Propositional Logic

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Propositional Logic

• Reading Ch. 7, AIMA 2nd Ed.
• (skip 7.6-7.7)

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Logic for knowledge representation

• Important problem knowledge representation
  solving the problems of
• How to represent knowledge about problem domain
• How to reason using this knowledge in order to
  answer queries or make decisions
• Knowledge-based agents
• Have knowledge representation in a formal
  language
• Can reason about world using inference in the
  language
• Can decide what action to take by inferring that
  the action is good
• Declarative agents
• Declare to agents facts about world
• Pose questions to get answers

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Declarative knowledge-based agents
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Wumpus world

• Performance measureGold 1000, death 1000, step
  1, arrow 10
• Environment- squares adjacent to wumpus are
  smelly- squares adjacent to pits are breezy-
  glitter iff gold is in the same square- shooting
  kills wumpus if you are facing it- shooting uses
  up the only arrow- grabbing picks up gold if in
  the same square- releasing drops the gold in
  same square
• SensorsBreeze, glitter, smell
• ActuatorsLeft, right turn, forward, grab,
  release, shoot

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Wumpus world characterization

• Observable
• No only local perception
• Deterministic
• Yes outcomes explicite
• Episodic
• No sequential actions
• Discrete
• Yes
• Single-agent
• Yes

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Exploring wumpus world

A
7
Exploring wumpus world
B
A

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Exploring wumpus world
B

A
9
Exploring wumpus world
B
S

A
10
Exploring wumpus world
P
ok
B
S

W
A
How can we make these inferences automatically?
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Logic

• Logic is a formal language for representing
  information such that conclusions can be drawn
• A logic includes
• Syntax specifies symbols in the language and how
  they can be combined to form sentences
• Semantics specifies what facts in the world a
  semantics refers to.Assigns truth values to
  sentences based on their meaning in the world.
• Inference procedure mechanical method for
  computing (deriving) new (true) sentences from
  existing sentences

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Example

• Language of arithmetic
• 4xy gt 0 is a sentence, 4xlty0 is not
• 4xy gt 0 is true iff number 4xy is greater than
  zero
• 4xy gt 0 is true in a world where x0, y 1
• 4xy gt 0 is false in a world where x0, y 0
• Hence, to build a logic-based representation
• Define a set of primitive symbols and the
  associated semantics
• Logic defines the ways of putting there symbols
  together in order to represent true facts about
  world
• Logic defines ways of inferring new sentences
  from existing ones

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Propositional (Boolean) logic

• Simple language (but useful for key ideas and
  definitions)
• Language syntax
• Atoms/Symbols P12, B11, W33,, IS_HOT, IS_BREEZY,
  ?, User defines meaning of symbols.
• Connectives ? (and), ? (or), ? (implies), ?
  (iff), ? (not)
• Sentences or Well-formed-formulae (wff)
• A symbol is a sentence
• If S is a sentence, ?S (negation) is a sentence
• If S and T are sentences, S?T (conjunction), S?T
  (disjunction), S?T (implication), S?T
  (equivalence) are sentences
• A finite number of applications of (1)-(3) is a
  sentence

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Symbols Sentences of Wumpus world

• Pij is pit in (i,j)
• Bij is breeze in (i,j)
• Wij is wumpus in (i,j)
• ?B11
• Pits cause breezes in adjacent squares
• B11 ? ( P12 ? P21 )
• B12 ? ( P11 ? P22 ? P13 )

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Semantics

• Association of elements of logical language
  (atoms sentences) with real world
• Propositional logic associate atoms with
  propositionsE.g.,
• P12 is associated with pit is in cell (1,2)
• B11 is associated with breeze is felt in cell
  (1,1)
• W33 is associated with wumpus is in (3,3)
• IS_HOT is associated with I am taking cs440
• Association of atoms with propositions
  interpretationIf atom ? has value TRUE (1),
  then its interpretation P is true in the world
  otherwise ? has value FALSE (0)
• E.g., P12 1 means pit is in cell (1,2) is true

