem

• Closely related to representational frame problem
  is inferential frame problem.
• Inferential frame problem is the problem to
  project the results of a t-step sequence of
  actions in time O(Et), rather than O(Ft) or
  O(AEt).

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Solving representational frame problem

• Use successor-state axioms
• Action is possible ? (Fluent is true in result
  state IF AND ONLY IF Actions effect made it true
  OR It was true before and action left it alone).
• Also need to say about implicit effect of an
  action ? ramification problem
• Also need to say that inference processes must be
  able to prove nonidentities ? use unique names
  axiom (can be assumed by the theorem prover or
  written explicitly in the KB).

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Solving inferential frame problem

• Successor-state axioms solve the representational
  frame problem but not the inferential frame
  problem.
• Consider a t-step plan p such that St
  Result(p,S0). To decide which fluents are true in
  St, we need to consider oeach the F frame axioms
  on each of the t time steps. Because the axioms
  have the average size AE this gives us O(AEt)
  inferential work. Most of the work involves
  copying fluents unchanged from one situation to
  the next.

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Solving inferential frame problem

• Two possibilities to solve this problem
• Discard situation calculus! Invent a new
  formalism to write axioms ? fluent calculus.
• Or alter the inference mechanism to handle frame
  axioms more efficiently.

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Solving inferential frame problem

• Poss(a,s) ?
• (Fi(Result(a,s)) ?
• (PosEffect(a,Fi) \/
• Fi(s) /\ NegEffect(a,Fi)))
• PosEffect(A1,Fi)
• PosEffect(A2,Fi)
• NegEffect(A3,Fi)
• NegEffect(A4,Fi)

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Solving inferential frame problem

• To make efficient inference procedure for
  previous axiom schema, we need to do three
  things
• Index the PosEffect and NegEffect predicates by
  their first argument so that when we are given an
  action that occurs at time t, we can find its
  effects in O(1) time.
• Index the axioms so that once its known that Fi
  is an effect of an action, we can find the axiom
  for Fi in O(1) time. Then you need not even
  consider the axioms for fluents that are not an
  effect of the action.
• Represent each situation as a previous situation
  plus a delta. If nothing changes from one step to
  the next then we do no work at all.
• Thus at each time step, we look at current
  action, fetch its effects, and update the set of
  true fluents.

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Time and event calculus

• Situation calculus works well when there is a
  single agent performing instantaneous, discrete
  actions.
• When actions have duration and can overlap with
  each other, situation calculus becomes awkward.
• Therefore, use event calculus ? based on points
  in time rather than on situations.

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Time and event calculus

• In event calculus, fluents hold at points in time
  rather than at situations.
• Reasoning is done over intervals of time.
• Event calculus axiom a fluent is true at a
  point in time if the fluent was initiated by an
  event at some time in the past and was not
  terminated by an intervening event.

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Time and event calculus

• Initiates (e,f,t) ? the occurrence of event e at
  time t causes fluent f to become true.
• Terminates(w,f,t) ? at time t, event e the fluent
  f ceases to be true.
• Happens(e,t) ? event e happens at time t.
• Clipped(f,t,t2) ? f is terminated by some event
  sometime between t and t2.

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Time and event calculus

• EVENT CALCULUS AXIOM
• T(f,t2) ? e,t Happens(e,t) /\
  Initiates(e,f,t) /\ (t lt t2) /\
  Clipped(f,t,t2)
• Clipped(f,t,t2) ? e,t1 Happens(e,t1) /\
  Terminates(e,f,t1) /\
• (t lt t1) /\ (t lt t2).

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Generalized events

• Generalized event is composed from aspects of
  some space-time chunk a piece of this
  multidimensional space-time universe.
• This abstraction generalizes most of the concepts
  so far, including actions, locations, times,
  fluents and physical objects.
• Generalized event can be broken down into
  subevents, e.g.
• SubEvent(BattleOfBritain, WorldWarII).
• SubEvent(WorldWarII, TwentiethCentury).

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Generalized events

• The 20th century is an interval of time.
  Intervals are chunks of space-time that include
  all of space between two time points.
• Period(e) ? smallest interval enclosing the event
  e
• Duration(e) ? the length of time occupied by an
  interval.
• E.g. Duration(Period(WorldWarII)) gt Years(5)

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Generalized events

• Australia is a place chunks of space time with
  some fixed spatial borders. The borders can vary
  over time, due to geological or political
  changes.
• In(e1,e2) ?subevent relation that holds when one
  events spatial projection is part of anothers.
• Location(e) ? the smallest place that encloses
  the event e.

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Processes

• Process liquid events (contrast to discrete
  events).
• Liquid events describe processes of continuous
  change and processes of continuous non-change.
• The latter is called states
• Fluent Calculus approach for knowledge
  representation that reifies combinations of
  fluents not just individual fluents. Here complex
  states and events are form by combining primitive
  ones.

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Intervals

• Representing times moments and extended
  intervals. Both form partition of time.
• Moments ? zero duration
• Extended intervals ? non zero duration.
• Notation Start, End, Time, Date, Seconds, Meet,
  Before, After, During, Overlap

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Mental Events

• Believes, Knows, Wants ? propositional attitudes.
• Reification turning a proposition into an
  object.

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Referential transparency

• In logic it is common to substitute a term freely
  for an equal term. This property is called
  referential transparency. Every relation in FOL
  is referentially transparent.
• On the other hand, we would like to define
  propositional attitudes as relations whose 2nd
  argument is opaque. It means that one cannot
  substitute an equal term for that 2nd argument
  without changing the meaning.

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Referentially opaque relation

• Achieving a referentially opaque 2nd argument for
  propositional attitude is done by
• Modal logic ? propositional attitudes become
  modal operator
• Syntactic theory of mental objects ? mental
  objects are represented by strings.

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Syntactic theory

• Unique string axiom strings are identical if and
  only if they consist of identical characters.
• Need a function that maps a string to the object
  ? Den
• Need a function that maps an object to a string
  which is the name of a constant that denote the
  object ? Name
• Define inference rules for string representation
  language. E.g., modus ponens.
• Final result answer of the question, given an
  agent knows the premises, can it draw the
  conclusion? ? according to axioms, an agent can
  deduce any consequence of its beliefs infallibly
  ? logical omnisience.
• Example read subchapter 10.5 Internet Shopping
  World