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# Chapter 7 - 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

• 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 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

• 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

• 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

• 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

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

• 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

• 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

• 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

Introduction to Artificial Intelligence - APSU

• KB = wumpus-world rules + observations

Introduction to Artificial Intelligence - APSU

• 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

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

Introduction to Artificial Intelligence - APSU

• 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

Introduction to Artificial Intelligence - APSU

• 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 rules

R1:

R2:

R3:

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

R4:

R5:

Introduction to Artificial Intelligence - APSU

KB:

Note KB is true in three models

Does KB entails the following sentences?

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

• 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

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

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

written as an implication

Introduction to Artificial Intelligence - APSU

• 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

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

• 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

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

Introduction to Artificial Intelligence - APSU

• 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

• 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