Logics for Data and Knowledge Representation

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Logics for Data and KnowledgeRepresentation

• Exercises Propositional Logic

Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese
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Outline

• Syntax
• Symbols and formation rules
• Modeling a problem
• Semantics
• Assignments, models
• Logical implication
• Reasoning
• Satisfiability and validity
• Some nice problems

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Symbols in PL
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Which of the following symbols are used in PL?
• ? ? ? ? ? ? ? ? ? ? ?
• Which of the following symbols are in well formed
  formulas?
• ? ? ? ? ? ? ? ? ? ? ?

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Symbols in PL (solution)
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Which of the following symbols are used in PL?
• ? ? ? ? ? ? ? ? ? ? ?
• Which of the following symbols are in well formed
  formulas?
• ? ? ? ? ? ? ? ? ? ? ?
• Remember the BNF grammar
• ltAtomic Formulagt A B ... P Q ...
  ? ?
• ltwffgt ltAtomic Formulagt ltwffgt ltwffgt?
  ltwffgt ltwffgt? ltwffgt ltwffgt ? ltwffgt ltwffgt ?
  ltwffgt

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Formation rules
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Which of the following is not a wff?
• ? MonkeyLow ? BananaHigh
• ? ? MonkeyLow ? BananaHigh
• MonkeyLow ? ? BananaHigh
• MonkeyLow ? ? GetBanana
• NUM. 3 IS WRONG!

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Modeling Bananas
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Highlight relevant words, define a PL language
  and a theory for the problem below.

• Bananas may differ in many ways. However, there
  are red and yellow bananas. I like bananas, but I
  eat only yellow bananas. If I do not eat at least
  a banana I get crazy.

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Modeling Bananas (possible solution)
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Bananas may differ in many ways. However, there
  are red and yellow bananas. I like bananas, but I
  eat only yellow bananas. If I do not eat at least
  a banana I get crazy.

L RedBanana, YellowBanana, EatBanana,
GetCrazy T EatBanana ? YellowBanana, ?
EatBanana ? GetCrazy
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Truth valuations and Truth Tables
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• A truth valuation on a PL language L is a mapping
  ? that assigns to each formula P of L a truth
  value ?(P).
• A truth table is composed of one column for each
  input variable and one (or more) final column for
  all of the possible results of the logical
  operation that the table is meant to represent.
  Each row of the truth table therefore contains
  one possible assignment of the input variables,
  and the result of the operation for those values.

LOGICAL OPERATION
VARIABLES
A B A?B
T T T
T F F
F T F
F F F
POSSIBLE ASSIGNEMENTS
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Example
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Calculate the Truth Table of the following
  formulas
• (1) A ? B (2) P ? Q (3) X ? Y.

VARIABLES
(1)
(2)
(3)
A B A?B A?B A?B
T T T T T
T F F T F
F T F T F
F F F F T
POSSIBLE ASSIGNEMENTS
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Exercise Truth Valuations
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Compute truth valuations for the formulas
• (A?B)?C
• (A?B)?C
• (A?B?C?D?E)?(F?A)?(F?G?H?F)?(I?(D?J))?(J?D
  ?E)?F

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Provide the models for the propositions
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• RECALL a truth valuation ? is a model for a
  proposition P iff ?(P) true
• List the models for the following formulas
• A ? B
• (A ? B) ? (B ? C)
• A ? B ? C
• A ? B ? C

A B A ? B
T T F
T F T
F T F
F F F
MODEL
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Entailment
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• RECALL ? ? ? iff all models satisfying the
  formulas in ? also satisfy ?
• Let A, B, C be propositional sentences.
• (a) If A ? B ? C, then A ? B or A ? C or both?
• (b) What if A ? B ? C?
• Proof The only model satisfying B ? C is ?
  BT, CT.
• If A ? B ? C then ? should be also a model of
  A. However, since ? assigns true to B, ? is a
  model of B. Similarly, ? is also a model of
  C. So, (a) above is true for both.
• Assume now that A ? B ? C. A model of B ? C
  is ? BT, CF.
• ? is not a model of C, therefore A ? C. The
  other cases are similar.

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Exercise prove entailment
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Given that
• P (A ? B ) ? (?C ? ?D ? E)
• Q1 A ? B
• Q2 (A ? B ? C) ? ((B ? C ? D) ? E)
• Q3 (A ? B) ? (?D ? E)
• Does P ? Qi ?
• Proof Let X A ? B, Y ?D ? E, then we can
  rewrite
• P X ? (C ? Y) Q1 X Q2 (X ? C) ? (B ?
  C ? Y) Q3 X ? Y
• P ? Q1 is obvious.
• Since X ? X ? C and ( C ? Y) ? (B ? C ?
  Y), then P ? Q2.
• Since Y ? (C ? Y), then Q3 ? P (and not vice
  versa).

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Exercise prove entailment using truth tables
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Given that
• P (A ? B ) ? (?C ? ?D ? E)
• Q1 A ? B
• Q2 (A ? B ? C) ? (B ? C ? D ? E)
• Q3 (A ? B) ? (?D ? E)
• (1) List all truth assignments such that P ? Qi
• (2) Is there any assignment such that P ? Qi for
  all i?
• Solution to (1) First compute the truth tables
  for all the propositions above. Then, list all
  rows for which both P and Qi are true.
• Solution to (2) Check whether there is any
  assignment for which all the sentences above are
  true.

