Logics for Data and Knowledge Representation

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Logics for Data and KnowledgeRepresentation

• Exercises of Propositional Logic

Vincenzo Maltese
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Modeling the world
Mental Model
Language L
World
Monkey
SEMANTIC GAP
PROPOSITION
SENTENCE
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The famous triangle
CONCEPT (what we have in mind)
refers to
symbolizes
Monkey
SYMBOL (what we write)
REFERENT (the real object)
stands for
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Example what is a Banana?
THE WORLD
MENTAL MODEL
MENTAL MODEL
I mean banana as something yellow or red, which
can be one or many together, no matter how big
they are, and whether they are opened or not
I mean banana as something yellow and curve, no
matter if partially opened
T
F
T
F
FORMAL MODEL M1
FORMAL MODEL M2
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Symbols in PL
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Which of the following symbols are used in PL?
• ? ? ? ? ? ? ? ? ? ? ?

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

• Which of the following symbols are used in PL?
• ? ? ? ? ? ? ? ? ? ? ?

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Formation Rules (FR) well formed formulas
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Well formed formulas (wff) in PL can be described
  by the following BNF grammar (codifying the FR)
• ltAtomic Formulagt A B ... P Q ...
  ? ?
• ltwffgt ltAtomic Formulagt ltwffgt ltwffgt?
  ltwffgt ltwffgt ? ltwffgt
• ltwffgt? ltwffgt ltwffgt ? ltwffgt
• S0 FR define a propositional language
• A formula is correct if and only if it is a wff

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

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

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Translate NL sentences into PL
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• If David comes to the party then Bruno and Carlo
  come too
• D ? B ? C
• Carlo comes to the party only if Angelo and Bruno
  do not come
• (?A ? ?B) ? C
• Carlo comes to the party given that David doesn't
  come, but, if David comes, then Bruno doesn't
  come
• (C ? ?D) ? (D ? ?B)
• A necessary condition for Angelo coming to the
  party, is that, if
• Bruno and Carlo aren't coming, David comes
• A ? ((?B ? ?C) ? D)

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Modeling the problem of passing the exam
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• If you play and you study you'll pass the exams,
  while if you play and don't study you won't pass.
  Thus, if you play, then you study and you'll pass
  the exams, or you don't study and you won't pass.

• L P, S, E
• P you play, S you study E you pass
  the exam
• (P ? S) ? E
• (P ? ?S) ? ?E
• P ? (S ? E) ? (?S ? ?E)

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

• 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, Banana T RedBanana ? Banana,
YellowBanana ? Banana, EatBanana ?
YellowBanana, ? EatBanana ? GetCrazy
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Truth-values
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Definition a truth valuation on a propositional
  language L is a mapping ? assigning to each
  formula A of L a truth value ?(A), namely in the
  domain D T, F
• ?(A) T or F according to the modeler, with A
  atomic
• ?(A) T iff ?(A) F
• ?(A?B) T iff ?(A) T and ?(B) T
• ?(A?B) T iff ?(A) T or ?(B) T
• ?(?) F (since ?df P?P)
• ?(?) T (since ?df ?)

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Truth valuations and Truth Tables
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• 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

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

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

• 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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Exercise Truth Valuations
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Compute truth valuations for the formula P
  (A?B)? ?C
• a) Rewrite as ? (A ? B) ? ? C
• b) Rewrite as (? A ? ? B) ? ? C

A B C P
T T T F
T F T F
F T T F
F F T T
T T F T
T F F T
F T F T
F F F T
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Exercise Truth Valuations
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Compute truth valuations for the formula P (A ?
  B) ? B
• Notice that the formula is unsatisfiable. In
  fact

A B P
T T F
T F F
F T F
F F F
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Exercise Truth Valuations
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Compute truth valuations for the formula P (A
  ? B) ? (A ? B)
• Notice that the formula is valid. In fact
• a) Rewrite as (A ? B) ? (A ? B)
• b) Rewrite as (A ? B) ? (A ? B)

A B P
T T T
T F T
F T T
F F T
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Entailment
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• RECALL ? ? ? iff all models satisfying the
  formulas in ? also satisfy ?
• Let A, B, C be propositional sentences.
• If A ? B ? C, then A ? B or A ? C or both?
• Proof If A ? B ? C, for all ? such that v(A)
  T it should be v(B ? C) T. However, by
  definition this means that v(B) T and v(C) T.
  Therefore it is true for both.

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

• 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

• 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

• 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 (deduction
  theorem).
• For instance (A ? B) ? (A ? B) iff (A ? B)
  ? (A ? B)
• Well-known tautologies (e.g. the De Morgan Law
  above) can be used

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

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

• Suppose p, q, r, s are four propositional
  sentences, is the following sentence valid?
• A (p ? r) ? (?p ? r)
• 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) ? (p ? r)
• (p ? r) ? (p ? r)
• (p ? r) ? (p ? r) ?

