CS 2710, ISSP 2610

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CS 2710, ISSP 2610

• Chapter 7
• Propositional Logic
• Reasoning

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Knowledge Based Agents

• Central component knowledge base, or KB.
• A set of sentences in a knowledge representation
  language
• Generic Functions
• TELL (add a fact to the knowledge base)
• ASK (get next action based on info in KB)
• Both often involve inference, which is?
• Deriving new sentences from old

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Note

• Were all familiar with logic from mathematics
• In AI, we need to think about things we usually
  can take for granted
• To use the notion of prime number, you dont have
  to think about what prime means or number, for
  that matter
• In AI, we create knowledge representation
  schemes, which define the objects, predicates,
  and functions which exist

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The Wumpus World

• Game is played in a MxN grid
• One player, one wumpus, one or more pits
• Goal find gold while avoiding wumpus and pits
• Percepts
• Glitter (gold is in this square)
• Stench (wumpus is within 1 square N,E,S,W)
• Breeze (pit is within 1 square N, E, S, W)
• Bump (agent walks into a wall)
• Scream (when Wumpus dies hear it anywhere)

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Wumpus Example
0
0
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Wumpus

• Main difficulty player doesnt know the
  configuration
• Reason about configuration
• Knowledge evolves as new percepts arrive and
  actions are taken.

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Examples of reasoning

• If the player is in square (1, 0) and the percept
  is breeze, then there must be a pit in (0,0) or a
  pit in (2,0) or a pit in (1,1).
• If the player is now in (0,0) and still alive,
  there is not a pit in (0,0).
• If there is no breeze percept in (0,0), there is
  no pit in (0,1)
• If there is also no breeze in (0,1) then there is
  no pit in (1,1).
• Therefore, there must be a pit in (2,0)

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Fundamental Concepts of logical representation
and reasoning

• Information is represented in sentences, which
  must have correct syntax
• ( 1 2 ) 7 21 vs. 2 ) 7 ( 1 21
• The semantics of a sentence defines its truth
  with respect to each possible world an
  interpretation assigning T or F to all
  propositions
• W is a model of S means that sentence S is true
  under interpretation W
• What do the following mean?
• X Y
• X entails Y
• Y logically follows from X

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Which are true?Which are not true but useful?

• Man, Man ? Mortal Mortal
• Raining,Dog? Mammal Mammal
• Raining,Raining ? Wet Wet
• Smoke, Fire ? Smoke Fire
• Tall Silly Tall
• Tall v Silly Silly
• Tall, Silly Tall Silly
• (On board E.g. with Wumpus world)

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Entailment

• A B
• Under all interpretations in which A is true, B
  is true as well
• All models of A are models of B
• Whenever A is true, B is true as well
• A entails B
• B logically follows from A

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Inference

• KB -i A
• Inference algorithm i can derive A from KB
• i derives A from KB
• i can derive A from KB
• A can be inferred from KB by i

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Inference Algorithm Examples

• A,B - (intro) A B
• A, A?B - (MP) B
• A, B ? A - (abduction) B
• A - (some religions) GodExists
• SunnyDay - (some students) NoClass
• (Italics used for constants here)

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Inference Algorithms

• A. Definition of soundness of inference algorithm
  i?
• B. Definition of completeness of inference
  algorithm i?
• Notes
• implication, piece of syntax
• entailment, used to describe semantics
• Inference algorithm, inference procedure, rule of
  inference, inference rule procedure that
  derives sentences from sentences

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Monotonicity

• A logic is monotonic if

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Propositional Logic Syntax

• Sentence -gt AtomicSent complexSent
• AtomicSent -gt truefalse P, Q, R
• ComplexSent -gt
• ?sentence
• ( sentence ? sentence )
• ( sentence ? sentence )
• ( sentence ?sentence )
• ( sentence ? sentence )
• ( sentence )
• no predicate or function symbols

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Propositional Logic Sentences

• If there is a pit at 1,1, there is a breeze at
  1,0
• P11 ? B10
• There is a breeze at 2,2, if and only if there
  is a pit in the neighborhood
• B22 ? ( P21 ? P23 ? P12 ? P32 )
• There is no breeze at 2,2
• ?B22

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Semantics of Prop Logic

• In model-theoretic semantics, an interpretation
  assigns elements of the world to sentences, and
  defines the truth values of sentences
• Propositional logic easy! Assign T or F to each
  proposition symbol then assign truth values to
  complex sentences in the obvious way

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Logical Equivalences

• Sentences A and B are logically equivalent if
• they are true under exactly the same
  interpretations
• A B and B A

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Validity

• A sentence (or set of sentences) is valid if
• it is true under all interpretations
• P v P

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Satisfiability

• A sentence (or set of sentences) is satisfiable
  if
• there exists some interpretation that makes it
  true
• An interpretation satisfies a set of sentences if
  it makes them true

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Entailment

• A B
• In all worlds in which A is true, B is true as
  well
• All models of A are models of B
• Whenever A is true, B is true as well
• A entails B
• B logically follows from A
• All interpretations that satisfy A also satisfy B

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Propositional Logic Inference

• Question Does KB entail S?
• Method 1 Truth Table Entailment
• Construct a truth table whose columns are all
  propositions used in the sentences in KB.
• If S is true everywhere all sentences in KB are
  true, then KB entails S (otherwise not)
• Method 2 Proof
• Proof by deduction
• Proof by contradiction
• Etc.

