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    Logic

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    Knowledge-based agents

    Knowledge base (KB) = set ofsentences in a formal language

    Declarative approach to building an agent (or other system):

    Tell it what it needs to know

    What the agent can do is determined by what it knows Fullest realization of this philosophy was in the field of expert

    systems or knowledge-based systems in the 1970s and 1980s

    Inference engine

    Knowledge base

    Domain-independent algorithms

    Domain-specific content

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    What is logic?

    Logic is a formal system for manipulating facts so

    that true conclusions may be drawn

    The tool for distinguishing between the true and the

    false Averroes (12th

    cen.) Syntax: rules for constructing valid sentences

    E.g., x + 2 y is a valid arithmetic sentence, x2y + is not

    Semantics: meaning of sentences, or relationship

    between logical sentences and the real world

    Specifically, semantics defines truth of sentences

    E.g., x + 2 y is true in a world where x = 5 and y = 7

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    Overview

    Propositional logic

    Inference rules and theorem proving

    First order logic

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    Propositional logic: Syntax

    Atomic sentence: A proposition symbol representing a true or false statement

    Negation: IfP is a sentence, P is a sentence

    Conjunction: IfP and Q are sentences, P Q is a sentence

    Disjunction: IfP and Q are sentences, P Q is a sentence

    Implication:

    IfP and Q are sentences, P Q is a sentence Biconditional:

    IfP and Q are sentences, P Q is a sentence

    , , , , are called logical connectives

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    Propositional logic: Semantics

    A model specifies the true/false status of each

    proposition symbol in the knowledge base E.g., P is true, Q is true, R is false

    How many possible models are there with three symbols?

    What about n symbols?

    Rules for evaluating truth with respect to a model:P is true iff P is falseP Q is true iff P is true and Q is trueP Q is true iff P is true or Q is trueP Q is true iff P is false or Q is trueP Q is true iff P Q is true and Q P is true

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    Truth tables

    A truth table specifies the truth value of a

    composite sentence for each possible

    assignments of truth values to its atoms

    The truth value of a more complex sentence can

    be evaluated recursively orcompositionally

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    Logical equivalence

    Two sentences are logically equivalent iff they are true in

    same models

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    Validity, satisfiability

    A sentence is valid if it is true in all models,

    e.g., True,A A,A A, (A (A B)) B

    A sentence is satisfiable if it is true in some model

    e.g., AB, C

    A sentence is unsatisfiable if it is true in no models

    e.g.,AA

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    Entailment

    Entailment means that a sentence follows from

    the premises contained in the knowledge base:

    KB

    Knowledge base KB entails sentence if and

    only if is true in all models where KB is true

    E.g., x = 0 entails x * y = 0

    Can be true when KB is false?

    KB iff(KB) is valid KB iff(KB) is unsatisfiable

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    Inference

    Logical inference: a procedure for generating

    sentences that follow from a knowledge base KB

    An inference procedure is sound if whenever it

    derives a sentence , KB A sound inference procedure can derive only entailed

    sentences

    An inference procedure is complete if whenever

    KB , can be derived by the procedure A complete inference procedure can derive every

    entailed sentence

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    Inference

    How can we check whether a sentence is entailed by KB? How about we enumerate all possible models of the KB

    (truth assignments of all its symbols), and check that istrue in every model in which KB is true?

    Is this sound?

    Is this complete?

    Problem: if KB contains n symbols, the truth table will be of

    size 2n

    Better idea: use inference rules, or sound procedures to

    generate new sentences orconclusions given the premises

    in the KB

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    Inference rules

    Modus Ponens

    And-elimination

    ,

    premises

    conclusion

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    Inference rules

    And-introduction

    Or-introduction

    ,

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    Inference rules

    Double negative elimination

    Unit resolution

    ,

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    Resolution

    Example:

    : The weather is dry

    : The weather is cloudy

    : I take my umbrella

    ,

    ,or

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    Resolution is complete

    To prove KB , assume KB and derive acontradiction

    Rewrite KB as a conjunction ofclauses,or disjunctions ofliterals

    Conjunctive normal form (CNF)

    Keep applying resolution to clauses that contain

    complementary literals and adding resulting clauses

    to the list

    If there are no new clauses to be added, then KB does not entail If two clauses resolve to form an empty clause, we have a

    contradiction and KB

    ,

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    Complexity of inference

    Propositional inference is co-NP-complete

    Complement of the SAT problem: if andonly if the sentence is unsatisfiable

    Every known inference algorithm has worst-

    case exponential running time

    Efficient inference possible for restricted

    cases

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    Definite clauses

    A definite clause is a disjunction with exactly

    one positive literal

    Equivalent to (P1 Pn) Q

    Basis of logic programming (Prolog) Efficient (linear-time) complete inference through

    forward chaining and backward chaining

    premise orbodyconclusion

    orhead

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    Forward chaining

    Idea: find any rule whose premises are satisfied in

    the KB, add its conclusion to the KB, and keep

    going until query is found

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    Forward chaining example

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    Forward chaining example

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    Forward chaining example

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    Forward chaining example

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    Forward chaining example

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    Forward chaining example

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    Forward chaining example

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    Forward chaining example

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    Backward chaining

    Idea: work backwards from the query q:

    to prove q by BC,

    check ifq is known already, or

    prove by BC all premises of some rule concluding q

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    Backward chaining example

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    Backward chaining example

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    Backward chaining example

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    Backward chaining example

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    Backward chaining example

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    Backward chaining example

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    Backward chaining example

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    Backward chaining example

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    Backward chaining example

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    Backward chaining example

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    Forward vs. backward chaining

    Forward chaining is data-driven, automatic

    processing

    May do lots of work that is irrelevant to the goal

    Backward chaining is goal-driven, appropriate

    for problem-solving

    Complexity can be much less than linear in size of KB

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    Summary

    Logical agents apply inference to a knowledge base to

    derive new information and make decisions

    Basic concepts of logic:

    syntax: formal structure ofsentences

    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 Resolution is complete for propositional logic