CS451CS551EE565 ARTIFICIAL INTELLIGENCE

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CS451/CS551/EE565ARTIFICIAL INTELLIGENCE

• Logic Agents
• 10-09-2006
• 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)
• IBM talk Global Technology Outlook
• Tues, 10/10, 530 pm, SC160
• HW4 due Wed., 10/11
• Exam1 Wednesday, 10/18,

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What is a logic?

• A study of correct inference (truth-preserving)
• Truth-preserving inference
• If there is a potato in the tailpipe, the car
  will not start.
• There is a potato in the tailpipe.
• Therefore, the car will not start.
• Non-Truth-preserving inference
• If there is a potato in the tailpipe, the car
  will not start.
• My car will not start.
• Therefore, there is a potato in the tailpipe.

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Formal Logic

• Three components
• The syntax of a formal language.
• i.e. what constitutes a well-formed sentence.
• The semantics of a formal language.
• What is the meanings of the well-formed
  sentences i.e. under what conditions is a
  sentence true?
• A proof theory
• A formal specification of what constitutes
  correct inference i.e. a set of axioms and a set
  of inference rules.

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Inference
Representation
World

• Logical inference generates new sentences that
  are entailed by existing sentences.
• KB entails a is denoted by KB a

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Inference

• KB i a sentence a can be derived from KB by
  procedure I
• Soundness i is sound if whenever KB i a, it is
  also true that KB a
• Completeness i is complete if whenever KB a, it
  is also true that KB i a

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Inference Procedure

• An inference rule is complete if, given a set S
  of sentences, it can infer every sentence that
  logically follows from S.
• given KB, infer all sentences a entailed from it
• given a, discover whether or not KB a

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Propositional logic Syntax

• Propositional logic is the simplest logic
  illustrates basic ideas
• The proposition symbols P1, P2, etc. are
  sentences
• If S is a sentence, ?S is a sentence (negation)
• If S1 and S2 are sentences
• S1 ? S2 is a sentence (conjunction)
• S1 ? S2 is a sentence (disjunction)
• S1 ? S2 is a sentence (implication)
• S1 ? S2 is a sentence (biconditional)

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Propositional logic Semantics

• Each model specifies true/false for each
  proposition symbol
• e.g. P1,2 P2,2 P3,1
• false true false
• With these symbols, 8 possible models, can be
  enumerated automatically.
• Rules for evaluating truth with respect to a
  model m
• ?S is true iff S is false
• S1 ? S2 is true iff S1 is true and S2
  is true
• S1 ? S2 is true iff S1is true or S2
  is true
• S1 ? S2 is true iff S1 is false or S2
  is true
• i.e., is false iff S1 is true
  and S2 is false
• S1 ? S2 is true iff S1?S2 is true
  and S2?S1 is true
• Simple recursive process evaluates an arbitrary
  sentence, e.g.,
• ?P1,2 ? (P2,2 ? P3,1) true ? (true ? false)
  true ? true true

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Truth tables for connectives
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Wumpus world sentences

• Let Pi,j be true if there is a pit in i, j.
• Let Bi,j be true if there is a breeze in i, j.
• ?P1,1
• ?B1,1
• B2,1
• "Pits cause breezes in adjacent squares"
• B1,1 ? (P1,2 ? P2,1)
• B2,1 ? (P1,1 ? P2,2 ? P3,1)

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Truth tables for inference
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Inference by enumeration

• Depth-first enumeration of all models is sound
  complete
• For n symbols, time complexity is O(2n)
  and space complexity is O(n)

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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