CS451CS551EE565 ARTIFICIAL INTELLIGENCE

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CS451/CS551/EE565ARTIFICIAL INTELLIGENCE

• First Order Logic
• 10-11-2006
• Prof. Janice T. Searleman
• jets_at_clarkson.edu, jetsza

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Outline

• Knowledge-Based Agents
• - Propositional (Boolean) logic
• - Equivalence, validity, satisfiability
• - Inference rules and theorem proving
• First order logic
• Reading Assignment AIMA
• Chapters 7, 8 9 (FOL inference)
• HW4 due today
• Exam1 Wednesday, 10/18, 700 pm, SC356

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Logical equivalence

• Two sentences are logically equivalent iff true
  in same models a ß iff a ß and ß a

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Validity and satisfiability

• A sentence is valid if it is true in all models,
• e.g., True, A ??A, A ? A, (A ? (A ? B)) ? B
• Validity is connected to inference via the
  Deduction Theorem
• KB a if and only if (KB ? a) is valid
• A sentence is satisfiable if it is true in some
  model
• e.g., A? B, C
• A sentence is unsatisfiable if it is true in no
  models
• e.g., A??A
• Satisfiability is connected to inference via the
  following
• KB a if and only if (KB ??a) is unsatisfiable

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Proof methods

• Proof methods divide into (roughly) two kinds
• Application of inference rules
• Legitimate (sound) generation of new sentences
  from old
• Proof a sequence of inference rule
  applications Can use inference rules as
  operators in a standard search algorithm
• Typically require transformation of sentences
  into a normal form
• Model checking
• truth table enumeration (always exponential in n)
• improved backtracking, e.g., Davis--Putnam-Logeman
  n-Loveland (DPLL)
• heuristic search in model space (sound but
  incomplete)
• e.g., min-conflicts-like hill-climbing
  algorithms

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Forward and backward chaining

• Horn Form (restricted)
• KB conjunction of Horn clauses
• Horn clause
• proposition symbol or
• (conjunction of symbols) ? symbol
• e.g., (C ? B ? A) ? (C ? D ? B)
• Modus Ponens (for Horn Form) complete for Horn
  KBs
• a1, ,an, a1 ? ? an ? ß
• ß
• Can be used with forward chaining or backward
  chaining.
• These algorithms are very natural and run in
  linear time

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Forward chaining

• Idea fire any rule whose premises are satisfied
  in the KB add its conclusion to the KB repeat
  until query is found

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Forward chaining algorithm

• Forward chaining is sound and complete for Horn KB

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Proof of completeness

• FC derives every atomic sentence that is entailed
  by KB
• FC reaches a fixed point where no new atomic
  sentences are derived
• Consider the final state as a model m, assigning
  true/false to symbols
• Every clause in the original KB is true in m
• a1 ? ? ak ? b
• Hence m is a model of KB
• If KB q, q is true in every model of KB,
  including m

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Backward chaining

• Idea work backwards from the query q
• to prove q by BC,
• check if q is known already, or
• prove by BC all premises of some rule concluding
  q
• Avoid loops check if new subgoal is already on
  the goal stack
• Avoid repeated work check if new subgoal
• has already been proved true, or
• has already failed

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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 PhD
  program?
• Complexity of BC can be much less than linear in
  size of KB

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Inference-based agents in the wumpus world

• A wumpus-world agent using propositional logic
• ?P1,1
• ?W1,1
• Bx,y ? (Px,y1 ? Px,y-1 ? Px1,y ? Px-1,y)
• Sx,y ? (Wx,y1 ? Wx,y-1 ? Wx1,y ? Wx-1,y)
• W1,1 ? W1,2 ? ? W4,4
• ?W1,1 ? ?W1,2
• ?W1,1 ? ?W1,3
• ? 64 distinct proposition symbols, 155 sentences

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Pros and cons of propositional logic

• ? Propositional logic is declarative
• ? Propositional logic allows partial/disjunctive/n
  egated information
• (unlike most data structures and databases)
• Propositional logic is compositional
• meaning of B1,1 ? P1,2 is derived from meaning of
  B1,1 and of P1,2
• ? Meaning in propositional logic is
  context-independent
• (unlike natural language, where meaning depends
  on context)
• ? Propositional logic has very limited expressive
  power
• (unlike natural language)
• e.g., cannot say "pits cause breezes in adjacent
  squares except by writing one sentence for each
  square

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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 of sentence
• 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

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Summary (cont.)

• Wumpus world requires the ability to represent
  partial and negated information, reason by cases,
  etc.
• Resolution is complete for propositional
  logicForward, backward chaining are linear-time,
  complete for Horn clauses
• Propositional logic lacks expressive power

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Expressiveness

• Consider expressing the idea Johns father is a
  diplomat.
• Propositional logic symbol JFIAD represents
  this (either is true or false)
• problem cant reason about any of the parts
• e.g. fatherhood, being a diplomat, John, etc.
• solution First Order Logic (aka the Predicate
  Calculus)

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First Order Logic

• Prof. Janice T. Searleman
• jets_at_clarkson.edu

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First-order logic

• Whereas propositional logic assumes the world
  contains facts,
• first-order logic (like natural language) assumes
  the world contains
• Objects people, houses, numbers, colors,
  baseball games, wars,
• Relations red, round, prime, brother of, bigger
  than, part of, comes between,
• Functions father of, best friend, one more than,
  plus,

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Expressiveness

• Johns father is a diplomat.
• Propositional logic symbol JFIAD represents
  this (either is true or false)
• First Order Logic diplomat(father-of(John)).
• variable X refers to an element of the domain
• predicate diplomat(X) means X is a diplomat
• function father-of(X) returns the person who is
  Xs father
• constant the symbol John refers to a particular
  person named John in the domain

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Syntax of FOL Basic elements

• Constants KingJohn, 2, NUS,...
• Predicates Brother, gt,...
• Functions Sqrt, LeftLegOf,...
• Variables x, y, a, b,...
• Connectives ?, ?, ?, ?, ?
• Equality
• Quantifiers ?, ?

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Atomic sentences

• Atomic sentence predicate (term1,...,termn)
  or term1 term2
• Term function (term1,...,termn)
  or constant or variable
• e.g. Brother(KingJohn,RichardTheLionheart) gt
  (Length(LeftLegOf(Richard)), Length(LeftLegOf(King
  John)))

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Complex sentences

• Complex sentences are made from atomic sentences
  using connectives
• ?S, S1 ? S2, S1 ? S2, S1 ? S2, S1 ? S2,
• E.g. Sibling(KingJohn,Richard) ?
  Sibling(Richard,KingJohn)
• gt(1,2) ? (1,2)
• gt(1,2) ? ? gt(1,2)

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Truth in first-order logic

• Sentences are true with respect to a model and an
  interpretation
• Model contains objects (domain elements) and
  relations among them
• Interpretation specifies referents for
• constant symbols ? objects
• predicate symbols ? relations
• function symbols ? functional relations
• An atomic sentence predicate(term1,...,termn) is
  true
• iff the objects referred to by term1,...,termn
• are in the relation referred to by predicate

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Interpretation

• Must interpret variables, constants, functions,
  and predicate symbols by associating them with
  objects, functions and relations in the world.
• Truth values are relative to an interpretation
  (model)
• Example
• p(X,Y) means X is above Y
• p(X,Y) means X is on top of and touching Y