Propositional Logic

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Propositional Logic

• Russell and Norvig Chapter 6Chapter 7, Sections
  7.17.4Slides adapted from
• robotics.stanford.edu/latombe/cs121/2003/home.htm
  

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Knowledge-Based Agent
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Types of Knowledge

• Procedural, e.g. functions Such knowledge can
  only be used in one way -- by executing it
• Declarative, e.g. constraints It can be used
  to perform many different sorts of inferences

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Logic

• Logic is a declarative language to
• Assert sentences representing facts that hold in
  a world W (these sentences are given the value
  true)
• Deduce the true/false values to sentences
  representing other aspects of W

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Connection World-Representation
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Examples of Logics

• Propositional calculus A ? B ? C
• First-order predicate calculus ( x)( y)
  Mother(y,x)
• Logic of Belief B(John,Father(Zeus,Cronus))

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Model

• Assignment of a truth value true or false to
  every atomic sentence
• Examples
• Let A, B, C, and D be the propositional symbols
• m Atrue, Bfalse, Cfalse, Dtrue is a model
• m Atrue, Bfalse, Cfalse is not a model
• With n propositional symbols, one can define 2n
  models

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Model of a KB

• Let KB be a set of sentences
• A model m is a model of KB iff it is a model
  of all sentences in KB, that is, all sentences
  in KB are true in m
• Given a vocabulary A, B, C and D, how many models
  for AB -gt C are there?
• for AB -gt B?

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Satisfiability of a KB
A KB is satisfiable iff it admits at least one
model otherwise it is unsatisfiable
KB1 P, ?Q?R is satisfiableKB2 ?P?P is
satisfiable KB3 P, ?P is unsatisfiable
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Logical Entailment

• KB set of sentences
• ? arbitrary sentence
• KB entails ? written KB ? iff every model
  of KB is also a model of ?
• Alternatively, KB ? iff
• KB,?? is unsatisfiable
• KB ? ? is valid

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Inference Rule

• An inference rule ?, ? ? consists of 2
  sentence patterns ? and ? called the conditions
  and one sentence pattern ? called the conclusion
• If ? and ? match two sentences of KB then the
  corresponding ? can be inferred according to the
  rule

?
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Inference

• I Set of inference rules
• KB Set of sentences
• Inference is the process of applying successive
  inference rules from I to KB, each rule adding
  its conclusion to KB

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Example Modus Ponens

• Battery-OK ? Bulbs-OK ? Headlights-Work
• Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
  Engine-Starts
• Engine-Starts ? ?Flat-Tire ? Car-OK
• Battery-OK ? Bulbs-OK

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? Connective symbol (implication) Logical
entailment Inference
?
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Soundness

• An inference rule is sound if it generates only
  entailed sentences
• All inference rules previously given are sound,
  e.g.modus ponens ? ? ? , ? ?
• The following rule ? ? ? , ? ? is
  unsound, which does not mean it is useless

?
?
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Completeness

• A set of inference rules is complete if every
  entailed sentences can be obtained by applying
  some finite succession of these rules
• Modus ponens alone is not complete, e.g.from A
  ? B and ?B, we cannot get ?A

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Proof
The proof of a sentence ? from a set of
sentences KB is the derivation of ? by applying
a series of sound inference rules
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Proof
The proof of a sentence ? from a set of
sentences KB is the derivation of ? by applying
a series of sound inference rules

• Battery-OK ? Bulbs-OK ? Headlights-Work
• Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
  Engine-Starts
• Engine-Starts ? ?Flat-Tire ? Car-OK
• Headlights-Work
• Battery-OK
• Starter-OK
• ?Empty-Gas-Tank
• ?Car-OK
• Battery-OK ? Starter-OK ? (56)
• Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
  (97)
• Engine-Starts ? (210)
• Engine-Starts ? Flat-Tire ? (38)
• Flat-Tire ? (1112)

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Inference Problem

• Given
• KB a set of sentence
• ? a sentence
• Answer
• KB ? ?

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Deduction vs. Satisfiability Test
KB ? iff KB,?? is unsatisfiable

• Hence
• Deciding whether a set of sentences entails
  another sentence, or not
• Testing whether a set of sentences is
  satisfiable, or not
• are closely related problems

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Complementary Literals

• A literal is a either an atomic sentence or the
  negated atomic sentence, e.g.
  P or ?P
• Two literals are complementary if one is the
  negation of the other, e.g.
  P and ?P

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Unit Resolution Rule

• Given two sentences L1 ? ? Lp and
  M where Li,, Lp and M are all literals,
  and M and Li are complementary literals
• Infer L1 ? ? Li-1 ? Li1 ? ? Lp

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Examples

• From?Engine-Starts ? Car-OK
• Engine-Starts
• InferCar-OK

Modus ponens

• From?Engine-Starts ? Car-OK
• ?Car-OK
• Infer ?Engine-Starts

Modus tolens
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Shortcoming of Unit Resolution

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? Empty-Gas-Tank
• we can infer nothing!

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Full Resolution Rule

• Given two sentences L1 ? ? Lp and
  M1 ? ? Mq where L1,, Lp, M1,, Mq are all
  literals, and Li and Mj are complementary
  literals
• Infer L1? ? Li-1?Li1??Lk?M1? ?
  Mj-1?Mj1??Mk in which only one copy of each
  literal is retained (factoring)

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Example

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? Empty-Gas-Tank
• Infer
• Empty-Gas-Tank ? Flat-Tire ? Car-OK

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Example

• From
• ?P ? Q (? P ? Q)
• ?Q ? R (? Q ? R)
• Infer
• ?P ? R (? P ? R)

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Not All Inferences are Useful!

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? ?Flat-Tire
• Infer
• ?Flat-Tire ? Flat-Tire ? Car-OK

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Not All Inferences are Useful!

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? ?Flat-Tire
• Infer
• ?Flat-Tire ? Flat-Tire ? Car-OK

tautology
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Not All Inferences are Useful!

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? ?Flat-Tire
• Infer
• ?Flat-Tire ? Flat-Tire ? Car-OK ? True

tautology
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Full Resolution Rule

• Given two clauses L1 ? ? Lp and
  M1 ? ? Mq
• Infer the clause L1? ? Li-1?Li1??Lk?M1? ?
  Mj-1?Mj1??Mk

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Sentence ? Clause Form
Example (A ? ?B) ? (C ? D) 1. Eliminate ?
?(A ? ?B) ? (C ? D)2. Reduce scope of ? (?A ?
B) ? (C ? D)3. Distribute ? over ? (?A ? (C ?
D)) ? (B ? (C ? D)) (?A ? C) ? (?A ? D) ? (B ?
C) ? (B ? D) Set of clauses ?A ? C , ?A ? D ,
B ? C , B ? D
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Resolution Refutation Algorithm

• RESOLUTION-REFUTATION(KB,a)
• clauses ? set of clauses obtained from KB and ?a
• new ?
• Repeat
• For each C, C in clauses do res ?
  RESOLVE(C,C) If res contains the empty clause
  then return yes
• new ? new U resIf new ? clauses then return no
• clauses ? clauses U new

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Example

• ?Battery-OK ? ?Bulbs-OK ? Headlights-Work
• ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank ?
  Engine-Starts
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Headlights-Work
• Battery-OK
• Starter-OK
• ?Empty-Gas-Tank
• ?Car-OK
• ?Flat-Tire

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Summary

• Propositional Logic
• Model of a KB
• Logical entailment
• Inference rules
• Resolution rule
• Clause form of a set of sentences
• Resolution refutation algorithm