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Chapter 7 Logic Agents Outline Knowledge-based agents Wumpus world Logic in general – models and entailment Propositional (Boolean) Logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution Knowledge bases

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outline
Outline
  • Knowledge-based agents
  • Wumpus world
  • Logic in general – models and entailment
  • Propositional (Boolean) Logic
  • Equivalence, validity, satisfiability
  • Inference rules and theorem proving
    • forward chaining
    • backward chaining
    • resolution

Introduction to Artificial Intelligence - APSU

knowledge bases
Knowledge bases
  • Knowledge base (KB)
    • a set of sentences in a formal language
  • Declarative approach to building an agent
    • TELL it what it needs to know (add new sentences to the KB)
    • ASK what it knows (query KB) and the answer should follow from what has been told
    • involve inference – deriving new sentences from old
  • Agents can be viewed at the knowledge level
    • what they know, regardless of how implemented

or at the implementation level

    • data structures in KB
    • algorithms that manipulate them

Introduction to Artificial Intelligence - APSU

slide4
Define game as a search problem
  • The agent must be able to:
    • represent states, actions, etc.
    • incorporate new percepts
    • update internal representations of the world
    • deduce hidden properties of the world
    • deduce appropriate actions

Introduction to Artificial Intelligence - APSU

wumpus world peas description
Wumpus world PEAS description
  • Performance measure
    • gold +1000, death -1000
    • -1 per step, -10 for using the arrow
  • Environment
    • Squares adjacent to wumpus are stench
    • Squares adjacent to pit are breezy
    • Glitter iff gold is in the same square
    • Shooting kills wumpus if you are facing it
    • Shooting uses up the only arrow
    • Grabbing picks up gold if in same square
    • Releasing drops the gold in same square
    • Wumpus emits a woeful scream when it is killed
    • Agent dies if entering a square with a pit or wumpus
  • Actuators
    • Left turn, Right turn, Forward, Grab, Release, Shoot
  • Sensors
    • Stench, Breeze, Glitter, Bump, Woeful scream

Introduction to Artificial Intelligence - APSU

properties of wumpus world
Properties of wumpus world
  • Observable?
  • No – only local perception
  • Deterministic?
  • Yes – outcomes exactly specified
  • Episodic?
  • No – sequential at the level of actions
  • Static?
  • Yes – wumpus and pits do not move
  • Discrete?
  • Yes
  • Single agent?
  • Yes – wumpus is essentially a natural feature

Introduction to Artificial Intelligence - APSU

slide7
Exploring a wumpus world

Introduction to Artificial Intelligence - APSU

slide8
Exploring a wumpus world (cont’d)

Introduction to Artificial Intelligence - APSU

slide9
Logic in general
  • Logics are formal languages for representing information such that conclusions can be drawn
  • Syntax defines the (well-formed) sentences in the language
  • Semantics defines the “meaning” of sentences, i.e., defines truth of a sentence in each possible world
    • Every sentence must be either true or false in each possible world
  • Examples: the language of arithmetic

Introduction to Artificial Intelligence - APSU

slide10
Model and Entailment
  • Logicians call each possible world Model
    • We say m is a model of a sentence aifa is true in m.
  • Entailment refers a sentence follows logically from another sentence
    • a |= b represents the sentence aentails the sentence b
    • a |= b if and only if, in every model in whichais true, b is also true
    • e.g. x + y = 4 entails 4 = x + y
    • M(a) is the set of all models of a

M(b) is the set of all models of b

  • Knowledge base KB entails sentence a
    • KB |= a iff a is true in all models where KB is true
    • M(KB) is the set of all models of KB

M(a) is the set of all models of a

Introduction to Artificial Intelligence - APSU

slide11
Entailment in the wumpus world
  • Situation after detecting nothing in [1, 1],

moving right, breeze in [2, 1]

  • Consider possible models for ?s

assuming only pits

  • 3 Boolean choices 8 possible models

Introduction to Artificial Intelligence - APSU

slide12
Wumpus models

Introduction to Artificial Intelligence - APSU

slide13
Wumpus models (cont’d)
  • KB = wumpus-world rules + observations

Introduction to Artificial Intelligence - APSU

slide14
Wumpus models (cont’d)
  • a1 = “There is no pit in [1, 2].” KB |= a1 proved by model checking
  • enumerates all possible models to check that a1 is true in all models in which KB is true