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AC, C does not entail B?C
A,B, Entails A?B
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Rules for Deductive Proofs

• Modus Ponens
• Given S1 ? S2 and S1, derive S2
• And-elimination
• Given S1 ? S2, derive S1
• Given S1 ? S2, derive S2
• DeMorgans Law
• Given ?( A ? B) derive ?A ? ?B
• Given ?( A ? B) derive ?A ? ?B
• See p. 249 (for review if needed)

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Example Proof by Deduction

• Knowledge
• S1 B22 ? ( P21 ? P23 ? P12 ? P32 ) rule
• S2 ?B22 observation
• Inferences
• S3 (B22 ? (P21 ? P23 ? P12 ? P32 ))? ((P21
  ? P23 ? P12 ? P32 ) ? B22) S1,bi elim
• S4 ((P21 ? P23 ? P12 ? P32 ) ? B22) S3, and
  elim
• S5 (?B22 ? ?( P21 ? P23 ? P12 ? P32 )) contrapos
• S6 ?(P21 ? P23 ? P12 ? P32 )
  S2,S5, MP
• S7 ?P21 ? ?P23 ? ?P12 ? ?P32 S6,
  DeMorg

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Proofs

• A derivation
• A sequence of applications of (usually sound)
  rules of inference
• Reasoning by Search
• Successor function all possible applications of
  inference rules
• Monotonicity means search can be local, and more
  efficient

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Resolution

• Resolution allows a complete inference mechanism
  (search-based) using only one rule of inference
• Resolution rule
• Given P1 ? P2 ? P3 ? Pn, and ?P1 ? Q1 ? Qm
• Conclude P2 ? P3 ? Pn ? Q1 ? Qm
• Complementary literals P1 and ?P1 cancel out

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Resolution

• winter v summer
• winter v cold
• Either winter or winter is true, so we know that
  summer or cold is true
• Resolution rule
• Given P1 ? P2 ? P3 ? Pn, and ?P1 ? Q1 ? Qm
• Conclude P2 ? P3 ? Pn ? Q1 ? Qm
• Complementary literals P1 and ?P1 cancel out

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Resolution in Wumpus World

• There is a pit at 2,1 or 2,3 or 1,2 or 3,2
• P21 ? P23 ? P12 ? P32
• There is no pit at 2,1
• ?P21
• Therefore (by resolution) the pit must be at 2,3
  or 1,2 or 3,2
• P23 ? P12 ? P32

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Resolution

• Any complete search algorithm, applying only the
  resolution rule, can derive any conclusion
  entailed by any KB in propositional logic.

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Proof using Resolution

• To prove P, apply res until either
• No new clauses can be added, (KB does not entail
  P)
• The empty clause is derived (KB does entail P)
• Proof by contradiction prove KB ? ?P is
  contradictory (empty clause) to prove P
• Sentences need to be in CNF
• To carry out the proof, need a search mechanism
  that will enumerate all possible resolutions.

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B22 ? ( P21 ? P23 ? P12 ? P32 )

• Eliminate ? , replacing with two implications
• (B22 ? ( P21 ? P23 ? P12 ? P32 )) ? ((P21 ? P23 ?
  P12 ? P32 ) ? B22)
• Replace implication (A ? B) by ?A ? B
• (?B22 ? ( P21 ? P23 ? P12 ? P32 )) ? (?(P21 ? P23
  ? P12 ? P32 ) ? B22)
• Move ? inwards (unnecessary parens removed)
• (?B22 ? P21 ? P23 ? P12 ? P32 ) ? ( (?P21 ? ?P23
  ? ?P12 ? ?P32 ) ? B22)
• 4. Distributive Law
• (?B22 ? P21 ? P23 ? P12 ? P32 ) ? (?P21 ? B22) ?
  (?P23 ? B22) ? (?P12 ? B22) ? (?P32 ? B22)

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Previous Slide Sentence is in CNF

• Next step (with simpler example)
• (P1 v P2 v P3) P4 P5 (P2 v P3)
• Create a separate clause corresponding to each
  conjunct
• P1 v P2 v P3
• P4
• P5
• P2 v P3

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Finally

• Add the negation of the goal to the set of
  clauses, and perform resolution. If you reach
  the empty clause, you have proved the goal

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Simple Resolution EG

• When the agent is in 1,1, there is no breeze, so
  there can be no pits in neighboring squares
• (B11 ?? (P12 v P21)) B11
• Prove P12.

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Horn Clauses

• A Horn Clause is a CNF clause with at most one
  positive literal
• Horn Clauses form the basis of forward and
  backward chaining
• The Prolog language is based on Horn Clauses
• Deciding entailment with Horn Clauses is linear
  in the size of the knowledge base

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Reasoning with Horn Clauses

• Forward Chaining
• For each new piece of data, generate all new
  facts, until the desired fact is generated
• Data-directed reasoning
• Backward Chaining
• To prove the goal, find a clause that contains
  the goal as its head, and prove the body
  recursively
• (Backtrack when you chose the wrong clause)
• Goal-directed reasoning