Logics for Data and Knowledge Representation

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Logics for Data and Knowledge Representation

• ClassL (part 1) syntax and semantics

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Outline

• Introduction
• Syntax
• Alphabet
• Formation rules
• Semantics
• Class-valuation
• Venn diagrams
• Satisfiability
• Validity
• Reasoning
• Comparing PL and ClassL
• ClassL reasoning using DPLL

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Introduction ClassL, the logic of classes
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PL AND CLASSL CLASSL REASONING USING DPLL

• It is a propositional logic
• Sentences expressing propositions (something true
  or false)
• It is also called Propositional Description Logic
  (DL) or ALC DL
• Different alphabet and semantics w.r.t. PL
  (notational variant)
• The logical constants (operators) are ?
  (and, intersection), ? (or, disjunction), ?
  (not)
• Meta-logical symbols ?, ?
• Extensional interpretation
• The domain is a set of objects. Propositions are
  interpreted using an extensional interpretation.

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Intensional vs Extensional interpretation
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PL AND CLASSL CLASSL REASONING USING DPLL
Intentional interpretation D T, F
Lion
Monkey
Tree
T
F
.
.
Lion2
Lion1
The World
The Mental Model
The Formal Model
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Intensional vs Extensional interpretation
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PL AND CLASSL CLASSL REASONING USING DPLL
Extensional interpretation D Cita, Kimba,
Simba
Lion
Monkey
Tree
Kimba
.
Cita
.
Simba
.
Lion2
Lion1
The World
The Mental Model
The Formal Model
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Language (Syntax)
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• The syntax of ClassL is similar to PL
• Alphabet of symbols S0

NOTE not only characters but also words
(composed by several characters) like monkey
are descriptive symbols
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Additional Symbols
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• Auxiliary symbols
• Parentheses ( )
• Additional logical constants
• Logical constants are, for all propositions P
• ? (falsehood symbol, false, bottom) ? ? P ? P
• T (truth symbol, true, top) T ? ?
• Note that differently from PL, in ClassL they are
  not defined symbols but they are logical facts,
  i.e. theorems

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Formation Rules (FR) well formed formulas
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• Well formed formulas (wff) in ClassL can be
  described by the following BNF grammar (codifying
  the rules)
• ltAtomic Formulagt A B ... P Q ...
  ? ?
• ltwffgt ltAtomic Formulagt ltwffgt ltwffgt ?
  ltwffgt ltwffgt ? ltwffgt
• Atomic formulas are also called atomic
  propositions
• Wff are class-propositional formulas (or just
  propositions)
• A formula is correct if and only if it is a wff
• S0 FR define a propositional language

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Semantics means providing an interpretation
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PL AND CLASSL CLASSL REASONING USING DPLL

• So far the elements of our propositional language
  are simply strings of symbols
• without formal meaning
• The meanings which are intended to be attached to
  the symbols and propositions form the intended
  interpretation s (sigma) of the language

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Extensional Semantics Extensions
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PL AND CLASSL CLASSL REASONING USING DPLL

• The semantics of a propositional language of
  classes L are extensional (semantics)
• The extensional semantics of L is based on the
  notion of extension of a formula (proposition)
  in L
• The extension of a proposition is the totality,
  or class, or set of all objects D (domain
  elements) to which the proposition applies

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Extensions - Remarks
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• If a proposition applies to an individual object,
  its extension is simply the one object designated
  (denoted) by the proposition.
• If a proposition applies to a group of objects,
  its extension is the class consisting of all the
  objects, if any, to which it applies.
• In ClassL, a proposition is also called a concept

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Examples
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• Take the proposition lion
• its extension may include (according to the
  modeler) not only living lions, but also all the
  lions of the past, and those of the future
• Take the proposition Rome
• its extension can be simply the singleton set
  whose element is the city of Rome (notice that
  several cities may have the same name, so we need
  to specify which Rome)
• Take the proposition red ? apple
• its extension can the class containing all the
  red apples

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Extensional Interpretation
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• Given a domain (or universe) of interpretation U,
  the extensional interpretation I of a proposition
  P, denoted by I(P) or PI is a subset of U
• This is fundamental to make the language formal.
• NOTE By assuming one world, i.e. one domain,
  the extension of a proposition is unique.