Introduction to Artificial Intelligence - APSU

slide15
Wumpus models (cont’d)
  • a2 = “There is no pit in [2, 2].” KB a2

Introduction to Artificial Intelligence - APSU

slide16
Inference
  • Inference is to derive conclusions from KB

i.e., an inference algorithm derives only entailed sentences

i.e., an inference algorithm can derive any sentence that is entailed

Introduction to Artificial Intelligence - APSU

slide17
Propositional logic: Syntax

Introduction to Artificial Intelligence - APSU

slide18
Propositional logic: Semantics
  • The semantics defines the rules for determining the truth of a sentence w.r.t. a particular model
    • Each model specifies true/false for each propositional symbol
  • 8 possible models can be enumerated automatically with these symbols
  • Rules for evaluating truth w.r.t. a model m:

Note

means “If P is true, then I am claiming that Q is true. Otherwise I am making no claim.”

  • simple recursive process evaluates an arbitrary sentence, e.g.,

Introduction to Artificial Intelligence - APSU

slide19
Wumpus world sentences

Wumpus world rules

R1:

R2:

R3:

The sentences from the breeze percepts for the first two squares visited

R4:

R5:

Introduction to Artificial Intelligence - APSU

slide20
Truth tables for inference

KB:

Note KB is true in three models

Does KB entails the following sentences?

Introduction to Artificial Intelligence - APSU

slide21
Inference by enumeration

Introduction to Artificial Intelligence - APSU

slide22
Logical equivalence

Introduction to Artificial Intelligence - APSU

slide23
Validity and satisfiability

Introduction to Artificial Intelligence - APSU

slide24
Proof methods

Introduction to Artificial Intelligence - APSU

slide25
Inference Rules
  • Modus Ponens
  • And-Elimination
  • All of the logical equivalences
  • Example on P212
  • Monotonicity: the set of entailed sentences can only increase as information is added to the knowledge base
    • inference rules can be applied whenever suitable premises are found in the knowledge base – the conclusion of rule must follow regardless of what else is in the knowledge base.

Introduction to Artificial Intelligence - APSU

slide26
Resolution

Introduction to Artificial Intelligence - APSU

slide27
Conversion to CNF

Introduction to Artificial Intelligence - APSU

slide28
Resolution algorithm

Introduction to Artificial Intelligence - APSU

slide29
Resolution example

Introduction to Artificial Intelligence - APSU

slide30
Forward and backward chaining

A disjunction of literals of which at most one is positive.

written as an implication

Introduction to Artificial Intelligence - APSU

slide31
Forward chaining
  • Idea: Fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found

Data-driven reasoning

Introduction to Artificial Intelligence - APSU

slide32
Forward chaining algorithm

Introduction to Artificial Intelligence - APSU

slide33
Forward chaining example

Introduction to Artificial Intelligence - APSU

slide34
Forward chaining example

Introduction to Artificial Intelligence - APSU

slide35
Forward chaining example

Introduction to Artificial Intelligence - APSU

slide36
Backward chaining
  • Idea: Work backwards from the query q:

to prove q by BC,

1) check if q is known already, or

2) prove by BC all premises of some rule concluding q

3) until reaches a set of known facts

Goal-driven reasoning

Introduction to Artificial Intelligence - APSU

slide37
Backward chaining example

Introduction to Artificial Intelligence - APSU

slide38
Backward chaining example

Introduction to Artificial Intelligence - APSU

slide39
Backward chaining example

Introduction to Artificial Intelligence - APSU

slide40
Backward chaining example

Introduction to Artificial Intelligence - APSU

slide41
Forward vs. backward chaining
  • FC is data-driven: automatic, unconscious processing,

e.g., object recognition, routine decisions

May do lots of work that is irrelevant to the goal

  • BC is goal-driven: appropriate for problem-solving,

e.g., Where are my keys? How do I get into a graduate program

Complexity of BC can be much less than linear in size of KB

Introduction to Artificial Intelligence - APSU

slide42
Summary
  • Logical agents apply inference to a knowledge base

to derive new information and make decisions

  • Basic concepts of logic:
    • syntax: formal structure of sentences
    • semantics: truth of sentences wrt models
    • entailment: necessary truth of one sentence given another
    • inference: deriving sentences from other sentences
    • soundness: derivations produce only entailed sentences
    • completeness: derivations can produce all entailed sentences
  • Wumpus world requires the ability to represent partial and negated information, reason by cases, etc.
  • Forward, backwardchaining are linear-time, complete for Horn clauses
  • Resolution is complete for propositional logic
  • Propositional logic lacks expressive power

Introduction to Artificial Intelligence - APSU

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