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Logical implication and deduction
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Deduction (?) is the process of inferring new
  knowledge from known facts.
• Logical implication (?) is a way to include
  deduction directly in the language. It is an
  alternative way to implement deduction.
• For instance (A ? B) ? (A ? B) iff (A ? B)
  ? (A ? B)
• We provided some well known tautologies in the
  theoretical part (e.g. the De Morgan Law above)

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Prove using truth tables, the following deductions
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Double negative elimination
•  ??P ? P
• Conjunction introduction / elimination
• (a) P, Q ? P?Q (b) P ? Q ? P (c) P ? Q ? Q.
• Disjunction introduction / elimination 
• (a) P ?P ? Q (a) Q ? P ? Q (c) P ? Q, P ? R,
  Q ? R ? R
• Bi-conditional introduction / elimination
• (P ? Q) ? (Q ? P) ? (P ? Q) 
• De Morgan
• (a) ?(P ? Q) ? ?P ? ?Q (b) ?(P ? Q) ? ?P ? ?Q

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Proofs of the Deduction Rules
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL
P Q P?Q P?Q P?Q
T T T T T
T F F T F
F T F T F
F F F F T
P Q P P?Q Q?P (P?Q) P?Q (P?Q) P?Q
T T T T T F F F F
T F T F T T T F F
F T F T F T T F F
F F F T T T T T T
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Soundness and Completeness
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• A deductive system is sound if any sentence P
  that is derivable from a set ? of sentences is
  also a logical consequence of that set ?.
• A deductive system is complete if every sentence
  P that is a semantic consequence of a set of
  sentences G can be derived in the deduction
  system from that set.
• A soundness property provides the initial reason
  for counting a logical system as desirable. The
  completeness property means that every validity
  (truth) is provable.

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Prove validity (I)
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Suppose p, q, r, s are four propositional
  sentences, is the following sentence valid?
• A (p ? r) ? (q ? s) ? (?r ? ?s) ?(?p ? ?q)
• A way to prove validity is to show that the
  proposition entails ?. This can be done by
  applying well known tautologies (e.g. De Morgan).
  In alternative we can show using truth tables
  that all the assignments are true.
• (p ? r) ? (q ? s) ? (r ? s) ? (p ? q)
• ((p ? r) ? (q ? s) ? (r ? s)) ? p ? q
• (p ? r) ? (q ? s) ? (r ? s) ? p ? q
• ?

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Prove validity (II)
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL
p q r s A
T T T T T
T T T F T
T T F T T
T T F F T
T F T T T
T F T F T
T F F T T
T F F F T
F T T T T
F T T F T
F T F T T
F T F F T
F F T T T
F F T F T
F F F T T
F F F F T
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Calculus with Tableaux
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL
G A ??B, B, ?A
G B ?A, B
G ?(A ? ?B), A
?A
B
A
B
?B ? A
?A ? B
A
?B
?A
B
A ? ?B
closed
closed
A
?B
closed
closed
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Calculus with Tableaux
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL
G ?(A ??B) ? (B ? ?A), B
B
??(A ? ?B) ? (? B ? ?A)
??(A ? ?B)
? B ? ?A
A ? ?B
?B
?A
closed
A
?B
closed
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Are you Sherlock Holmes?
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• There was a robbery in which a lot of goods were
  stolen. The robber(s) left in a truck. It is
  known that
• (1) Nobody else could have been involved other
  than A, B and C.
• (2) C never commits a crime without A's
  participation.
• (3) B does not know how to drive.
• Is A innocent or guilty?
• Proof The 3 points above can be translated in
  PL as follows
• (1) ?1 A ? B ? C
• (2) ?2 C ? A
• (3) ?3 B ? (B ? A) ? (B ? C)
• Does ?1 , ?2 , ?3 ? A ? Yes!
• We can prove it by showing that ?1 ? ?2 ? ?3 ? A
  is a tautology.

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Knights and Knaves
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• A very special island is inhabited only by
  knights and knaves. Knights always tell the
  truth, and knaves always lie.
• You meet two inhabitants Zoey and Mel.
• Zoey tells you that Mel is a knave.
• Mel says Neither Zoey nor I are knaves.
• Can you determine what are they? (who is a
  knight and who is a knave?)
• Proof The two sentences above can be translated
  in PL as follows
• (1) Z ? ?M (2) M ? Z ? M.
• We can use truth tables to prove that there are
  two possible situations
• - Both lie (they are both knaves)
• - Zoey tells the true (is a Knight) and Mel lies
  (is a knave)

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Recall the DPLL algorithm
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVTIONS CONCLUSIONS
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• Input a proposition P in CNF
• Output true if "P satisfiable" or false if "P
  unsatisfiable"
• boolean function DPLL(P)
• if consistent(P) then return true
• if hasEmptyClause(P) then return false
• foreach unit clause C in P do
• P unit-propagate(C, P)
• foreach pure-literal L in P do
• P pure-literal-assign(L, P)
• L choose-literal(P)
• return DPLL(P ? L) OR DPLL(P ? ?L)

Consistency Check
Empty Close Test
Unit Propagation
Pure Literal Elimination
Splitting rule
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Use DPLL to prove satisfiability
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS DPLL

• B ? A ? (C ? A) ? (B ? C)
• 1. Is it consistent? YES, there are no
  contradictions
• 2. Are there any empty clauses? No, go ahead.
• 3. Assign the right truth value to all literals
  and simplify the formula
• Assign B T, A F and then the formula
  simplifies to C
• 4. Assign the right value to pure literals
• Assign C F. Done.
• 5. No need for the splitting rule
• The formula is satisfiable at least for the
  assignment (a model for it)
• A F, B T, C F