Take P airplane. I(airplane)
Boeing747-3001, Boeing747-300n, piper1, piperk,
... all airplanes occurring in the
part of the world being modeled
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Class-valuation s
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PL AND CLASSL CLASSL REASONING USING DPLL

• In extensional semantics, the first central
  semantic notion is that of class-valuation (the
  interpretation function)
• Given a Class Language L
• Given a domain of interpretation U
• A class valuation s of a propositional language
  of classes L is a mapping (function) assigning to
  each formula ? of L a set s(?) of objects
  (truth-set) in U
• s L ? U

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Class-valuation s
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• s(?) Ø
• s(?) U (Universal Class, or Universe)
• s(P) ? U, as defined by s
• s(P) a ? U a ? s(P) comp(s(P))
  (Complement)
• s(P ? Q) s(P) n s(Q) (Intersection)
• s(P ? Q) s(P) ? s(Q) (Union)

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Example
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• Suppose Person and Female are atomic formulas
  (also called concepts)
• Person ? Female
• denotes those persons that are female
• Person ? ?Female
• denotes those that are not female
• Person ? ?Person
• is the concept describing the whole world (?)

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Venn Diagrams and Class-Values
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PL AND CLASSL CLASSL REASONING USING DPLL

• By regarding propositions as classes, it is very
  convenient to use Venn diagrams
• Venn diagrams are used to represent extensional
  semantics of propositions in analogy of how
  truth-tables are used to represent intentional
  semantics
• Venn diagrams allow to compute a class valuation
  ss value in polynomial time
• In Venn diagrams we use intersecting circles to
  represent the extension of a proposition, in
  particular of each atomic proposition
• The key idea is to use Venn diagrams to symbolize
  the extension of a proposition P by the device of
  shading the region corresponding to the
  proposition, as to indicate that P has a meaning
  (i.e., the extension of P is not empty).

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Venn Diagram of P, ?
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PL AND CLASSL CLASSL REASONING USING DPLL
Venn diagrams are built starting from a main
box which is used to represent the Universe U.
s(P)
P
s(?)
The falsehood symbol corresponds to the empty set.
?
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Venn Diagram of P,?
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PL AND CLASSL CLASSL REASONING USING DPLL
P corresponds to the complement of P w.r.t. the
universe U.
s(P)
P
s(?)
The truth symbol corresponds to the universe U.
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Venn Diagram of P ? Q and P ? Q
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The intersection of P and Q
s(P ? Q)
P
Q
s(P ? Q)
The union of P and Q
P
Q
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How to use Venn diagrams exercise 1
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PL AND CLASSL CLASSL REASONING USING DPLL

• Prove by Venn diagrams that s(P) s(P)
• Case s(P) Ø

s(P)
?
s(P)
s(P)
?
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How to use Venn diagrams exercise 1
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PL AND CLASSL CLASSL REASONING USING DPLL

• Prove by Venn diagrams that s(P) s(P)
• Case s(P) U

s(P)
s(P)
?
s(P)
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How to use Venn diagrams exercise 1
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PL AND CLASSL CLASSL REASONING USING DPLL

• Prove by Venn diagrams that s(P) s(P)
• Case s(P) not empty and different from U

s(P)
P
s(P)
P
s(P)
P
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How to use Venn diagrams exercise 2
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PL AND CLASSL CLASSL REASONING USING DPLL

• Prove by Venn diagrams that s((A ? B)) s( A ?
  B)
• Case s(A) and s(B) not empty (other cases as
  homework)

s((A ? B))
s(A ? B)
A
B
A
B
s( A ? B)
s( A)
A
B
A
B
s( B)
A
B
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Truth Relation (Satisfaction Relation)
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PL AND CLASSL CLASSL REASONING USING DPLL

• Let s be a class-valuation on language L, we
  define the truth-relation (or class-satisfaction
  relation) ? and write
• s ? P
• (read s satisfies P) iff s(P) ? Ø
• Given a set of propositions G, we define
• s ? G
• iff s ? ? for all formulas ? ? G

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Model and Satisfiability
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• Let s be a class valuation on language L. s is a
  model of a proposition P (set of propositions G)
  iff s satisfies P (G).
• P (G) is class-satisfiable if there is a class
  valuation s such that s ? P (s ? G).

