Chapter 7 Logic Agents

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# Chapter 7 Logic Agents - PowerPoint PPT Presentation

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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## Chapter 7 Logic Agents

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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

Introduction to Artificial Intelligence - APSU

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

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
• 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
• 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

Exploring a wumpus world

Introduction to Artificial Intelligence - APSU

Exploring a wumpus world (cont’d)

Introduction to Artificial Intelligence - APSU

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

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

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

Wumpus models

Introduction to Artificial Intelligence - APSU

Wumpus models (cont’d)

• KB = wumpus-world rules + observations

Introduction to Artificial Intelligence - APSU

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

Wumpus models (cont’d)

• a2 = “There is no pit in [2, 2].” KB a2

Introduction to Artificial Intelligence - APSU

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

Propositional logic: Syntax

Introduction to Artificial Intelligence - APSU

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

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

Truth tables for inference

KB:

Note KB is true in three models

Does KB entails the following sentences?

Introduction to Artificial Intelligence - APSU

Inference by enumeration

Introduction to Artificial Intelligence - APSU

Logical equivalence

Introduction to Artificial Intelligence - APSU

Validity and satisfiability

Introduction to Artificial Intelligence - APSU

Proof methods

Introduction to Artificial Intelligence - APSU

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

Resolution

Introduction to Artificial Intelligence - APSU

Conversion to CNF

Introduction to Artificial Intelligence - APSU

Resolution algorithm

Introduction to Artificial Intelligence - APSU

Resolution example

Introduction to Artificial Intelligence - APSU

Forward and backward chaining

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

written as an implication

Introduction to Artificial Intelligence - APSU

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

Forward chaining algorithm

Introduction to Artificial Intelligence - APSU

Forward chaining example

Introduction to Artificial Intelligence - APSU

Forward chaining example

Introduction to Artificial Intelligence - APSU

Forward chaining example

Introduction to Artificial Intelligence - APSU

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

Backward chaining example

Introduction to Artificial Intelligence - APSU

Backward chaining example

Introduction to Artificial Intelligence - APSU

Backward chaining example

Introduction to Artificial Intelligence - APSU

Backward chaining example

Introduction to Artificial Intelligence - APSU

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

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