Agents that Reason Logically

1
Agents that Reason Logically

• Logical agents have knowledge base, from which
  they draw conclusions
• TELL provide new facts to agent
• ASK decide on appropriate action

2
Sample Wumpus World

• Show original wumpus game
• goal is to shoot wumpus
• example of logical reasoning
• http//www.inthe70s.com/games/wumpus/index.shtml
• Our version
• Kill the wumpus.

3
A Wumpus Agent

• Agent does not perceive its own location (unlike
  sample game), but it can keep track of where it
  has been
• Percepts
• Stench wumpus is nearby
• Breeze pit is nearby

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Wumpus Agent

• Actuators
• Forward, Turn Left, Turn Right
• Shoot (shoots arrow forward until hits wumpus or
  wall)
• Environment
• 4x4 grid, start at (1,1) facing right

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Wumpus Agent

• Death
• Agent dies if it enters a pit or square with
  wumpus
• Goal kill wumpus

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Some complex reasoning examples

• Start in (1,1)
• Breeze in (1,2) and (2,1)
• Probably a pit in (2,2)
• Smell in (1,1) where can you go?
• Pick a direction shoot
• Walk in that direction
• Know where wumpus is

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The use of logic

• A logic formal language for representing
  information, rules for drawing conclusions
• Two kinds of logics
• Propositional Logic (Chap 7)
• Represents facts
• First Order Logic (Chap 8)
• Represents facts, objects, and relations

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Soundness

• Rules of inference allow us to derive new
  sentences entailed by a knowledge base
• Rules of inference must be sound sentences
  inferred by a KB should be entailed by that KB
• What is a non-sound inference?
• Video

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Entailment

• At any given time, we have a knowledge base of
  information
• If Jan likes ice cream, we would share tastes
• If we share tastes, I would like to know Jan
  better
• Jan likes ice cream
• The knowledge base KB entails a means a is true
  in all worlds where KB is true
• e.g. if a I would like to know Jan better
• KB a

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Propositional Logic Semantics
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Inference by Enumeration
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Enumeration Solution is a entailed by KB?
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Enumeration is too computationally intense

• For n proposition symbols, enumeration takes 2n
  rows (exponential)
• Inference rules allow you to deduce new sentences
  from the KB
• Can use inference rules as operators in a
  standard search algorithm
• Think of testing if something as true as
  searching for it

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Common inference rules for propositional logic

• Modus Ponens (Implication-Elimination)
• And-EliminationAnd-Introduction
• Or Introduction

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Common inference rules for propositional logic

• Double-Negation Elimination
• Unit Resolution
• Resolution

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Example of using logic in Wumpus World

• KB contains

Stench Agent
Start Breeze
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KB also contains knowledge of environment

• No stench ? no wumpus nearby
• Stench ? wumpus nearby

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We can determine where wumpus is!

• Method 1 Truth table
• At least 14 symbols currently S1,1, S2,1, S1,2,
  S2,2, W1,1, W2,1, W1,2, W2,2, W3,1, W1,3, B1,1,
  B2,1, B1,2, B2,2
• ? 214 rows, ouch!

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We can determine where wumpus is!

• Method 2 Inference
• Modus Ponens
• And-Elimination

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Inference continued...

• Modus Ponens and And-Elimination again
• One more Modus Ponens

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Inference continued...

• Unit Resolution

Wumpus is in (1,3)!!! Shoot it. Shoot where?
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Determining action based on knowledge

• Propositional logic cannot answer well the
  question What action should I take?
• It only answers Should I take action X?

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Propositional logic seems inefficient

• Rule Shoot if the wumpus is in front of you
• 16 x 4 64 rules for the 4x4 grid
• Ditto for pits
• First order logic is much more efficient

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First-Order Logic Better choice for Wumpus World

• Propositional logic represents facts
• First-order logic gives us
• Objects
• Relations how objects relate to each other
• Functions return value for given input

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Syntax and Semantics

• First-order logic has the following
• Constants represent objects
• book, A, cs327
• Predicates represent relations
• OlderSibling(Lisa,Bart)
• OlderSibling(Maggie,Lisa)
• OlderSibling(Maggie,Bart)
• Functions
• FatherOf(Luke) Anakin

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Syntax and Semantics

• Atomic Sentences
• Father(Luke,Anakin)
• Siblings(SonOf(Anakin), DaughterOf(Anakin))
• Complex Sentences
• and, or, not, implies, equivalence
• Equality

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Universal Quantification

• For all, for every
• Examples
• Usually use with
• Common mistake to use

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Existential Quantification

• There exists
• Typically use with
• Common mistake to use

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First-Order Logic in Wumpus World

• Suppose an agent perceives a stench, breeze, at
  time t 5
• Percept(Stench,Breeze,5)
• Stench,Breeze is a list
• Then want to query for an appropriate action.
  Find an a (ask the KB)