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Recall the DPLL algorithm
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• 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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Convert a formula in CNF
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• (A ? B) ? A
• (A ? B) ? A
• A ? B ? A
• (C ? A) ? (B ? A)
• (C ? A) ? (B ? A)
• (C ? A) ? (B ? A)
• (C ? A) ? (B ? A)

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Convert a formula in CNF
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• (A ? C) ? (C ? A)
• (A ? C) ? (C ? A)
• (A ? C) ? (C ? A)
• A ? C ? C ? A
• A ? C
• ((A ? B) ? A) ? A
• ?(?(? A ? B) ? A) ? A
• ?((A ? ?B) ? A) ? A
• ?((A ? A) ? (A ? ?B)) ? A
• ?A ? ?(A ? ?B) ? A
• ?(A ? ?B)
• ?A ? B

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Using DPLL for reasoning tasks
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Model checking Does ? satisfy P? (? ? P?)
• Check if ?(P) true
• Satisfiability Is there any ? such that ? ? P?
• Check that DPLL(P) succeeds and returns a ?
• Unsatisfiability Is it true that there are no ?
  satisfying P?
• Check that DPLL(P) fails
• Validity Is P a tautology? (true for all ?)
• Check that DPLL(?P) fails
• NOTE typical DPLL implementations take two
  parameters the proposition P and a model ?.
  Therefore, in case of model checking the real
  call would be DPLL(P, ?) and check that it
  succeeds

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Use DPLL to prove satisfiability
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• B ? A ? (C ? A) ? (B ? C)
• 1. Is it a consistent set of literals? There are
  no contradictions, but there are closes which are
  not atomic.
• 2. Are there any empty clauses? No, go ahead.
• 3. Assign the right truth value to all literals
  and simplify the formula
• Assign v(B) T, v(A) F and then the formula
  simplifies to C
• 4. Assign the right value to pure literals
• Assign v(C) F. Done.
• 5. No need for the splitting rule
• The formula is satisfiable at least for the
  assignment (a model for it)
• v(A) F, v(B) T, v(C) F

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Use DPLL to prove satisfiability
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• (C ? A) ? (C ? B) ? ?(A ? B)
• First convert in CNF (? C ? A) ? (? C ? B) ? (?
  A ? ? B)
• 1. Is it a consistent set of literals? There are
  no contradictions, but there are closes which are
  not atomic.
• 2. Are there any empty clauses? No, go ahead.
• 3. Assign the right truth value to all literals
  and simplify the formula.
• There are no atomic clauses. Go ahead.
• 4. Assign the right value to pure literals
• Assign v(C) T and simplify to (? A ? ? B)
• 5. Apply the splitting rule and return DPLL((? A
  ? ? B) ? A) or DPLL((? A ? ? B) ? ? A)

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Use DPLL to prove unsatisfiability
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Using DPLL, prove the unsatisfiability of (B ? A)
  ? (A ? B)
• First convert in CNF (?B ? A) ? A ? B
• Assign the right truth value to all literals and
  simplify the formula
• Assign v(A) F, v(B) T.
• Now, by propagation all the literals in the first
  clause become false.This generates an empty
  clause, thus the DPLL returns false.

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Use DPLL to prove validity
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Using DPLL, prove the validity of (A ? B) ? (B ?
  A)
• First negate the formula ((A ? B) ? (B ? A))
• Then convert in CNF ((A ? B) ? (B ? A))
• ((A ? B) ? (B ? A))
• (A ? B) ? (B ? A)
• (A ? B) ? B ? A
• Assign the right truth value to all literals and
  simplify the formula
• Assign v(A) T, v(B) F.
• Now, by propagation all the literals in the first
  clause become false.This generates an empty
  clause, thus the DPLL returns false.
• Thus, the original formula is valid.

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Use DPLL to prove validity
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• Using DPLL, prove the validity of (A ? B) ? (A ?
  B)
• First negate the formula ((A ? B) ? (A ? B))
• Then convert in CNF ((A ? B) ? (A ? B))
• (A ? B) ? (A ? B)
• (A ? B) ? (A ? B)
• (A ? B) ? (A ? B)
• (A ? B ? A) ? (A ? B ? B)
• (B ? ?) ? (A ? ?)
• This is clearly unsatisfiable even before calling
  DPLL.

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Are you Sherlock Holmes?
SYNTAX MODELING A PROBLEM SEMANTICS
LOGICAL IMPLICATION REASONING SOME NICE
PROBLEMS

• 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

• 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)