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Satisfiability, an example
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• Is the formula P (A ? B) satisfiable?
• In other words, there exist a s that satisfies
  P? YES!
• In order to prove it we use Venn diagrams and it
  is enough to find one.
• s is a model for P

A
B
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Truth, satisfiability and validity
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PL AND CLASSL CLASSL REASONING USING DPLL

• Let s be a class valuation on language L.
• P is true under s if P is satisfiable (s ? P)
• P is valid if s ? P for all s (notation ? P)
• In this case, P is called a tautology (always
  true)
• NOTE the notions of true and false are
  relative to some truth valuation.
• NOTE A proposition is true iff it is satisfiable

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Validity, an example
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• Is the formula P A ? A valid?
• In other words, is P true for all s? YES!
• In order to prove it we use Venn diagrams, but
  we need to discuss all cases.

Case s(A) empty if s(A) is empty, then s(A) is
the universe U
?
Case s(A) not empty if s(A) is not empty, s(A)
covers all the other elements of U
A
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Reasoning on Class-Propositions
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• Given a class-propositions P we want to reason
  about the following
• Model checking Does s satisfy P? (s ? P?)
• Satisfiability Is there any s such that s ? P?
• Unsatisfiability Is it true that there are no s
  satisfying P?
• Validity Is P a tautology? (true for all s)

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Class-Values and Truth-Values
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• Intensional Interpretation the intentional
  interpretation ? of a proposition P determines a
  truth-value ?(P)
• P holds
• Extensional Interpretation the extensional
  interpretation of s of P determines a class of
  objects s(P)
• x belongs to P or x in P or x is an
  instance of P
• What is the relation between ?(P) and s(P)? (see
  next slides)

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PL and ClassL table of the symbols
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PL AND CLASSL CLASSL REASONING USING DPLL
PL ClassL
Syntax ? ?
? ?
? ?
? ?
? ?
P, Q... P, Q...
Semantics ?true, false ?o, (compare models)

• RECALL A proposition P is true (in a model) iff
  P is satisfiable
• NOTE There is no logical implication (yet)

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PL and ClassL are notational variants
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PL AND CLASSL CLASSL REASONING USING DPLL

• Theorem P is satisfiable w.r.t. an intensional
  interpretation ? if and only if P is
  satifisfiable w.r.t. an extensional
  interpretation s
• ?(P) implies s(P)
• Build s(P) from ?(P) by substituting true with U
  and false with empty set.
• s(P) implies ?(P)
• Less trivial. Build first a s(P) which is
  equivalent to s(P) and which uses only U and
  empty set.

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ClassL reasoning using DPLL
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PL AND CLASSL CLASSL REASONING USING DPLL

• Given the theorem and the correspondences above,
  ClassL reasoning can be implemented using DPLL.
• The first step consists in translating P into P
  expressed in PL
• Model checking Does s satisfy P? (s ? P?)
• Find the corresponding model ? and check that
  v(P) true
• Satisfiability Is there any s such that s ? P?
• Check that DPLL(P) succeeds and returns a ?
• Unsatisfiability Is it true that there are no s
  satisfying P?
• Check that DPLL(P) fails
• Validity Is P a tautology? (true for all s)
• Check that DPLL(?P) fails

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Entailment in ClassL
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• Logical consequence (entailment) is not preserved
  in ClassL
• Intersection
• s ? P and s ? Q may not imply that s ? P ? Q
• implies s ? ? (P
  ? Q) (sometimes)
• Satisfiability in an extensional interpretation
  is richer than in an intensional interpretation
  
• NOTE about union
• s ? P and s ? Q, implies s ? P ? Q (always)
• NOTE about complement
• If s ? P implies s ? ?P (sometimes)

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Entailment in ClassL
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CLASSL CLASSL REASONING USING DPLL

• Suppose that Male and Female are satisfiable.
• Is Male ? Female satisfiable in PL?
• And Male ? Female in ClassL?
• It is clear that if we assume that they are
  disjoint
• 1. It cannot be ? ? Male and ? ? Female for the
  same ?
• 2. s ? Male and s ? Female do not imply that s ?
  Male ? Female
• 3. We have that s ? Male ? Female
• 4. We have that s ? Male and s ? ?Male
• IMPORTANT NOTE
• In PL, ? ? A and ? ? B implies that ? ? A ? B
  (same ?!)
• In ClassL, s ? A and s ? B may not imply that s ?
  A ? B (same s!)
• Think to the case Male and ?Male above.

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