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Propositional truth tables

• Used to compute values of any sentence, given
  values of atoms
• Establishes meaning of propositional connectives

• The basic truth table can be used to evaluate any
  sentence by applying the rules recursively ?A ?
  ( A?B ) ?0 ? ( 0?1 ) 1 ? ( 0?1 ) 1 ? 1 1

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Models

• A model is an interpretation of a set of
  sentences such that each sentence is True? is
  a model of a sentence S if S is true in ? ( an
  interpretation of S satisfies ? )
• A mathematical structure that represents the
  (problem in) real world.
• Some other notions
• Unsatisfiable there is no interpretation that
  satisfies S

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Models of Wumpus world

• Situation after detecting nothing in (1,1),
  moving up, breeze in (2,1)
• What are the possible models for P?, assuming
  only pits?
• P12, P22, P31 ? 0,1
• 8 possible models

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Models of Wumpus world
Are these the models of Wumpusworld?
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Wumpus Knowledge Base (KB)

• KB S1, S2, , SN set of all sentences
  describing our current knowledge of the world,
  where each sentence is in propositional logic
• Wumpus world KBS1 ?B11S2 B21S3 B11 ? ( P12
  ? P21 )S4 B21 ? ( P11 ? P22 ? P31 )S5 ?P11
• Remember, models are interpretations where all Si
  are true
• How to find models?

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Model checking Enumeration of symbols in
sentences

• Check for valid models by enumerating all
  possible symbols interpretations KB S1 ? S2 ?
  S3 ? S4 ? S5

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Model checking Enumeration of symbols in
sentences

• Models are shown in red!
• How many enumerations? 27

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Models of Wumpus world

• Rows of the truth table where the last column
  (KB) is true (I.e., all sentences are true)

KB
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Inference entailment

• Given KB, what else can we conclude about the
  world?E.g., does a goal (a.k.a. query,
  conclusion, theorem) sentence G follow from KB?
• Note we do not know semantics.Hence we have to
  determine if all models of KB are models of G.
• I.e., KB entails G, ( KB G ) iff G is true
  whenever KB is true
• KB G iff KB ? G is valid(A sentence is valid
  iff it is true under all possible
  interpretations)
• KB G iff KB ? ?G is unsatisfiable

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Wumpus world entailment

• G (1,2) is safe
• Does KB G ?

YES!
M(KB)?M(G)
KB
G
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Wumpus world entailment (II)

• G ?P12
• Column KB?G is all 1, hence it is valid. Thus,
  KB G.
• Conclusion G follows from KB no matter what the
  interpretations

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Inference by enumeration properties

• The truth table method of inference is complete
  for Propositional Logic because we can always
  enumerate all 2N rows for the N propositional
  symbols that occur.
• But this is exponential in N. In general, it has
  been shown that the problem of checking if a set
  of sentences in PL is satisfiable is NP-complete.
• Can be implemented using which search procedure?
• Depth-first search.
• (The truth table method of inference is not
  complete for First-Order Logic.)

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Inference procedures

• Inference methods
• Model checking
• Enumeration (seen previously)
• Improved backtracking local search
• Inference using sound rules of inference
• Derive new sentences that are true in all cases
  where premises are true.E.g., ( P 1 and P?Q
  1 ) ? Q 1
• Construct a proof that a given sentence G can be
  derived from KB using a sequence of inference
  rules
• Rules R are sound if, for a KB and sentence G, KB
  - G under rules R implies KB G
• If when KB G there exists a proof of G from KB
  using R, then the R is complete
• If R is sound and complete we can prove
  entailment by searching for a proof

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(Some) Sound rules of inference
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Using SRI in Wumpus world

• S1 ?B11S2 B21S3 B11 ? ( P12 ? P21 ) KBS4
  B21 ? ( P11 ? P22 ? P31 )S5 ?P11
• S6 (B11?(P12?P21))?( P12?P21)?B11) S3
  equivalence elimin.
• S7 (P12?P21)?B11 S6 and elimination
• S8 ? B11? ?( P12 ? P21 ) S7 negation
• S9 ?( P12 ? P21 ) S1, S8, modus ponens
• S10 ?P12 ? ?P21 S9 DeMorgan

• Monotonicity property
• adding new sentences to a KB does not change
  entailment ( KB G ? KB ? S G ). It can
  only lead to new conclusions.

