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

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

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Language (Syntax)
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL

• The syntax of ClassL is similar to PL
• Alphabet of symbols S0

• Auxiliary symbols parentheses ( )
• Defined symbols
• ? (falsehood symbol, false, bottom) ? df P ? P
• T (truth symbol, true, top) T df ?

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Formation Rules (FR) well formed formulas
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL

• 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
• () BNF BackusNaur form (formal grammar)

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Extensional Semantics Extensions
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL

• The meanings which are intended to be attached to
  the symbols and propositions form the intended
  interpretation s (sigma) of the language
• 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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Extensional interpretation
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
D Cita, Kimba, Simba
BeingLion
Monkey
Tree
Kimba
.
Cita
.
Simba
.
Lion2
Lion1
The World
The Mental Model
The Formal Model
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Class-valuation s
INTRODUCTION SYNTAX SEMANTICS REASONING
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 ? pow(U)

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Class-valuation s
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL

• 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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Venn Diagrams and Class-Values
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL

• By regarding propositions as classes, it is very
  convenient to use Venn diagrams

s(P)
s(P)
P
P
s(?)
s(?)
s(P ? Q)
s(P ? Q)
P
Q
P
Q
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Truth Relation (Satisfaction Relation)
INTRODUCTION SYNTAX SEMANTICS REASONING
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
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL

• 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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Truth, satisfiability and validity
INTRODUCTION SYNTAX SEMANTICS REASONING
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 by s (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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Reasoning on Class-Propositions
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL

• 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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PL and ClassL are notational variants
INTRODUCTION SYNTAX SEMANTICS REASONING
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

PL ClassL
Syntax ? ?
? ?
? ?
? ?
? ?
P, Q... P, Q...
Semantics ?true, false ?o, (compare models)
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ClassL reasoning using DPLL
INTRODUCTION SYNTAX SEMANTICS REASONING
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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