Reasoning in Predicate Logic

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Reasoning in Predicate Logic
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Learning Objectives

• Introduction to First-Order Logic (FOL)
• Syntax and Semantics of FOL
• Fun with sentences
• Wumpus world in FOL

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Acknowledgements

• These slides have been adapted from Stuart
  Russell and Peter Norvig.

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ReviewSimple Knowledge-Based Agent
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Pros and Cons of Propositional Logic
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First-Order Logic
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Logics in General
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Syntax of FOL Basic Elements
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Atomic Sentences
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Complex Sentences
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Truth in First-Order Logic
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Models for FOL Example
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Models for FOL A Lot!
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Universal Quantification
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A Common Mistake to Avoid
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Existential Quantification
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Another Common Mistake to Avoid
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Properties of Quantifiers
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Fun with Sentences
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Fun with Sentences
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Fun with Sentences
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Fun with Sentences
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Fun with Sentences
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Equality
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More Fun with Sentences

• Every Dog Chases Its Own Tail
• ? d . Chases (d, tail-of (d))
• Alternative Statement ? d . ? t . Tail-Of (t, d)
  ? Chases (d, t)
• Prefigures concept of Skolemization (Skolem
  variables / functions)
• Every Dog Chases Its Own (Unique) Tail
• ? d . ?1 t . Tail-Of (t, d) ? Chases (d, t) ?
  
  ? d . ? t . Tail-Of (t, d) ? Chases (d, t) ? ?
  t Chases (d, t) ? t t
• Only The Wicked Flee when No One Pursueth
• ? x . Flees (x) ? ? y Pursues (y, x) ? Wicked
  (x)
• Alternative ? x . ? y . Flees (x, y) ? ? z
  . Pursues (z, x) ? Wicked (x)

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Wumpus World RevisitedInteracting with FOL KBs
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Knowledge Base forThe Wumpus World
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Deducing Hidden Properties
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Keeping Track of ChangeSituation Calculus
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Describing Actions 1Frame, Qualification, and
Ramification Problems
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Describing Actions 2Successor State Axioms
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Making Plans
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Making PlansA Better Way
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First-Order LogicSummary
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A Look Ahead 1Automated Deduction
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A Look Ahead 2Classical Planning
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Summary Points

• Previously Logical Agents and Calculi
• Logic in general tools for KR, inference,
  planning
• Propositional calculus normal forms, sequent
  rules
• Predicate logic
• First-order logic (FOL) aka first-order predicate
  calculus (FOPC)
• Today FOL in Practice
• FOL agents
• Example Wumpus World in FOL
• Situation calculus
• Frame problem and variants (see RN sidebar)
• Representational vs. inferential frame problems
• Qualification problem what if?
• Ramification problem what else? (side effects)
• Successor-state axioms

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Terminology

• Logical Languages
• Propositional logic
• Predicates, terms, functions, atoms (atomic
  sentences / atomic WFFs), WFFs
• First-order logic (FOL, FOPC) universal and
  existential quantification
• Properties of Knowledge Bases (KBs)
• Satisfiability and validity
• Entailment and provability
• Properties of Proof Systems Soundness and
  Completeness
• Normal Forms CNF, DNF, Horn Clauses vs. Terms
• Situation Calculus
• Frame, Ramification, Qualification Problems
• Successor-State Axiomatization