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Inference using Resolution

• A single rule sufficient for complete inference
  procedure, when coupled with a complete search
  algorithm ( A ?
  B, ?A) leads to B ( A ?
  B, ?A ? C) leads to B ? C

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Resolution in Wumpus world

• Continue from previous example by moving A into
  (1,2) and not feeling breeze

• S1 ?B11, S2 B21, S3 B11 ? ( P12 ? P21 ), S4
  B21 ? ( P11 ? P22 ? P31 ), S5 ?P11 KB
• S10 ?P12 ? ?P21 previously inferred
• S11 ?B12 percept
• S12 B12 ? ( P11 ? P22 ? P13 ) rule
• S13 ? P22 S11 S12 equiv. elim., add elim,
  modus ponens
• S14 ? P13
• S15 P11 ? P22 ? P31 S4 S2 equiv. elim.
  modus ponens
• S16 P11 ? P31 S15, S13 resolution
• S17 P31 S16, S5 resolution

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Resolution algorithm CNF

• How to effectively use resolution? It only
  applies to disjunctions of symbols.
• CNF conjunctive normal forms
• Every KB can be represented as a CNF
• Every sentence can be represented as a
  conjunction of disjunctions of literals
• Method
• Eliminate equivalences (conjunction of
  implications)
• Eliminate implications (disjunctions of negation
  symbols/sent)
• Propagate negations to literals (DeMorgan)
• Done!

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CNF in Wumpus world

• S3 B11 ? ( P12 ? P21 )
• ( B11 ? ( P12 ? P21 ) ) ? ( ( P12 ? P21 ) ?B11 )
• ( ?B11 ? ( P12 ? P21 ) ) ? (?( P12 ? P21 ) ? B11
  )
• ( ?B11 ? P12 ? P21 ) ? ( (? P12 ? ? P21 ) ? B11 )
• ( ?B11 ? P12 ? P21 ) ? ( ? P12 ? B11 ) ? ( ? P21
  ? B11 )
• CNF

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Resolution refutation algorithm

• Relies on proof by contradiction
  (refutation)Assume goal sentence is false,
  prove KB does not hold.
• Remember, KB G iff KB ? ?G is unsatisfiable,
  i.e., KB ? ?G
• Algorithm
• Convert all sentences in KB to CNFs
• Resolve all pairs of CNFs into new clauses
• Check for contradiction
• Resolution refutation is complete

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Resolution refutation in Wumpus world
KB
G
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Special case Horn clauses and forward- backward
chaining

• Restricted set of clauses Horn
  clausesdisjunction of literals where at most
  one is positive, e.g.,?A ? ?B ? C or ?A ? ?B
• Why Horn clauses?
• Every Horn clause can be written as an
  implication, e.g.,?A ? ?B ? C ? ( A ? B ) ? C
  ( A ? B ) ? C ?A ? ?B ? ( A ? B ) ( A ? B
  ) ? 0 (integrity constraint)
• Inference in Horn clauses can be done using
  forward-backward (F-B) chaining in linear time

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Example of FC
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How good is PL as a representational language?

• Not very expressive
• Cannot express complex environments concisely
• E.g., need to write separate rules for every
  square in Wumpus world even though they do not
  change from square to squareBij ? ( Pi,j-1 ?
  Pi,j1 ? Pi-1,j ? Pi1,j ), for all (i,j)
• E.g., to specify there is exactly one wumpus in
  the world, need to specify
• There is at least one ?
• There is at most one ?

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Announcement

• New assignment type mini project-presentations
• Prepared by a team of two students
• Related to a topic discussed in class
• Presented in class, 15-20 mins
• Will be graded, 15 of total grade (new grade
  distribution final 30, midterm 30, hw 25,
  presentation 15)
• First presentationOct 15, Bayesian network
  software
• Prepare slides and handouts (web page, pdf file
  is ok) for the class